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

Average Error: 3.2 → 0.1

Time: 2.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -3.674152695645306 \cdot 10^{+16}:\\ \;\;\;\;x \cdot 1 + \left(z \cdot x\right) \cdot \left(y - 1\right)\\ \mathbf{elif}\;z \leq 0.003410362954560997:\\ \;\;\;\;x \cdot 1 + x \cdot \left(z \cdot \left(y - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 + z \cdot \left(x \cdot \left(y - 1\right)\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if ((z <= -3.674152695645306e+16)) {
		VAR = ((double) (((double) (x * 1.0)) + ((double) (((double) (z * x)) * ((double) (y - 1.0))))));
	} else {
		double VAR_1;
		if ((z <= 0.003410362954560997)) {
			VAR_1 = ((double) (((double) (x * 1.0)) + ((double) (x * ((double) (z * ((double) (y - 1.0))))))));
		} else {
			VAR_1 = ((double) (((double) (x * 1.0)) + ((double) (z * ((double) (x * ((double) (y - 1.0))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	3.2
Target	0.3
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 z < -36741526956453056

   1. Initial program Error: 9.3 bits

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\]
   2. Using strategy rm

   3. Applied sub-neg Error: 9.3 bits

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

   4. Applied distribute-lft-in Error: 9.3 bits

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

   5. Simplified Error: 9.3 bits

      \[\leadsto x \cdot 1 + \color{blue}{x \cdot \left(z \cdot \left(y - 1\right)\right)}\]
   6. Using strategy rm

   7. Applied associate-*r* Error: 0.1 bits

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

   if -36741526956453056 < z < 0.0034103629545609972

   1. Initial program Error: 0.1 bits

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\]
   2. Using strategy rm

   3. Applied sub-neg Error: 0.1 bits

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

   4. Applied distribute-lft-in Error: 0.1 bits

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

   5. Simplified Error: 0.1 bits

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

   if 0.0034103629545609972 < z

   1. Initial program Error: 7.5 bits

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt Error: 8.5 bits

      \[\leadsto x \cdot \left(1 - \color{blue}{\left(\sqrt[3]{\left(1 - y\right) \cdot z} \cdot \sqrt[3]{\left(1 - y\right) \cdot z}\right) \cdot \sqrt[3]{\left(1 - y\right) \cdot z}}\right)\]

   4. Taylor expanded around inf Error: 7.5 bits

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

   5. Simplified Error: 0.1 bits

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

4. Final simplification Error: 0.1 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.674152695645306 \cdot 10^{+16}:\\ \;\;\;\;x \cdot 1 + \left(z \cdot x\right) \cdot \left(y - 1\right)\\ \mathbf{elif}\;z \leq 0.003410362954560997:\\ \;\;\;\;x \cdot 1 + x \cdot \left(z \cdot \left(y - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 + z \cdot \left(x \cdot \left(y - 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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))))