Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1

Average Error: 9.3 → 0.1

Time: 6.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \mathbf{if}\;x \leq -1 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 31000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{x}{y}, x\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ 1.0 (/ x y)) (/ -1.0 y))))
   (if (<= x -1e+20)
     t_0
     (if (<= x 31000000000000.0) (/ (fma x (/ x y) x) (+ x 1.0)) 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 = (1.0 + (x / y)) + (-1.0 / y);
	double tmp;
	if (x <= -1e+20) {
		tmp = t_0;
	} else if (x <= 31000000000000.0) {
		tmp = fma(x, (x / y), x) / (x + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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(Float64(1.0 + Float64(x / y)) + Float64(-1.0 / y))
	tmp = 0.0
	if (x <= -1e+20)
		tmp = t_0;
	elseif (x <= 31000000000000.0)
		tmp = Float64(fma(x, Float64(x / y), x) / Float64(x + 1.0));
	else
		tmp = t_0;
	end
	return tmp
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[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e+20], t$95$0, If[LessEqual[x, 31000000000000.0], N[(N[(x * N[(x / y), $MachinePrecision] + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Error

Bits error versus x

Bits error versus y

Target

Original	9.3
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if x < -1e20 or 3.1e13 < x

   1. Initial program 22.7

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

   2. Simplified 22.7

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

   3. Taylor expanded in x around inf 0.0

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

   if -1e20 < x < 3.1e13

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \frac{x}{y}, x\right)}{x + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \mathbf{elif}\;x \leq 31000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{x}{y}, x\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) + \frac{-1}{y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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)))