Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.4 → 0.2

Time: 6.2s

Precision: binary64

Cost: 704

Math

\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * ((x / y) + 1.0d0)) / (x + 1.0d0)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + (x / y)) / ((1.0d0 + x) / x)
end function

Java

public static double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}

↓

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

Python

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

↓

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

Julia

function code(x, y)
	return Float64(Float64(x * Float64(Float64(x / y) + 1.0)) / Float64(x + 1.0))
end

↓

function code(x, y)
	return Float64(Float64(1.0 + Float64(x / y)) / Float64(Float64(1.0 + x) / x))
end

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	9.4
Target	0.1
Herbie	0.2

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

Derivation

1. Initial program 9.4

   \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]

2. Applied egg-rr 0.1

   \[\leadsto \color{blue}{\left(\frac{x}{y} + 1\right) \cdot \left(x \cdot \frac{1}{x + 1}\right)} \]

3. Applied egg-rr 0.2

   \[\leadsto \left(\frac{x}{y} + 1\right) \cdot \color{blue}{\frac{1}{\frac{x + 1}{x}}} \]

4. Applied egg-rr 0.2

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	1.3
Cost	968

\[\begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -29.197655605942373:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.022082053038973322:\\ \;\;\;\;x + x \cdot \left(x \cdot \left(\frac{1}{y} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	19.1
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -29.197655605942373:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 44587738.865049265:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+119}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Error	18.4
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 44587738.865049265:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+119}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Error	1.5
Cost	712

\[\begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -29.197655605942373:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.022082053038973322:\\ \;\;\;\;x + x \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	9.5
Cost	584

\[\begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -29.197655605942373:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 44587738.865049265:\\ \;\;\;\;\frac{x}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	28.5
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -21573125341413687000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.022082053038973322:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	36.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

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

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))