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

Average Error: 3.3 → 0.1

Time: 17.3s

Precision: binary64

Cost: 13960

Math

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

↓

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ t_1 := z \cdot \left(y \cdot x - x\right)\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+246}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+233}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, z, -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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)) (t_1 (* z (- (* y x) x))))
   (if (<= t_0 -2e+246)
     t_1
     (if (<= t_0 5e+233) (fma x (fma y z (- z)) x) t_1))))

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 t_1 = z * ((y * x) - x);
	double tmp;
	if (t_0 <= -2e+246) {
		tmp = t_1;
	} else if (t_0 <= 5e+233) {
		tmp = fma(x, fma(y, z, -z), x);
	} else {
		tmp = t_1;
	}
	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)
	t_1 = Float64(z * Float64(Float64(y * x) - x))
	tmp = 0.0
	if (t_0 <= -2e+246)
		tmp = t_1;
	elseif (t_0 <= 5e+233)
		tmp = fma(x, fma(y, z, Float64(-z)), x);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(z * N[(N[(y * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+246], t$95$1, If[LessEqual[t$95$0, 5e+233], N[(x * N[(y * z + (-z)), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Target

Original	3.3
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 2 regimes

2. if (*.f64 (-.f64 1 y) z) < -2.00000000000000014e246 or 5.00000000000000009e233 < (*.f64 (-.f64 1 y) z)

   1. Initial program 24.9

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

   2. Taylor expanded in z around inf 0.3

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

   3. Simplified 0.3

      \[\leadsto \color{blue}{z \cdot \left(y \cdot x - x\right)} \]
      Proof
      (*.f64 z (-.f64 (*.f64 y x) x)): 0 points increase in error, 0 points decrease in error
      (*.f64 z (-.f64 (*.f64 y x) (Rewrite<= *-lft-identity_binary64 (*.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (*.f64 z (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x (-.f64 y 1)))): 2 points increase in error, 1 points decrease in error
      (*.f64 z (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 y 1) x))): 0 points increase in error, 0 points decrease in error

   if -2.00000000000000014e246 < (*.f64 (-.f64 1 y) z) < 5.00000000000000009e233

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, z, -z\right), x\right)} \]
      Proof
      (fma.f64 x (fma.f64 y z (neg.f64 z)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y z) (neg.f64 z))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 x (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (*.f64 y z)))) (neg.f64 z)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 x (+.f64 (neg.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 y) z))) (neg.f64 z)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 x (+.f64 (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 (neg.f64 y) (neg.f64 z))) (neg.f64 z)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 x (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 (neg.f64 y) 1) (neg.f64 z))) x): 1 points increase in error, 0 points decrease in error
      (fma.f64 x (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (neg.f64 y))) (neg.f64 z)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 x (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 y)) (neg.f64 z)) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 x (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (-.f64 1 y) z))) x): 0 points increase in error, 0 points decrease in error
      (fma.f64 x (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (-.f64 1 y)) z)) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (*.f64 (neg.f64 (-.f64 1 y)) z)) x)): 2 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 x (*.f64 (neg.f64 (-.f64 1 y)) z)) (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-in_binary64 (*.f64 x (+.f64 (*.f64 (neg.f64 (-.f64 1 y)) z) 1))): 3 points increase in error, 2 points decrease in error
      (*.f64 x (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 (neg.f64 (-.f64 1 y)) z)))): 0 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 1 (*.f64 (-.f64 1 y) z)))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \leq -2 \cdot 10^{+246}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \mathbf{elif}\;\left(1 - y\right) \cdot z \leq 5 \cdot 10^{+233}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, z, -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	1352

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ t_1 := z \cdot \left(y \cdot x - x\right)\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+246}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+233}:\\ \;\;\;\;x \cdot \left(1 - t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	19.5
Cost	848

\[\begin{array}{l} t_0 := z \cdot \left(-x\right)\\ t_1 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -6.337637843331155 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.36629881051669 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.656329137159945:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	19.5
Cost	848

\[\begin{array}{l} t_0 := z \cdot \left(-x\right)\\ t_1 := z \cdot \left(y \cdot x\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -6.337637843331155 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.36629881051669 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.656329137159945:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	12.4
Cost	848

\[\begin{array}{l} t_0 := z \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -1.4775314561492657 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 228737.93265213745:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+127}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	12.5
Cost	848

\[\begin{array}{l} t_0 := z \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -1.4775314561492657 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 228737.93265213745:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+73}:\\ \;\;\;\;x \cdot \left(y \cdot z - z\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+127}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.2
Cost	840

\[\begin{array}{l} t_0 := x + z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{if}\;z \leq -2.6376144672932906 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.986159065514463 \cdot 10^{-87}:\\ \;\;\;\;x + x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	2.4
Cost	712

\[\begin{array}{l} t_0 := x + y \cdot \left(z \cdot x\right)\\ \mathbf{if}\;y \leq -25952.18790902413:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.0003412031599407887:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	1.1
Cost	712

\[\begin{array}{l} t_0 := z \cdot \left(y \cdot x - x\right)\\ \mathbf{if}\;z \leq -1.841073836502696 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.583013194625646 \cdot 10^{-6}:\\ \;\;\;\;x + x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	19.7
Cost	520

\[\begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -31.24259517742624:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	33.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022308 
(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))))