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

Average Error: 3.5 → 0.1

Time: 7.7s

Precision: binary64

Cost: 13960

Math

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

↓

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, z, -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \end{array} \]

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))
     (* y (* z x))
     (if (<= t_0 4e+212)
       (fma x (fma y z (- z)) x)
       (+ x (* 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 = y * (z * x);
	} else if (t_0 <= 4e+212) {
		tmp = fma(x, fma(y, z, -z), x);
	} else {
		tmp = x + (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(y * Float64(z * x));
	elseif (t_0 <= 4e+212)
		tmp = fma(x, fma(y, z, Float64(-z)), x);
	else
		tmp = Float64(x + 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[(y * N[(z * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+212], N[(x * N[(y * z + (-z)), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(z * N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $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) < -inf.0

   1. Initial program 64.0

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

   2. Taylor expanded in y around inf 0.3

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

   if -inf.0 < (*.f64 (-.f64 1 y) z) < 3.9999999999999996e212

   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): 1 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): 0 points increase in error, 3 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)): 3 points increase in error, 1 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))): 7 points increase in error, 3 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

   if 3.9999999999999996e212 < (*.f64 (-.f64 1 y) z)

   1. Initial program 20.8

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

   2. Taylor expanded in z around 0 0.7

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \leq -\infty:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;\left(1 - y\right) \cdot z \leq 4 \cdot 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, z, -z\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1864

\[\begin{array}{l} t_0 := x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;t_0 \leq 10^{+304}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot x\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	1352

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+215}:\\ \;\;\;\;x \cdot \left(1 + z \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 3
Error	20.0
Cost	1244

\[\begin{array}{l} t_0 := z \cdot \left(-x\right)\\ t_1 := y \cdot \left(z \cdot x\right)\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.18 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	20.1
Cost	1244

\[\begin{array}{l} t_0 := z \cdot \left(-x\right)\\ t_1 := y \cdot \left(z \cdot x\right)\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{+170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.06 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	12.5
Cost	976

\[\begin{array}{l} t_0 := x \cdot \left(z \cdot \left(y + -1\right)\right)\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{+170}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;z \leq -23.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.046:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	3.8
Cost	712

\[\begin{array}{l} t_0 := x + z \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.7:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	3.0
Cost	712

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

Alternative 8
Error	3.2
Cost	712

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

Alternative 9
Error	12.2
Cost	584

\[\begin{array}{l} t_0 := z \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{+181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	12.2
Cost	584

\[\begin{array}{l} t_0 := z \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -5.1 \cdot 10^{+179}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+47}:\\ \;\;\;\;x - z \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	18.9
Cost	520

\[\begin{array}{l} t_0 := z \cdot \left(-x\right)\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 0.165:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	33.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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))))