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

Percentage Accurate: 95.7% → 99.1%

Time: 7.0s

Precision: binary64

Cost: 7112

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + -1\right) \cdot z\\ \mathbf{if}\;x \leq -5 \cdot 10^{+95}:\\ \;\;\;\;x + x \cdot t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(\left(y + -1\right) \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + t_0\right)\\ \end{array} \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ y -1.0) z)))
   (if (<= x -5e+95)
     (+ x (* x t_0))
     (if (<= x 2e-120) (fma (* (+ y -1.0) x) z x) (* x (+ 1.0 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 + -1.0) * z;
	double tmp;
	if (x <= -5e+95) {
		tmp = x + (x * t_0);
	} else if (x <= 2e-120) {
		tmp = fma(((y + -1.0) * x), z, x);
	} else {
		tmp = x * (1.0 + 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 + -1.0) * z)
	tmp = 0.0
	if (x <= -5e+95)
		tmp = Float64(x + Float64(x * t_0));
	elseif (x <= 2e-120)
		tmp = fma(Float64(Float64(y + -1.0) * x), z, x);
	else
		tmp = Float64(x * Float64(1.0 + 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 + -1.0), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[x, -5e+95], N[(x + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-120], N[(N[(N[(y + -1.0), $MachinePrecision] * x), $MachinePrecision] * z + x), $MachinePrecision], N[(x * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	95.7%
Target	99.7%
Herbie	99.1%

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

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000025e95

   1. Initial program 100.0%

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

   2. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(z, y + -1, 1\right)} \]
      Step-by-step derivation

      [Start] 100.0%	\[ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
      sub-neg [=>] 100.0%	\[ x \cdot \color{blue}{\left(1 + \left(-\left(1 - y\right) \cdot z\right)\right)} \]
      +-commutative [=>] 100.0%	\[ x \cdot \color{blue}{\left(\left(-\left(1 - y\right) \cdot z\right) + 1\right)} \]
      distribute-rgt-neg-in [=>] 100.0%	\[ x \cdot \left(\color{blue}{\left(1 - y\right) \cdot \left(-z\right)} + 1\right) \]
      sub-neg [=>] 100.0%	\[ x \cdot \left(\color{blue}{\left(1 + \left(-y\right)\right)} \cdot \left(-z\right) + 1\right) \]
      +-commutative [=>] 100.0%	\[ x \cdot \left(\color{blue}{\left(\left(-y\right) + 1\right)} \cdot \left(-z\right) + 1\right) \]
      distribute-rgt1-in [<=] 100.0%	\[ x \cdot \left(\color{blue}{\left(\left(-z\right) + \left(-y\right) \cdot \left(-z\right)\right)} + 1\right) \]
      distribute-rgt-neg-in [<=] 100.0%	\[ x \cdot \left(\left(\left(-z\right) + \color{blue}{\left(-\left(-y\right) \cdot z\right)}\right) + 1\right) \]
      associate-+l+ [=>] 100.0%	\[ x \cdot \color{blue}{\left(\left(-z\right) + \left(\left(-\left(-y\right) \cdot z\right) + 1\right)\right)} \]
      associate-+l+ [<=] 100.0%	\[ x \cdot \color{blue}{\left(\left(\left(-z\right) + \left(-\left(-y\right) \cdot z\right)\right) + 1\right)} \]
      distribute-rgt-neg-in [=>] 100.0%	\[ x \cdot \left(\left(\left(-z\right) + \color{blue}{\left(-y\right) \cdot \left(-z\right)}\right) + 1\right) \]
      distribute-rgt1-in [=>] 100.0%	\[ x \cdot \left(\color{blue}{\left(\left(-y\right) + 1\right) \cdot \left(-z\right)} + 1\right) \]
      +-commutative [<=] 100.0%	\[ x \cdot \left(\color{blue}{\left(1 + \left(-y\right)\right)} \cdot \left(-z\right) + 1\right) \]
      sub-neg [<=] 100.0%	\[ x \cdot \left(\color{blue}{\left(1 - y\right)} \cdot \left(-z\right) + 1\right) \]
      distribute-rgt-neg-in [<=] 100.0%	\[ x \cdot \left(\color{blue}{\left(-\left(1 - y\right) \cdot z\right)} + 1\right) \]
      *-commutative [=>] 100.0%	\[ x \cdot \left(\left(-\color{blue}{z \cdot \left(1 - y\right)}\right) + 1\right) \]
      distribute-rgt-neg-in [=>] 100.0%	\[ x \cdot \left(\color{blue}{z \cdot \left(-\left(1 - y\right)\right)} + 1\right) \]
      fma-def [=>] 100.0%	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -\left(1 - y\right), 1\right)} \]

