Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.3 → 0.1

Time: 8.5s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

Python

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

↓

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

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

MATLAB

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

↓

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

Target

Original	9.3
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 9.3

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

2. Simplified 0.1

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

   [Start] 9.3	\[ \frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
   rational.json-simplify-49 [=>] 0.1	\[ \color{blue}{\left(\frac{x}{y} + 1\right) \cdot \frac{x}{x + 1}} \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.3
Cost	968

\[\begin{array}{l} t_0 := \left(\frac{x}{y} + 1\right) \cdot 1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.81:\\ \;\;\;\;x \cdot \left(\left(\frac{1}{y} - 1\right) \cdot x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	10.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{elif}\;x \leq 195:\\ \;\;\;\;x \cdot \left(\frac{x}{y} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Error	19.1
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -155:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Error	19.1
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -155:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5
Error	1.5
Cost	712

\[\begin{array}{l} t_0 := \frac{x}{y} + 1\\ t_1 := t_0 \cdot 1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	0.1
Cost	704

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

Alternative 7
Error	19.3
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -155:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 8
Error	19.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.86 \cdot 10^{+139}:\\ \;\;\;\;\frac{x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 9
Error	27.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -155:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	53.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023069 
(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)))