Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.2 → 0.1

Time: 3.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathsf{fma}\left(t_0, \frac{x}{y}, t_0\right) \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0)))) (fma t_0 (/ x y) t_0)))

C

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

↓

double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	return fma(t_0, (x / y), t_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)
	t_0 = Float64(x / Float64(x + 1.0))
	return fma(t_0, Float64(x / y), t_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_] := Block[{t$95$0 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * N[(x / y), $MachinePrecision] + t$95$0), $MachinePrecision]]

Target

Original	9.2
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 9.2

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

2. Simplified 9.2

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

3. Taylor expanded in y around 0 13.5

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

4. Applied egg-rr 0.1

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

5. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022210 
(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)))