Development.Shake.Progress:message from shake-0.15.5

Average Accuracy: 99.4% → 99.4%

Time: 4.8s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

(FPCore (x y) :precision binary64 (/ 100.0 (+ (/ y x) 1.0)))

C

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

↓

double code(double x, double y) {
	return 100.0 / ((y / x) + 1.0);
}

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 / ((y / x) + 1.0d0)
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 / ((y / x) + 1.0);
}

Python

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

↓

def code(x, y):
	return 100.0 / ((y / x) + 1.0)

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(y / x) + 1.0))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = 100.0 / ((y / x) + 1.0);
end

Wolfram

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

↓

code[x_, y_] := N[(100.0 / N[(N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Target

Original	99.4%
Target	99.8%
Herbie	99.4%

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

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 64.5%

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

4. Simplified 99.4%

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

   [Start] 64.5	\[ e^{\mathsf{log1p}\left(x \cdot \frac{100}{x + y}\right)} - 1 \]
   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/ [<=] 99.4	\[ \color{blue}{\frac{100}{\frac{x + y}{x}}} \]

5. Taylor expanded in x around 0 99.4%

   \[\leadsto \frac{100}{\color{blue}{1 + \frac{y}{x}}} \]

6. Simplified 99.4%

   \[\leadsto \frac{100}{\color{blue}{\frac{y}{x} + 1}} \]
   Proof

   [Start] 99.4	\[ \frac{100}{1 + \frac{y}{x}} \]
   +-commutative [<=] 99.4	\[ \frac{100}{\color{blue}{\frac{y}{x} + 1}} \]

7. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	74.5%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-25} \lor \neg \left(y \leq 2.7 \cdot 10^{-35}\right):\\ \;\;\;\;100 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;100\\ \end{array} \]

Alternative 2
Accuracy	74.4%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-30}:\\ \;\;\;\;x \cdot \frac{100}{y}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-35}:\\ \;\;\;\;100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 3
Accuracy	74.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{y \cdot 0.01}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-35}:\\ \;\;\;\;100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4
Accuracy	74.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{100 \cdot x}{y}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-35}:\\ \;\;\;\;100\\ \mathbf{else}:\\ \;\;\;\;100 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 5
Accuracy	50.9%
Cost	64

\[100 \]

Reproduce

herbie shell --seed 2023141 
(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)))