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

Percentage Accurate: 96.5% → 98.3%

Time: 8.4s

Precision: binary64

Cost: 6980

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq 6.3 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, y + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(\left(y + -1\right) \cdot x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z 6.3e-21) (* x (fma z (+ y -1.0) 1.0)) (+ x (* z (* (+ y -1.0) x)))))

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 tmp;
	if (z <= 6.3e-21) {
		tmp = x * fma(z, (y + -1.0), 1.0);
	} else {
		tmp = x + (z * ((y + -1.0) * x));
	}
	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)
	tmp = 0.0
	if (z <= 6.3e-21)
		tmp = Float64(x * fma(z, Float64(y + -1.0), 1.0));
	else
		tmp = Float64(x + Float64(z * Float64(Float64(y + -1.0) * x)));
	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_] := If[LessEqual[z, 6.3e-21], N[(x * N[(z * N[(y + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(N[(y + -1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	96.5%
Target	99.7%
Herbie	98.3%

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

Derivation

1. Split input into 2 regimes

2. if z < 6.3e-21

   1. Initial program 97.9%

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

   2. Simplified 98.0%

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

      [Start] 97.9%	\[ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
      sub-neg [=>] 97.9%	\[ x \cdot \color{blue}{\left(1 + \left(-\left(1 - y\right) \cdot z\right)\right)} \]
      +-commutative [=>] 97.9%	\[ x \cdot \color{blue}{\left(\left(-\left(1 - y\right) \cdot z\right) + 1\right)} \]
      distribute-rgt-neg-in [=>] 97.9%	\[ x \cdot \left(\color{blue}{\left(1 - y\right) \cdot \left(-z\right)} + 1\right) \]
      sub-neg [=>] 97.9%	\[ x \cdot \left(\color{blue}{\left(1 + \left(-y\right)\right)} \cdot \left(-z\right) + 1\right) \]
      +-commutative [=>] 97.9%	\[ x \cdot \left(\color{blue}{\left(\left(-y\right) + 1\right)} \cdot \left(-z\right) + 1\right) \]
      distribute-rgt1-in [<=] 98.0%	\[ x \cdot \left(\color{blue}{\left(\left(-z\right) + \left(-y\right) \cdot \left(-z\right)\right)} + 1\right) \]
      distribute-rgt-neg-in [<=] 98.0%	\[ x \cdot \left(\left(\left(-z\right) + \color{blue}{\left(-\left(-y\right) \cdot z\right)}\right) + 1\right) \]
      associate-+l+ [=>] 98.0%	\[ x \cdot \color{blue}{\left(\left(-z\right) + \left(\left(-\left(-y\right) \cdot z\right) + 1\right)\right)} \]
      associate-+l+ [<=] 98.0%	\[ x \cdot \color{blue}{\left(\left(\left(-z\right) + \left(-\left(-y\right) \cdot z\right)\right) + 1\right)} \]
      distribute-rgt-neg-in [=>] 98.0%	\[ x \cdot \left(\left(\left(-z\right) + \color{blue}{\left(-y\right) \cdot \left(-z\right)}\right) + 1\right) \]
      distribute-rgt1-in [=>] 97.9%	\[ x \cdot \left(\color{blue}{\left(\left(-y\right) + 1\right) \cdot \left(-z\right)} + 1\right) \]
      +-commutative [<=] 97.9%	\[ x \cdot \left(\color{blue}{\left(1 + \left(-y\right)\right)} \cdot \left(-z\right) + 1\right) \]
      sub-neg [<=] 97.9%	\[ x \cdot \left(\color{blue}{\left(1 - y\right)} \cdot \left(-z\right) + 1\right) \]
      distribute-rgt-neg-in [<=] 97.9%	\[ x \cdot \left(\color{blue}{\left(-\left(1 - y\right) \cdot z\right)} + 1\right) \]
      *-commutative [=>] 97.9%	\[ x \cdot \left(\left(-\color{blue}{z \cdot \left(1 - y\right)}\right) + 1\right) \]
      distribute-rgt-neg-in [=>] 97.9%	\[ x \cdot \left(\color{blue}{z \cdot \left(-\left(1 - y\right)\right)} + 1\right) \]
      fma-def [=>] 98.0%	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -\left(1 - y\right), 1\right)} \]

