Clojure Mental math

Stack-math compendium

To wrap-up this series on stack-based mental math algorithms, below are approximations for three dozen conversions. If you find a better algorithm for any of those listed, or you'd like to suggest additional conversions, please email me.

(For the implementation of the times operation, see the end of the post.)

	Conversion	Estimate	Error
Celsius → Fahrenheit	`[9 * 5 / 32 +]`	`[double subtract-tenth 32 +]`	0.0%
Fahrenheit → Celsius	`[32 - 5 * 9 /]`	`[32 - halve add-tenth]`	1.0%
15% tip	`[0.15 *]`	`[add-half tenth]`	0.0%
multiply by π	`[3.1416 *]`	`[triple dup halve tenth +]`	0.3%
divide by π	`[3.1416 /]`	`[third dup halve tenth -]`	0.5%
kilometers → miles	`[0.6214 *]`	`[dup [triple] dip tenth + double tenth]`	0.2%
miles → kilometers	`[1.609 *]`	`[dup [add-half] dip tenth +]`	0.6%
meters → feet	`[3.281 *]`	`[triple add-tenth]`	0.6%
feet → meters	`[0.3048 *]`	`[third subtract-tenth]`	1.6%
miles → feet	`[5280 *]`	`[dup tenth halve + halve 10000 *]`	0.6%
feet → miles	`[5280 /]`	`[dup tenth halve - double 10000 /]`	0.3%
kilograms → pounds	`[2.205 *]`	`[double add-tenth]`	0.2%
pounds → kilograms	`[2.205 /]`	`[halve subtract-tenth]`	0.8%
hours → seconds	`[3600 *]`	`[double double subtract-tenth 1000 *]`	0.0%
seconds → hours	`[3600 /]`	`[halve halve add-tenth 1000 /]`	1.0%
meters per second → miles per hour	`[2.237 *]`	`[double add-tenth]`	1.7%
miles per hour → meters per second	`[0.447 *]`	`[halve subtract-tenth]`	0.7%
miles per hour → feet per second	`[1.467 *]`	`[add-half]`	2.2%
feet per second → miles per hour	`[0.6818 *]`	`[third double]`	2.2%
square miles → acres	`[640 *]`	`[7 * subtract-tenth 100 *]`	1.6%
acres → square miles	`[640 /]`	`[7 / add-tenth 100 /]`	0.6%
acres → square feet	`[43560 *]`	`[double double add-tenth 10000 *]`	1.0%
square feet → acres	`[43560 /]`	`[halve halve subtract-tenth 10000 /]`	2.0%
square kilometers → acres	`[247.1 *]`	`[halve halve 1000 *]`	1.2%
pounds → troy ounces	`[14.58 *]`	`[dup [add-half tenfold] dip halve -]`	0.5%
troy ounces → pounds	`[14.58 /]`	`[third double tenth]`	2.8%
US cups → fluid ounces	`[8 *]`	`[[double] 3 times]`	0.0%
fluid ounces → US cups	`[8 /]`	`[[halve] 3 times]`	0.0%
liquid gallons → US cups	`[16 *]`	`[[double] 4 times]`	0.0%
US cups → liquid gallons	`[16 /]`	`[[halve] 4 times]`	0.0%
liquid gallons → liters	`[3.785 *]`	`[dup tenth halve - double double]`	0.4%
liters → liquid gallons	`[3.785 /]`	`[dup tenth halve + halve halve]`	0.6%
sphere radius → surface area	`[dup * 3.1416 * 4 *]`	`[dup * tenfold dup halve halve +]`	0.5%
sphere radius → volume	`[dup dup * * 3.1416 * 4 * 3 /]`	`[dup dup * * double double dup halve tenth +]`	0.3%
sphere surface area → radius	`[4 / 3.1416 / sqrt]`	`[[double] 3 times tenth tenth sqrt]`	0.3%
sphere volume → radius	`[3 * 4 / 3.1416 / cube-root]`	`[halve halve cube-root]`	1.5%

For the conversion of sphere surface area and volume to radius, accuracy naturally depends on your choice of algorithm for sqrt and cube-root. The error percentages listed above use high-precision accuracy, but for practical mental math uses, these algorithms should be good enough.

Here's the implementation of the times operation:

(def times
  ^:stack-op
  (fn [stack]
    (let [n (peek stack)
          ops (peek (pop stack))
          stack (pop (pop stack))]
      (nth (iterate #(execute % ops ->operations) stack) n))))

May 2022