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

Average Error: 5.6% → 0.14%

Time: 7.9s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+276}:\\ \;\;\;\;x - t_0 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y + -1\right) \cdot \left(z \cdot x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 y) z)))
   (if (<= t_0 (- INFINITY))
     (* z (* y x))
     (if (<= t_0 5e+276) (- x (* t_0 x)) (* (+ y -1.0) (* z 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 t_0 = (1.0 - y) * z;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = z * (y * x);
	} else if (t_0 <= 5e+276) {
		tmp = x - (t_0 * x);
	} else {
		tmp = (y + -1.0) * (z * x);
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z) {
	double t_0 = (1.0 - y) * z;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = z * (y * x);
	} else if (t_0 <= 5e+276) {
		tmp = x - (t_0 * x);
	} else {
		tmp = (y + -1.0) * (z * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	return x * (1.0 - ((1.0 - y) * z))

↓

def code(x, y, z):
	t_0 = (1.0 - y) * z
	tmp = 0
	if t_0 <= -math.inf:
		tmp = z * (y * x)
	elif t_0 <= 5e+276:
		tmp = x - (t_0 * x)
	else:
		tmp = (y + -1.0) * (z * 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)
	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+276)
		tmp = Float64(x - Float64(t_0 * x));
	else
		tmp = Float64(Float64(y + -1.0) * Float64(z * x));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * (1.0 - ((1.0 - y) * z));
end

↓

function tmp_2 = code(x, y, z)
	t_0 = (1.0 - y) * z;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = z * (y * x);
	elseif (t_0 <= 5e+276)
		tmp = x - (t_0 * x);
	else
		tmp = (y + -1.0) * (z * x);
	end
	tmp_2 = 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+276], N[(x - N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision], N[(N[(y + -1.0), $MachinePrecision] * N[(z * x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	5.6%
Target	0.28%
Herbie	0.14%

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100

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

   2. Taylor expanded in y around inf 0.42

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

   3. Simplified 0.35

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

      [Start] 0.42	\[ y \cdot \left(z \cdot x\right) \]
      associate-*r* [=>] 100	\[ \color{blue}{\left(y \cdot z\right) \cdot x} \]
      *-commutative [=>] 100	\[ \color{blue}{\left(z \cdot y\right)} \cdot x \]
      associate-*l* [=>] 0.35	\[ \color{blue}{z \cdot \left(y \cdot x\right)} \]

   if -inf.0 < (*.f64 (-.f64 1 y) z) < 5.00000000000000001e276

   1. Initial program 0.13

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

   2. Applied egg-rr 0.12

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

   if 5.00000000000000001e276 < (*.f64 (-.f64 1 y) z)

   1. Initial program 63.03

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

   2. Simplified 63.03

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

      [Start] 63.03	\[ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
      cancel-sign-sub-inv [=>] 63.03	\[ x \cdot \color{blue}{\left(1 + \left(-\left(1 - y\right)\right) \cdot z\right)} \]
      +-commutative [=>] 63.03	\[ x \cdot \color{blue}{\left(\left(-\left(1 - y\right)\right) \cdot z + 1\right)} \]
      *-commutative [=>] 63.03	\[ x \cdot \left(\color{blue}{z \cdot \left(-\left(1 - y\right)\right)} + 1\right) \]
      fma-def [=>] 63.03	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -\left(1 - y\right), 1\right)} \]
      sub-neg [=>] 63.03	\[ x \cdot \mathsf{fma}\left(z, -\color{blue}{\left(1 + \left(-y\right)\right)}, 1\right) \]
      +-commutative [=>] 63.03	\[ x \cdot \mathsf{fma}\left(z, -\color{blue}{\left(\left(-y\right) + 1\right)}, 1\right) \]
      distribute-neg-in [=>] 63.03	\[ x \cdot \mathsf{fma}\left(z, \color{blue}{\left(-\left(-y\right)\right) + \left(-1\right)}, 1\right) \]
      remove-double-neg [=>] 63.03	\[ x \cdot \mathsf{fma}\left(z, \color{blue}{y} + \left(-1\right), 1\right) \]
      metadata-eval [=>] 63.03	\[ x \cdot \mathsf{fma}\left(z, y + \color{blue}{-1}, 1\right) \]

   3. Taylor expanded in z around inf 0.31

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

   4. Simplified 0.29

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

      [Start] 0.31	\[ z \cdot \left(\left(y - 1\right) \cdot x\right) \]
      *-commutative [=>] 0.31	\[ z \cdot \color{blue}{\left(x \cdot \left(y - 1\right)\right)} \]
      sub-neg [=>] 0.31	\[ z \cdot \left(x \cdot \color{blue}{\left(y + \left(-1\right)\right)}\right) \]
      metadata-eval [=>] 0.31	\[ z \cdot \left(x \cdot \left(y + \color{blue}{-1}\right)\right) \]
      associate-*r* [=>] 0.29	\[ \color{blue}{\left(z \cdot x\right) \cdot \left(y + -1\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.14

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \leq -\infty:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;\left(1 - y\right) \cdot z \leq 5 \cdot 10^{+276}:\\ \;\;\;\;x - \left(\left(1 - y\right) \cdot z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y + -1\right) \cdot \left(z \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	45.03%
Cost	1904

\[\begin{array}{l} t_0 := x \cdot \left(y \cdot z\right)\\ t_1 := y \cdot \left(z \cdot x\right)\\ t_2 := z \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{+129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+74}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{+26}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -16500000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-80}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4.9 \cdot 10^{-181}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-287}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-200}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-105}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-51}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3100000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.15%
Cost	1352

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+276}:\\ \;\;\;\;x \cdot \left(1 - t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + -1\right) \cdot \left(z \cdot x\right)\\ \end{array} \]

Alternative 3
Error	18.75%
Cost	1112

\[\begin{array}{l} t_0 := x \cdot \left(y \cdot z\right)\\ t_1 := y \cdot \left(z \cdot x\right)\\ t_2 := x \cdot \left(1 - z\right)\\ \mathbf{if}\;y \leq -2.05 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.75 \cdot 10^{+26}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -16500000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3100000000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	18.72%
Cost	1112

\[\begin{array}{l} t_0 := y \cdot \left(z \cdot x\right)\\ t_1 := x \cdot \left(1 - z\right)\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{+137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{+54}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq -2.65 \cdot 10^{+26}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -15500000:\\ \;\;\;\;x \cdot \left(z \cdot \left(y + -1\right)\right)\\ \mathbf{elif}\;y \leq 5900000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	14.47%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -19000 \lor \neg \left(z \leq 1.9 \cdot 10^{-14}\right):\\ \;\;\;\;\left(y + -1\right) \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 6
Error	2.11%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1.9 \cdot 10^{-14}\right):\\ \;\;\;\;\left(y + -1\right) \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(y \cdot z\right)\\ \end{array} \]

Alternative 7
Error	31.35%
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1.9 \cdot 10^{-14}\right):\\ \;\;\;\;z \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	52.47%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(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))))