   if 6.3e-21 < z

   1. Initial program 92.8%

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

   2. Simplified 92.8%

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

      [Start] 92.8%	\[ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
      sub-neg [=>] 92.8%	\[ x \cdot \color{blue}{\left(1 + \left(-\left(1 - y\right) \cdot z\right)\right)} \]
      +-commutative [=>] 92.8%	\[ x \cdot \color{blue}{\left(\left(-\left(1 - y\right) \cdot z\right) + 1\right)} \]
      distribute-rgt-neg-in [=>] 92.8%	\[ x \cdot \left(\color{blue}{\left(1 - y\right) \cdot \left(-z\right)} + 1\right) \]
      sub-neg [=>] 92.8%	\[ x \cdot \left(\color{blue}{\left(1 + \left(-y\right)\right)} \cdot \left(-z\right) + 1\right) \]
      +-commutative [=>] 92.8%	\[ x \cdot \left(\color{blue}{\left(\left(-y\right) + 1\right)} \cdot \left(-z\right) + 1\right) \]
      distribute-rgt1-in [<=] 92.8%	\[ x \cdot \left(\color{blue}{\left(\left(-z\right) + \left(-y\right) \cdot \left(-z\right)\right)} + 1\right) \]
      distribute-rgt-neg-in [<=] 92.8%	\[ x \cdot \left(\left(\left(-z\right) + \color{blue}{\left(-\left(-y\right) \cdot z\right)}\right) + 1\right) \]
      associate-+l+ [=>] 92.8%	\[ x \cdot \color{blue}{\left(\left(-z\right) + \left(\left(-\left(-y\right) \cdot z\right) + 1\right)\right)} \]
      associate-+l+ [<=] 92.8%	\[ x \cdot \color{blue}{\left(\left(\left(-z\right) + \left(-\left(-y\right) \cdot z\right)\right) + 1\right)} \]
      distribute-rgt-neg-in [=>] 92.8%	\[ x \cdot \left(\left(\left(-z\right) + \color{blue}{\left(-y\right) \cdot \left(-z\right)}\right) + 1\right) \]
      distribute-rgt1-in [=>] 92.8%	\[ x \cdot \left(\color{blue}{\left(\left(-y\right) + 1\right) \cdot \left(-z\right)} + 1\right) \]
      +-commutative [<=] 92.8%	\[ x \cdot \left(\color{blue}{\left(1 + \left(-y\right)\right)} \cdot \left(-z\right) + 1\right) \]
      sub-neg [<=] 92.8%	\[ x \cdot \left(\color{blue}{\left(1 - y\right)} \cdot \left(-z\right) + 1\right) \]
      distribute-rgt-neg-in [<=] 92.8%	\[ x \cdot \left(\color{blue}{\left(-\left(1 - y\right) \cdot z\right)} + 1\right) \]
      *-commutative [=>] 92.8%	\[ x \cdot \left(\left(-\color{blue}{z \cdot \left(1 - y\right)}\right) + 1\right) \]
      distribute-rgt-neg-in [=>] 92.8%	\[ x \cdot \left(\color{blue}{z \cdot \left(-\left(1 - y\right)\right)} + 1\right) \]
      fma-def [=>] 92.8%	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -\left(1 - y\right), 1\right)} \]

   3. Applied egg-rr 99.9%

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

      [Start] 92.8%	\[ x \cdot \mathsf{fma}\left(z, y + -1, 1\right) \]
      fma-udef [=>] 92.8%	\[ x \cdot \color{blue}{\left(z \cdot \left(y + -1\right) + 1\right)} \]
      distribute-rgt-in [=>] 92.8%	\[ \color{blue}{\left(z \cdot \left(y + -1\right)\right) \cdot x + 1 \cdot x} \]
      *-commutative [=>] 92.8%	\[ \color{blue}{\left(\left(y + -1\right) \cdot z\right)} \cdot x + 1 \cdot x \]
      associate-*r* [<=] 99.9%	\[ \color{blue}{\left(y + -1\right) \cdot \left(z \cdot x\right)} + 1 \cdot x \]
      *-commutative [<=] 99.9%	\[ \left(y + -1\right) \cdot \color{blue}{\left(x \cdot z\right)} + 1 \cdot x \]
      associate-*r* [=>] 99.9%	\[ \color{blue}{\left(\left(y + -1\right) \cdot x\right) \cdot z} + 1 \cdot x \]
      *-un-lft-identity [<=] 99.9%	\[ \left(\left(y + -1\right) \cdot x\right) \cdot z + \color{blue}{x} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.3 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, y + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(\left(y + -1\right) \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.3%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;z \leq 6.3 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, y + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(\left(y + -1\right) \cdot x\right)\\ \end{array} \]

Alternative 2
Accuracy	97.7%
Cost	6848

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

Alternative 3
Accuracy	98.0%
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot \left(1 - y\right) \leq 10^{+298}:\\ \;\;\;\;x \cdot \left(1 + \left(y + -1\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4
Accuracy	64.8%
Cost	848

\[\begin{array}{l} t_0 := x \cdot \left(-z\right)\\ t_1 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0.00075:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	64.3%
Cost	716

\[\begin{array}{l} t_0 := y \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.00075:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+231}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]

Alternative 6
Accuracy	95.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2200000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x \cdot \left(1 + y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \]

Alternative 7
Accuracy	96.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2200000000:\\ \;\;\;\;x \cdot \left(1 + y \cdot z\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 8
Accuracy	98.3%
Cost	708

\[\begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \left(1 + \left(y + -1\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(\left(y + -1\right) \cdot x\right)\\ \end{array} \]

Alternative 9
Accuracy	84.0%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+69}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+47}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 10
Accuracy	65.6%
Cost	521

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

Alternative 11
Accuracy	38.2%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023272 
(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))))