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

Average Error: 3.5 → 0.4

Time: 8.9min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \le -1.78066746148344196 \cdot 10^{255}:\\ \;\;\;\;x \cdot 1 + z \cdot \left(\left(y - 1\right) \cdot x\right)\\ \mathbf{elif}\;\left(1 - y\right) \cdot z \le 9.59721426538854149 \cdot 10^{74}:\\ \;\;\;\;x \cdot 1 + \left(z \cdot \left(y - 1\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 + \left(x \cdot z\right) \cdot \left(y - 1\right)\\ \end{array}\]

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 ((((double) (((double) (1.0 - y)) * z)) <= -1.780667461483442e+255)) {
		VAR = ((double) (((double) (x * 1.0)) + ((double) (z * ((double) (((double) (y - 1.0)) * x))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (1.0 - y)) * z)) <= 9.597214265388541e+74)) {
			VAR_1 = ((double) (((double) (x * 1.0)) + ((double) (((double) (z * ((double) (y - 1.0)))) * x))));
		} else {
			VAR_1 = ((double) (((double) (x * 1.0)) + ((double) (((double) (x * z)) * ((double) (y - 1.0))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	3.5
Target	0.2
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt -1.618195973607049 \cdot 10^{50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \lt 3.8922376496639029 \cdot 10^{134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* (- 1.0 y) z) < -1.78066746148344196e255

   1. Initial program 32.2

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

   3. Applied sub-neg 32.2

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

   4. Applied distribute-lft-in 32.2

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

   5. Simplified 32.2

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

   7. Applied associate-*l* 0.2

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

   if -1.78066746148344196e255 < (* (- 1.0 y) z) < 9.59721426538854149e74

   1. Initial program 0.1

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

   3. Applied sub-neg 0.1

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

   4. Applied distribute-lft-in 0.1

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

   5. Simplified 0.1

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

   if 9.59721426538854149e74 < (* (- 1.0 y) z)

   1. Initial program 8.8

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

   3. Applied sub-neg 8.8

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

   4. Applied distribute-lft-in 8.8

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

   5. Simplified 8.8

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

   6. Taylor expanded around inf 8.8

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

   7. Simplified 1.9

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \le -1.78066746148344196 \cdot 10^{255}:\\ \;\;\;\;x \cdot 1 + z \cdot \left(\left(y - 1\right) \cdot x\right)\\ \mathbf{elif}\;\left(1 - y\right) \cdot z \le 9.59721426538854149 \cdot 10^{74}:\\ \;\;\;\;x \cdot 1 + \left(z \cdot \left(y - 1\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1 + \left(x \cdot z\right) \cdot \left(y - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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) (* (neg z) x))) (if (< (* x (- 1.0 (* (- 1.0 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1.0 y) (* (neg z) x)))))

  (* x (- 1.0 (* (- 1.0 y) z))))