Development.Shake.Progress:message from shake-0.15.5

Average Accuracy: 99.4% → 99.3%

Time: 3.5s

Precision: binary64

Cost: 448

Math

\[\frac{x \cdot 100}{x + y} \]

↓

\[\frac{100}{\frac{x + y}{x}} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* x 100.0) (+ x y)))

↓

(FPCore (x y) :precision binary64 (/ 100.0 (/ (+ x y) x)))

C

double code(double x, double y) {
	return (x * 100.0) / (x + y);
}

↓

double code(double x, double y) {
	return 100.0 / ((x + y) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * 100.0d0) / (x + y)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 100.0d0 / ((x + y) / x)
end function

Java

public static double code(double x, double y) {
	return (x * 100.0) / (x + y);
}

↓

public static double code(double x, double y) {
	return 100.0 / ((x + y) / x);
}

Python

def code(x, y):
	return (x * 100.0) / (x + y)

↓

def code(x, y):
	return 100.0 / ((x + y) / x)

Julia

function code(x, y)
	return Float64(Float64(x * 100.0) / Float64(x + y))
end

↓

function code(x, y)
	return Float64(100.0 / Float64(Float64(x + y) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * 100.0) / (x + y);
end

↓

function tmp = code(x, y)
	tmp = 100.0 / ((x + y) / x);
end

Wolfram

code[x_, y_] := N[(N[(x * 100.0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(100.0 / N[(N[(x + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Target

Original	99.4%
Target	99.8%
Herbie	99.3%

\[\frac{x}{1} \cdot \frac{100}{x + y} \]

Derivation

1. Initial program 99.4%

   \[\frac{x \cdot 100}{x + y} \]

2. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{x}{\frac{x + y}{100}}} \]
   Proof

   [Start] 99.4	\[ \frac{x \cdot 100}{x + y} \]
   associate-/l* [=>] 99.7	\[ \color{blue}{\frac{x}{\frac{x + y}{100}}} \]

3. Applied egg-rr 64.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{100}{x + y}\right)} - 1} \]
   Proof

   [Start] 99.7	\[ \frac{x}{\frac{x + y}{100}} \]
   expm1-log1p-u [=>] 98.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\frac{x + y}{100}}\right)\right)} \]
   expm1-udef [=>] 64.6	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\frac{x + y}{100}}\right)} - 1} \]
   div-inv [=>] 64.6	\[ e^{\mathsf{log1p}\left(\color{blue}{x \cdot \frac{1}{\frac{x + y}{100}}}\right)} - 1 \]
   clear-num [<=] 64.6	\[ e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{100}{x + y}}\right)} - 1 \]

4. Simplified 99.3%

   \[\leadsto \color{blue}{\frac{100}{\frac{x + y}{x}}} \]
   Proof

   [Start] 64.6	\[ e^{\mathsf{log1p}\left(x \cdot \frac{100}{x + y}\right)} - 1 \]
   expm1-def [=>] 98.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{100}{x + y}\right)\right)} \]
   expm1-log1p [=>] 99.8	\[ \color{blue}{x \cdot \frac{100}{x + y}} \]
   *-commutative [=>] 99.8	\[ \color{blue}{\frac{100}{x + y} \cdot x} \]
   associate-/r/ [<=] 99.3	\[ \color{blue}{\frac{100}{\frac{x + y}{x}}} \]

5. Final simplification 99.3%

   \[\leadsto \frac{100}{\frac{x + y}{x}} \]

Alternatives

Alternative 1
Accuracy	74.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-23}:\\ \;\;\;\;100\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+18}:\\ \;\;\;\;100 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;100\\ \end{array} \]

Alternative 2
Accuracy	74.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-23}:\\ \;\;\;\;100\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y \cdot 0.01}\\ \mathbf{else}:\\ \;\;\;\;100\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	448

\[x \cdot \frac{100}{x + y} \]

Alternative 4
Accuracy	50.3%
Cost	64

\[100 \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x y)
  :name "Development.Shake.Progress:message from shake-0.15.5"
  :precision binary64

  :herbie-target
  (* (/ x 1.0) (/ 100.0 (+ x y)))

  (/ (* x 100.0) (+ x y)))