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

Average Error: 3.5 → 0.1

Time: 10.1s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+198}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, z, 1\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 y) z)))
   (if (<= t_0 (- INFINITY))
     (* z (* y x))
     (if (<= t_0 5e+198) (* x (- (fma y z 1.0) z)) (* z (* x (+ y -1.0)))))))

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 = (1.0 - y) * z;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = z * (y * x);
	} else if (t_0 <= 5e+198) {
		tmp = x * (fma(y, z, 1.0) - z);
	} else {
		tmp = z * (x * (y + -1.0));
	}
	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(1.0 - y) * z)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(z * Float64(y * x));
	elseif (t_0 <= 5e+198)
		tmp = Float64(x * Float64(fma(y, z, 1.0) - z));
	else
		tmp = Float64(z * Float64(x * Float64(y + -1.0)));
	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[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+198], N[(x * N[(N[(y * z + 1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(z * N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	3.5
Target	0.3
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 3 regimes

2. if (*.f64 (-.f64 1 y) z) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(y, z, 1\right) - z\right)} \]
      Proof

      [Start] 64.0	\[ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
      *-commutative [=>] 64.0	\[ x \cdot \left(1 - \color{blue}{z \cdot \left(1 - y\right)}\right) \]
      sub-neg [=>] 64.0	\[ x \cdot \left(1 - z \cdot \color{blue}{\left(1 + \left(-y\right)\right)}\right) \]
      distribute-rgt-in [=>] 64.0	\[ x \cdot \left(1 - \color{blue}{\left(1 \cdot z + \left(-y\right) \cdot z\right)}\right) \]
      associate--r+ [=>] 64.0	\[ x \cdot \color{blue}{\left(\left(1 - 1 \cdot z\right) - \left(-y\right) \cdot z\right)} \]
      *-lft-identity [=>] 64.0	\[ x \cdot \left(\left(1 - \color{blue}{z}\right) - \left(-y\right) \cdot z\right) \]
      sub-neg [=>] 64.0	\[ x \cdot \left(\color{blue}{\left(1 + \left(-z\right)\right)} - \left(-y\right) \cdot z\right) \]
      distribute-lft-out-- [<=] 64.0	\[ \color{blue}{x \cdot \left(1 + \left(-z\right)\right) - x \cdot \left(\left(-y\right) \cdot z\right)} \]
      distribute-lft-in [=>] 64.0	\[ \color{blue}{\left(x \cdot 1 + x \cdot \left(-z\right)\right)} - x \cdot \left(\left(-y\right) \cdot z\right) \]
      *-rgt-identity [=>] 64.0	\[ \left(\color{blue}{x} + x \cdot \left(-z\right)\right) - x \cdot \left(\left(-y\right) \cdot z\right) \]
      +-commutative [=>] 64.0	\[ \color{blue}{\left(x \cdot \left(-z\right) + x\right)} - x \cdot \left(\left(-y\right) \cdot z\right) \]
      associate-+r- [<=] 64.0	\[ \color{blue}{x \cdot \left(-z\right) + \left(x - x \cdot \left(\left(-y\right) \cdot z\right)\right)} \]
      *-commutative [=>] 64.0	\[ x \cdot \left(-z\right) + \left(x - \color{blue}{\left(\left(-y\right) \cdot z\right) \cdot x}\right) \]
      cancel-sign-sub-inv [=>] 64.0	\[ x \cdot \left(-z\right) + \color{blue}{\left(x + \left(-\left(-y\right) \cdot z\right) \cdot x\right)} \]
      distribute-rgt-neg-in [=>] 64.0	\[ x \cdot \left(-z\right) + \left(x + \color{blue}{\left(\left(-y\right) \cdot \left(-z\right)\right)} \cdot x\right) \]
      distribute-rgt1-in [=>] 64.0	\[ x \cdot \left(-z\right) + \color{blue}{\left(\left(-y\right) \cdot \left(-z\right) + 1\right) \cdot x} \]
      *-commutative [=>] 64.0	\[ x \cdot \left(-z\right) + \color{blue}{x \cdot \left(\left(-y\right) \cdot \left(-z\right) + 1\right)} \]
      +-commutative [=>] 64.0	\[ \color{blue}{x \cdot \left(\left(-y\right) \cdot \left(-z\right) + 1\right) + x \cdot \left(-z\right)} \]

   3. Taylor expanded in y around inf 0.3

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

   4. Simplified 0.3

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

      [Start] 0.3	\[ y \cdot \left(z \cdot x\right) \]
      associate-*r* [=>] 64.0	\[ \color{blue}{\left(y \cdot z\right) \cdot x} \]
      *-commutative [=>] 64.0	\[ \color{blue}{x \cdot \left(y \cdot z\right)} \]
      associate-*r* [=>] 0.3	\[ \color{blue}{\left(x \cdot y\right) \cdot z} \]

   if -inf.0 < (*.f64 (-.f64 1 y) z) < 5.00000000000000049e198

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{fma}\left(y, z, 1\right) - z\right)} \]
      Proof

