Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.1 → 0.1

Time: 2.9s

Precision: binary64

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]

Error

Bits error versus x

Bits error versus y

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 9.1

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

3. Applied fma-udef_binary64 9.1

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

4. Taylor expanded in y around 0 13.8

   \[\leadsto \color{blue}{\frac{{x}^{2}}{\left(1 + x\right) \cdot y} + \frac{x}{1 + x}} \]

5. Simplified 0.1

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

6. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022131 
(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)))