Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.1 → 0.1

Time: 4.7s

Precision: binary64

Cost: 704

Math

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

↓

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

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(x / Float64(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[(x / N[(N[(x + 1.0), $MachinePrecision] / N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	9.1
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 9.1

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

2. Simplified 0.1

   \[\leadsto \color{blue}{\frac{x}{\frac{x + 1}{\frac{x}{y} + 1}}} \]
   Proof
   (/.f64 x (/.f64 (+.f64 x 1) (+.f64 (/.f64 x y) 1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x (+.f64 (/.f64 x y) 1)) (+.f64 x 1))): 45 points increase in error, 15 points decrease in error

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	18.7
Cost	1244

\[\begin{array}{l} t_0 := x \cdot \frac{x}{y}\\ t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -0.06:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+110}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Error	18.7
Cost	1244

\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -0.06:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-43}:\\ \;\;\;\;x \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{y} \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Error	18.1
Cost	1236

\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ t_1 := \frac{x}{y} \cdot t_0\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -0.31:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Error	18.1
Cost	1112

\[\begin{array}{l} t_0 := \frac{x}{y + \frac{y}{x}}\\ t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -3.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -0.108:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+110}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5
Error	9.6
Cost	1104

\[\begin{array}{l} t_0 := \left(1 + \frac{x}{y}\right) - \frac{1}{y}\\ t_1 := \frac{x}{y} \cdot \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -7900:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3700:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	9.6
Cost	1104

\[\begin{array}{l} t_0 := \left(1 + \frac{x}{y}\right) - \frac{1}{y}\\ \mathbf{if}\;x \leq -820:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{x + 1}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 80000:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	19.2
Cost	984

\[\begin{array}{l} t_0 := x \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -950000000000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+110}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 8
Error	19.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+110}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 9
Error	27.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	54.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022329 
(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)))