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

Average Error: 3.5 → 0.1

Time: 9.0s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+175}:\\ \;\;\;\;y \cdot \left(z \cdot x\right) - z \cdot x\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+261}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, y + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \]

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 -4e+175)
     (- (* y (* z x)) (* z x))
     (if (<= t_0 5e+261) (* x (fma z (+ y -1.0) 1.0)) (* z (- (* y x) 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 <= -4e+175) {
		tmp = (y * (z * x)) - (z * x);
	} else if (t_0 <= 5e+261) {
		tmp = x * fma(z, (y + -1.0), 1.0);
	} else {
		tmp = z * ((y * x) - 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 <= -4e+175)
		tmp = Float64(Float64(y * Float64(z * x)) - Float64(z * x));
	elseif (t_0 <= 5e+261)
		tmp = Float64(x * fma(z, Float64(y + -1.0), 1.0));
	else
		tmp = Float64(z * Float64(Float64(y * x) - 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[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+175], N[(N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision] - N[(z * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+261], N[(x * N[(z * N[(y + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	3.5
Target	0.2
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) < -3.9999999999999997e175

   1. Initial program 16.0

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

   2. Taylor expanded in y around 0 0.6

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

   3. Taylor expanded in z around inf 0.6

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

   4. Simplified 0.6

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

      [Start] 0.6	\[ y \cdot \left(z \cdot x\right) + -1 \cdot \left(z \cdot x\right) \]
      mul-1-neg [=>] 0.6	\[ y \cdot \left(z \cdot x\right) + \color{blue}{\left(-z \cdot x\right)} \]
      distribute-rgt-neg-in [=>] 0.6	\[ y \cdot \left(z \cdot x\right) + \color{blue}{z \cdot \left(-x\right)} \]

   if -3.9999999999999997e175 < (*.f64 (-.f64 1 y) z) < 5.0000000000000001e261

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
      cancel-sign-sub-inv [=>] 0.1	\[ x \cdot \color{blue}{\left(1 + \left(-\left(1 - y\right)\right) \cdot z\right)} \]
      +-commutative [=>] 0.1	\[ x \cdot \color{blue}{\left(\left(-\left(1 - y\right)\right) \cdot z + 1\right)} \]
      *-commutative [=>] 0.1	\[ x \cdot \left(\color{blue}{z \cdot \left(-\left(1 - y\right)\right)} + 1\right) \]
      fma-def [=>] 0.1	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -\left(1 - y\right), 1\right)} \]
      sub-neg [=>] 0.1	\[ x \cdot \mathsf{fma}\left(z, -\color{blue}{\left(1 + \left(-y\right)\right)}, 1\right) \]
      +-commutative [=>] 0.1	\[ x \cdot \mathsf{fma}\left(z, -\color{blue}{\left(\left(-y\right) + 1\right)}, 1\right) \]
      distribute-neg-in [=>] 0.1	\[ x \cdot \mathsf{fma}\left(z, \color{blue}{\left(-\left(-y\right)\right) + \left(-1\right)}, 1\right) \]
      remove-double-neg [=>] 0.1	\[ x \cdot \mathsf{fma}\left(z, \color{blue}{y} + \left(-1\right), 1\right) \]
      metadata-eval [=>] 0.1	\[ x \cdot \mathsf{fma}\left(z, y + \color{blue}{-1}, 1\right) \]

   if 5.0000000000000001e261 < (*.f64 (-.f64 1 y) z)

   1. Initial program 31.1

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

   2. Simplified 31.1

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

      [Start] 31.1	\[ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
      cancel-sign-sub-inv [=>] 31.1	\[ x \cdot \color{blue}{\left(1 + \left(-\left(1 - y\right)\right) \cdot z\right)} \]
      +-commutative [=>] 31.1	\[ x \cdot \color{blue}{\left(\left(-\left(1 - y\right)\right) \cdot z + 1\right)} \]
      *-commutative [=>] 31.1	\[ x \cdot \left(\color{blue}{z \cdot \left(-\left(1 - y\right)\right)} + 1\right) \]
      fma-def [=>] 31.1	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -\left(1 - y\right), 1\right)} \]
      sub-neg [=>] 31.1	\[ x \cdot \mathsf{fma}\left(z, -\color{blue}{\left(1 + \left(-y\right)\right)}, 1\right) \]
      +-commutative [=>] 31.1	\[ x \cdot \mathsf{fma}\left(z, -\color{blue}{\left(\left(-y\right) + 1\right)}, 1\right) \]
      distribute-neg-in [=>] 31.1	\[ x \cdot \mathsf{fma}\left(z, \color{blue}{\left(-\left(-y\right)\right) + \left(-1\right)}, 1\right) \]
      remove-double-neg [=>] 31.1	\[ x \cdot \mathsf{fma}\left(z, \color{blue}{y} + \left(-1\right), 1\right) \]
      metadata-eval [=>] 31.1	\[ x \cdot \mathsf{fma}\left(z, y + \color{blue}{-1}, 1\right) \]

   3. Taylor expanded in z around inf 0.4

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

   4. Simplified 0.4

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

      [Start] 0.4	\[ z \cdot \left(\left(y - 1\right) \cdot x\right) \]
      *-commutative [=>] 0.4	\[ z \cdot \color{blue}{\left(x \cdot \left(y - 1\right)\right)} \]
      sub-neg [=>] 0.4	\[ z \cdot \left(x \cdot \color{blue}{\left(y + \left(-1\right)\right)}\right) \]
      metadata-eval [=>] 0.4	\[ z \cdot \left(x \cdot \left(y + \color{blue}{-1}\right)\right) \]
      distribute-rgt-in [=>] 0.4	\[ z \cdot \color{blue}{\left(y \cdot x + -1 \cdot x\right)} \]
      mul-1-neg [=>] 0.4	\[ z \cdot \left(y \cdot x + \color{blue}{\left(-x\right)}\right) \]
      unsub-neg [=>] 0.4	\[ z \cdot \color{blue}{\left(y \cdot x - x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \leq -4 \cdot 10^{+175}:\\ \;\;\;\;y \cdot \left(z \cdot x\right) - z \cdot x\\ \mathbf{elif}\;\left(1 - y\right) \cdot z \leq 5 \cdot 10^{+261}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, y + -1, 1\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	1864

\[\begin{array}{l} t_0 := x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot x\right)\\ \end{array} \]

Alternative 2
Error	9.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+19} \lor \neg \left(z \leq 1.8 \cdot 10^{+15}\right):\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 3
Error	2.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -44 \lor \neg \left(y \leq 1.5 \cdot 10^{-23}\right):\\ \;\;\;\;x + y \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 4
Error	9.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+19}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \]

Alternative 5
Error	0.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -0.98:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(1 + y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \]

Alternative 6
Error	12.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+151}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 7
Error	19.5
Cost	521

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

Alternative 8
Error	33.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023041 
(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))))