Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J

Average Error: 3.4 → 0.1

Time: 4.1s

Precision: binary64

Math

\[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

↓

\[\begin{array}{l} t_0 := y \cdot \left(x \cdot z\right) + x \cdot \left(1 - z\right)\\ \mathbf{if}\;x \leq -1.7507905798756693 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (- 1.0 (* (- 1.0 y) z))))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* y (* x z)) (* x (- 1.0 z)))))
   (if (<= x -1.7507905798756693e-76)
     t_0
     (if (<= x 5e+38) (fma z (- (* x y) x) x) t_0))))

C

double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * z));
}

↓

double code(double x, double y, double z) {
	double t_0 = (y * (x * z)) + (x * (1.0 - z));
	double tmp;
	if (x <= -1.7507905798756693e-76) {
		tmp = t_0;
	} else if (x <= 5e+38) {
		tmp = fma(z, ((x * y) - x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
end

↓

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(x * z)) + Float64(x * Float64(1.0 - z)))
	tmp = 0.0
	if (x <= -1.7507905798756693e-76)
		tmp = t_0;
	elseif (x <= 5e+38)
		tmp = fma(z, Float64(Float64(x * y) - x), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7507905798756693e-76], t$95$0, If[LessEqual[x, 5e+38], N[(z * N[(N[(x * y), $MachinePrecision] - x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Target

Original	3.4
Target	0.2
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if x < -1.7507905798756693e-76 or 4.9999999999999997e38 < x

   1. Initial program 0.3

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   2. Taylor expanded in y around 0 0.1

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

   if -1.7507905798756693e-76 < x < 4.9999999999999997e38

   1. Initial program 5.7

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]

   2. Simplified 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7507905798756693 \cdot 10^{-76}:\\ \;\;\;\;y \cdot \left(x \cdot z\right) + x \cdot \left(1 - z\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(z, x \cdot y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right) + x \cdot \left(1 - z\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :herbie-target
  (if (< (* x (- 1.0 (* (- 1.0 y) z))) -1.618195973607049e+50) (+ x (* (- 1.0 y) (* (- z) x))) (if (< (* x (- 1.0 (* (- 1.0 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1.0 y) (* (- z) x)))))

  (* x (- 1.0 (* (- 1.0 y) z))))