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

Average Accuracy: 94.3% → 99.9%

Time: 7.9s

Precision: binary64

Cost: 8136

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{z}{\frac{-1}{x}}\\ \mathbf{elif}\;t_0 \leq 10^{+248}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, y + -1, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ 1.0 (* z (+ y -1.0))))))
   (if (<= t_0 (- INFINITY))
     (* (- 1.0 y) (/ z (/ -1.0 x)))
     (if (<= t_0 1e+248) t_0 (fma (* 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 t_0 = x * (1.0 + (z * (y + -1.0)));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (1.0 - y) * (z / (-1.0 / x));
	} else if (t_0 <= 1e+248) {
		tmp = t_0;
	} else {
		tmp = fma((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)
	t_0 = Float64(x * Float64(1.0 + Float64(z * Float64(y + -1.0))))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(1.0 - y) * Float64(z / Float64(-1.0 / x)));
	elseif (t_0 <= 1e+248)
		tmp = t_0;
	else
		tmp = fma(Float64(x * z), 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_] := Block[{t$95$0 = N[(x * N[(1.0 + N[(z * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 - y), $MachinePrecision] * N[(z / N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+248], t$95$0, N[(N[(x * z), $MachinePrecision] * N[(y + -1.0), $MachinePrecision] + x), $MachinePrecision]]]]

Target

Original	94.3%
Target	99.7%
Herbie	99.9%

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 0.0%

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

   2. Taylor expanded in z around inf 99.6%

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

   3. Simplified 0.0%

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

      [Start] 99.6	\[ z \cdot \left(\left(y - 1\right) \cdot x\right) \]
      associate-*r* [=>] 0.0	\[ \color{blue}{\left(z \cdot \left(y - 1\right)\right) \cdot x} \]
      sub-neg [=>] 0.0	\[ \left(z \cdot \color{blue}{\left(y + \left(-1\right)\right)}\right) \cdot x \]
      metadata-eval [=>] 0.0	\[ \left(z \cdot \left(y + \color{blue}{-1}\right)\right) \cdot x \]
      *-commutative [<=] 0.0	\[ \color{blue}{x \cdot \left(z \cdot \left(y + -1\right)\right)} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{x \cdot z}{\frac{1}{y + -1}}} \]

   5. Applied egg-rr 99.6%

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

   if -inf.0 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z))) < 1.00000000000000005e248

   1. Initial program 99.9%

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

   if 1.00000000000000005e248 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z)))

   1. Initial program 65.8%

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

   2. Applied egg-rr 65.4%

      \[\leadsto \color{blue}{{\left(\sqrt{x \cdot \left(1 - \left(1 - y\right) \cdot z\right)}\right)}^{2}} \]

   3. Taylor expanded in y around 0 99.8%

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

   4. Simplified 99.8%

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

      [Start] 99.8	\[ y \cdot \left(z \cdot x\right) + \left(1 - z\right) \cdot x \]
      associate-*r* [=>] 65.8	\[ \color{blue}{\left(y \cdot z\right) \cdot x} + \left(1 - z\right) \cdot x \]
      distribute-rgt-out [=>] 65.8	\[ \color{blue}{x \cdot \left(y \cdot z + \left(1 - z\right)\right)} \]
      +-commutative [<=] 65.8	\[ x \cdot \color{blue}{\left(\left(1 - z\right) + y \cdot z\right)} \]
      sub-neg [=>] 65.8	\[ x \cdot \left(\color{blue}{\left(1 + \left(-z\right)\right)} + y \cdot z\right) \]
      mul-1-neg [<=] 65.8	\[ x \cdot \left(\left(1 + \color{blue}{-1 \cdot z}\right) + y \cdot z\right) \]
      *-commutative [<=] 65.8	\[ x \cdot \left(\left(1 + \color{blue}{z \cdot -1}\right) + y \cdot z\right) \]
      associate-+r+ [<=] 65.8	\[ x \cdot \color{blue}{\left(1 + \left(z \cdot -1 + y \cdot z\right)\right)} \]
      *-commutative [=>] 65.8	\[ x \cdot \left(1 + \left(z \cdot -1 + \color{blue}{z \cdot y}\right)\right) \]
      distribute-lft-in [<=] 65.8	\[ x \cdot \left(1 + \color{blue}{z \cdot \left(-1 + y\right)}\right) \]
      distribute-rgt-in [=>] 65.8	\[ \color{blue}{1 \cdot x + \left(z \cdot \left(-1 + y\right)\right) \cdot x} \]
      *-lft-identity [=>] 65.8	\[ \color{blue}{x} + \left(z \cdot \left(-1 + y\right)\right) \cdot x \]
      +-commutative [=>] 65.8	\[ \color{blue}{\left(z \cdot \left(-1 + y\right)\right) \cdot x + x} \]
      *-commutative [<=] 65.8	\[ \color{blue}{x \cdot \left(z \cdot \left(-1 + y\right)\right)} + x \]
      associate-*r* [=>] 99.8	\[ \color{blue}{\left(x \cdot z\right) \cdot \left(-1 + y\right)} + x \]
      *-commutative [<=] 99.8	\[ \color{blue}{\left(z \cdot x\right)} \cdot \left(-1 + y\right) + x \]
      fma-def [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(z \cdot x, -1 + y, x\right)} \]
      +-commutative [=>] 99.8	\[ \mathsf{fma}\left(z \cdot x, \color{blue}{y + -1}, x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right) \leq -\infty:\\ \;\;\;\;\left(1 - y\right) \cdot \frac{z}{\frac{-1}{x}}\\ \mathbf{elif}\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right) \leq 10^{+248}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, y + -1, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.9%
Cost	1352

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;t_0 \leq 10^{+272}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 2
Accuracy	68.4%
Cost	981

\[\begin{array}{l} t_0 := x \cdot \left(-z\right)\\ t_1 := y \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -180000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+130} \lor \neg \left(z \leq 4.4 \cdot 10^{+147}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	68.5%
Cost	981

\[\begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -80000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-78}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+130} \lor \neg \left(z \leq 5 \cdot 10^{+147}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 4
Accuracy	81.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{+48}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+140}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 5
Accuracy	81.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{x \cdot z}{\frac{1}{y}}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+139}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 6
Accuracy	69.4%
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{-7} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	47.3%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023129 
(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))))