Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.0 → 0.1

Time: 7.5s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

Python

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

↓

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

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(x / Float64(x + 1.0)) * Float64(1.0 + Float64(x / y)))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = (x / (x + 1.0)) * (1.0 + (x / y));
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[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	9.0
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 9.0

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

2. Applied egg-rr 0.1

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

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	20.2
Cost	980

\[\begin{array}{l} t_0 := \frac{x + -1}{y}\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+246}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{+206}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -5.773894610190541:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.499890656474238 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.18725208196171444:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	19.8
Cost	848

\[\begin{array}{l} t_0 := \frac{x + -1}{y}\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+246}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{+206}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -5.773894610190541:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.18725208196171444:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	10.5
Cost	844

\[\begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;x \leq -2796971593112962.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.499890656474238 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{\frac{\left(x + 1\right) \cdot y}{x}}\\ \mathbf{elif}\;x \leq 0.18725208196171444:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	10.6
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -9098109552617528000:\\ \;\;\;\;1 + \frac{x}{y}\\ \mathbf{elif}\;x \leq -2.499890656474238 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{x}{\frac{y}{x}}}{x + 1}\\ \mathbf{elif}\;x \leq 0.18725208196171444:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]

Alternative 5
Error	1.2
Cost	840

\[\begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;x \leq -5.773894610190541:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.18725208196171444:\\ \;\;\;\;x + x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	1.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -5.773894610190541:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) - \frac{1}{y}\\ \mathbf{elif}\;x \leq 0.18725208196171444:\\ \;\;\;\;x + x \cdot \left(\frac{x}{y} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x + -1}{y}\\ \end{array} \]

Alternative 7
Error	19.9
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+246}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{+206}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -5.773894610190541:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 0.07269922170002763:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 8
Error	11.0
Cost	716

\[\begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -5.773894610190541:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.499890656474238 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.07269922170002763:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	10.8
Cost	716

\[\begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -5.773894610190541:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.499890656474238 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 0.07269922170002763:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	9.9
Cost	712

\[\begin{array}{l} t_0 := 1 + \frac{x + -1}{y}\\ \mathbf{if}\;x \leq -8.156010912272864:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.18725208196171444:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	0.1
Cost	704

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

Alternative 12
Error	10.1
Cost	584

\[\begin{array}{l} t_0 := 1 + \frac{x}{y}\\ \mathbf{if}\;x \leq -8.156010912272864:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.07269922170002763:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	28.5
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -5.773894610190541:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1647078740733973.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	36.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022317 
(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)))