   3. Applied egg-rr 100.0%

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

      [Start] 100.0%	\[ x \cdot \mathsf{fma}\left(z, y + -1, 1\right) \]
      fma-udef [=>] 100.0%	\[ x \cdot \color{blue}{\left(z \cdot \left(y + -1\right) + 1\right)} \]
      distribute-lft-in [=>] 100.0%	\[ \color{blue}{x \cdot \left(z \cdot \left(y + -1\right)\right) + x \cdot 1} \]
      *-commutative [<=] 100.0%	\[ x \cdot \left(z \cdot \left(y + -1\right)\right) + \color{blue}{1 \cdot x} \]
      *-un-lft-identity [<=] 100.0%	\[ x \cdot \left(z \cdot \left(y + -1\right)\right) + \color{blue}{x} \]

   if -5.00000000000000025e95 < x < 1.99999999999999996e-120

   1. Initial program 89.0%

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

   2. Simplified 89.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(z, y + -1, 1\right)} \]
      Step-by-step derivation

      [Start] 89.0%	\[ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
      sub-neg [=>] 89.0%	\[ x \cdot \color{blue}{\left(1 + \left(-\left(1 - y\right) \cdot z\right)\right)} \]
      +-commutative [=>] 89.0%	\[ x \cdot \color{blue}{\left(\left(-\left(1 - y\right) \cdot z\right) + 1\right)} \]
      distribute-rgt-neg-in [=>] 89.0%	\[ x \cdot \left(\color{blue}{\left(1 - y\right) \cdot \left(-z\right)} + 1\right) \]
      sub-neg [=>] 89.0%	\[ x \cdot \left(\color{blue}{\left(1 + \left(-y\right)\right)} \cdot \left(-z\right) + 1\right) \]
      +-commutative [=>] 89.0%	\[ x \cdot \left(\color{blue}{\left(\left(-y\right) + 1\right)} \cdot \left(-z\right) + 1\right) \]
      distribute-rgt1-in [<=] 89.0%	\[ x \cdot \left(\color{blue}{\left(\left(-z\right) + \left(-y\right) \cdot \left(-z\right)\right)} + 1\right) \]
      distribute-rgt-neg-in [<=] 89.0%	\[ x \cdot \left(\left(\left(-z\right) + \color{blue}{\left(-\left(-y\right) \cdot z\right)}\right) + 1\right) \]
      associate-+l+ [=>] 89.0%	\[ x \cdot \color{blue}{\left(\left(-z\right) + \left(\left(-\left(-y\right) \cdot z\right) + 1\right)\right)} \]
      associate-+l+ [<=] 89.0%	\[ x \cdot \color{blue}{\left(\left(\left(-z\right) + \left(-\left(-y\right) \cdot z\right)\right) + 1\right)} \]
      distribute-rgt-neg-in [=>] 89.0%	\[ x \cdot \left(\left(\left(-z\right) + \color{blue}{\left(-y\right) \cdot \left(-z\right)}\right) + 1\right) \]
      distribute-rgt1-in [=>] 89.0%	\[ x \cdot \left(\color{blue}{\left(\left(-y\right) + 1\right) \cdot \left(-z\right)} + 1\right) \]
      +-commutative [<=] 89.0%	\[ x \cdot \left(\color{blue}{\left(1 + \left(-y\right)\right)} \cdot \left(-z\right) + 1\right) \]
      sub-neg [<=] 89.0%	\[ x \cdot \left(\color{blue}{\left(1 - y\right)} \cdot \left(-z\right) + 1\right) \]
      distribute-rgt-neg-in [<=] 89.0%	\[ x \cdot \left(\color{blue}{\left(-\left(1 - y\right) \cdot z\right)} + 1\right) \]
      *-commutative [=>] 89.0%	\[ x \cdot \left(\left(-\color{blue}{z \cdot \left(1 - y\right)}\right) + 1\right) \]
      distribute-rgt-neg-in [=>] 89.0%	\[ x \cdot \left(\color{blue}{z \cdot \left(-\left(1 - y\right)\right)} + 1\right) \]
      fma-def [=>] 89.0%	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -\left(1 - y\right), 1\right)} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y + -1\right) \cdot x, z, x\right)} \]
      Step-by-step derivation