      [Start] 0.1	\[ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
      *-commutative [=>] 0.1	\[ x \cdot \left(1 - \color{blue}{z \cdot \left(1 - y\right)}\right) \]
      sub-neg [=>] 0.1	\[ x \cdot \left(1 - z \cdot \color{blue}{\left(1 + \left(-y\right)\right)}\right) \]
      distribute-rgt-in [=>] 0.1	\[ x \cdot \left(1 - \color{blue}{\left(1 \cdot z + \left(-y\right) \cdot z\right)}\right) \]
      associate--r+ [=>] 0.1	\[ x \cdot \color{blue}{\left(\left(1 - 1 \cdot z\right) - \left(-y\right) \cdot z\right)} \]
      *-lft-identity [=>] 0.1	\[ x \cdot \left(\left(1 - \color{blue}{z}\right) - \left(-y\right) \cdot z\right) \]
      sub-neg [=>] 0.1	\[ x \cdot \left(\color{blue}{\left(1 + \left(-z\right)\right)} - \left(-y\right) \cdot z\right) \]
      distribute-lft-out-- [<=] 0.1	\[ \color{blue}{x \cdot \left(1 + \left(-z\right)\right) - x \cdot \left(\left(-y\right) \cdot z\right)} \]
      distribute-lft-in [=>] 0.1	\[ \color{blue}{\left(x \cdot 1 + x \cdot \left(-z\right)\right)} - x \cdot \left(\left(-y\right) \cdot z\right) \]
      *-rgt-identity [=>] 0.1	\[ \left(\color{blue}{x} + x \cdot \left(-z\right)\right) - x \cdot \left(\left(-y\right) \cdot z\right) \]
      +-commutative [=>] 0.1	\[ \color{blue}{\left(x \cdot \left(-z\right) + x\right)} - x \cdot \left(\left(-y\right) \cdot z\right) \]
      associate-+r- [<=] 0.1	\[ \color{blue}{x \cdot \left(-z\right) + \left(x - x \cdot \left(\left(-y\right) \cdot z\right)\right)} \]
      *-commutative [=>] 0.1	\[ x \cdot \left(-z\right) + \left(x - \color{blue}{\left(\left(-y\right) \cdot z\right) \cdot x}\right) \]
      cancel-sign-sub-inv [=>] 0.1	\[ x \cdot \left(-z\right) + \color{blue}{\left(x + \left(-\left(-y\right) \cdot z\right) \cdot x\right)} \]
      distribute-rgt-neg-in [=>] 0.1	\[ x \cdot \left(-z\right) + \left(x + \color{blue}{\left(\left(-y\right) \cdot \left(-z\right)\right)} \cdot x\right) \]
      distribute-rgt1-in [=>] 0.1	\[ x \cdot \left(-z\right) + \color{blue}{\left(\left(-y\right) \cdot \left(-z\right) + 1\right) \cdot x} \]
      *-commutative [=>] 0.1	\[ x \cdot \left(-z\right) + \color{blue}{x \cdot \left(\left(-y\right) \cdot \left(-z\right) + 1\right)} \]
      +-commutative [=>] 0.1	\[ \color{blue}{x \cdot \left(\left(-y\right) \cdot \left(-z\right) + 1\right) + x \cdot \left(-z\right)} \]

   if 5.00000000000000049e198 < (*.f64 (-.f64 1 y) z)

   1. Initial program 19.3

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

   2. Taylor expanded in z around inf 0.5

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \leq -\infty:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;\left(1 - y\right) \cdot z \leq 5 \cdot 10^{+198}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, z, 1\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	1352

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+198}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 2
Error	20.6
Cost	981

\[\begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -0.0004:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-53}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+68} \lor \neg \left(z \leq 3.55 \cdot 10^{+109}\right) \land z \leq 7 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	20.5
Cost	981

\[\begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -0.0004:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+68}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+109} \lor \neg \left(z \leq 9.5 \cdot 10^{+138}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4
Error	19.9
Cost	716

\[\begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -0.0004:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-53}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	4.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.068\right):\\ \;\;\;\;x \cdot \left(1 + y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 6
Error	0.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.95 \lor \neg \left(z \leq 65000\right):\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + y \cdot z\right)\\ \end{array} \]

Alternative 7
Error	0.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.95 \lor \neg \left(z \leq 65000\right):\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(y \cdot z\right)\\ \end{array} \]

Alternative 8
Error	11.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{+70}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 3.35 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \end{array} \]

Alternative 9
Error	19.9
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -0.0004 \lor \neg \left(z \leq 2.3 \cdot 10^{+21}\right):\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	33.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023018 
(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))))