      [Start] 89.0%	\[ x \cdot \mathsf{fma}\left(z, y + -1, 1\right) \]
      fma-udef [=>] 89.0%	\[ x \cdot \color{blue}{\left(z \cdot \left(y + -1\right) + 1\right)} \]
      distribute-rgt-in [=>] 89.0%	\[ \color{blue}{\left(z \cdot \left(y + -1\right)\right) \cdot x + 1 \cdot x} \]
      *-commutative [=>] 89.0%	\[ \color{blue}{\left(\left(y + -1\right) \cdot z\right)} \cdot x + 1 \cdot x \]
      associate-*r* [<=] 94.4%	\[ \color{blue}{\left(y + -1\right) \cdot \left(z \cdot x\right)} + 1 \cdot x \]
      *-commutative [<=] 94.4%	\[ \left(y + -1\right) \cdot \color{blue}{\left(x \cdot z\right)} + 1 \cdot x \]
      associate-*r* [=>] 99.9%	\[ \color{blue}{\left(\left(y + -1\right) \cdot x\right) \cdot z} + 1 \cdot x \]
      *-un-lft-identity [<=] 99.9%	\[ \left(\left(y + -1\right) \cdot x\right) \cdot z + \color{blue}{x} \]
      fma-def [=>] 99.9%	\[ \color{blue}{\mathsf{fma}\left(\left(y + -1\right) \cdot x, z, x\right)} \]

   if 1.99999999999999996e-120 < x

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+95}:\\ \;\;\;\;x + x \cdot \left(\left(y + -1\right) \cdot z\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(\left(y + -1\right) \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(y + -1\right) \cdot z\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	7112

\[\begin{array}{l} t_0 := \left(y + -1\right) \cdot z\\ \mathbf{if}\;x \leq -5 \cdot 10^{+95}:\\ \;\;\;\;x + x \cdot t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(\left(y + -1\right) \cdot x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + t_0\right)\\ \end{array} \]

Alternative 2
Accuracy	97.9%
Cost	6848

\[\mathsf{fma}\left(y + -1, x \cdot z, x\right) \]

Alternative 3
Accuracy	99.7%
Cost	1353

\[\begin{array}{l} t_0 := z \cdot \left(1 - y\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+196} \lor \neg \left(t_0 \leq 10^{+151}\right):\\ \;\;\;\;z \cdot \left(\left(y + -1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(y + -1\right) \cdot z\right)\\ \end{array} \]

Alternative 4
Accuracy	64.3%
Cost	980

\[\begin{array}{l} t_0 := y \cdot \left(x \cdot z\right)\\ t_1 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{+105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1020000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-108}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-42}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+191}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	62.7%
Cost	848

\[\begin{array}{l} t_0 := x \cdot \left(-z\right)\\ t_1 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -820000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-72}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	99.1%
Cost	840

\[\begin{array}{l} t_0 := \left(y + -1\right) \cdot z\\ \mathbf{if}\;x \leq -1 \cdot 10^{+90}:\\ \;\;\;\;x + x \cdot t_0\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-120}:\\ \;\;\;\;x + z \cdot \left(\left(y + -1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + t_0\right)\\ \end{array} \]

Alternative 7
Accuracy	85.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-108} \lor \neg \left(z \leq 3.2 \cdot 10^{-42}\right):\\ \;\;\;\;z \cdot \left(\left(y + -1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	96.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 2.4 \cdot 10^{-32}\right):\\ \;\;\;\;x + y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \]

Alternative 9
Accuracy	84.5%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+111} \lor \neg \left(y \leq 8.6 \cdot 10^{+40}\right):\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \]

Alternative 10
Accuracy	64.4%
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -1.85 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	38.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023167 
(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))))