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

Average Error: 3.8 → 1.9

Time: 6.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq -\infty:\\ \;\;\;\;z \cdot \left(x \cdot y - x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq -6.567089915928027 \cdot 10^{+56}:\\ \;\;\;\;x + \left(x \cdot \left(y \cdot z\right) - x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(x \cdot y - x\right)\\ \end{array}\]

FPCore

(FPCore (x y z) :precision binary64 (* x (- 1.0 (* (- 1.0 y) z))))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x (- 1.0 (* (- 1.0 y) z))) (- INFINITY))
   (* z (- (* x y) x))
   (if (<= (* x (- 1.0 (* (- 1.0 y) z))) -6.567089915928027e+56)
     (+ x (- (* x (* y z)) (* x z)))
     (+ x (* z (- (* x y) 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 tmp;
	if ((x * (1.0 - ((1.0 - y) * z))) <= -((double) INFINITY)) {
		tmp = z * ((x * y) - x);
	} else if ((x * (1.0 - ((1.0 - y) * z))) <= -6.567089915928027e+56) {
		tmp = x + ((x * (y * z)) - (x * z));
	} else {
		tmp = x + (z * ((x * y) - x));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	3.8
Target	0.3
Herbie	1.9

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z))) < -inf.0

   1. Initial program 64.0

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

   3. Applied sub-neg_binary64_18485 64.0

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

   4. Applied distribute-rgt-in_binary64_18442 64.0

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

   5. Simplified 64.0

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

   6. Simplified 64.0

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

   7. Taylor expanded around inf 0.3

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

   if -inf.0 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z))) < -6.56708991592802665e56

   1. Initial program 0.1

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

   3. Applied sub-neg_binary64_18485 0.1

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

   4. Applied distribute-rgt-in_binary64_18442 0.1

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

   5. Simplified 0.1

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

   6. Simplified 0.1

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

   8. Applied sub-neg_binary64_18485 0.1

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

   9. Applied distribute-rgt-in_binary64_18442 0.1

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

   if -6.56708991592802665e56 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z)))

   1. Initial program 2.5

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

   3. Applied sub-neg_binary64_18485 2.5

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

   4. Applied distribute-rgt-in_binary64_18442 2.5

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

   5. Simplified 2.5

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

   6. Simplified 2.5

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

   7. Taylor expanded around 0 2.4

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq -\infty:\\ \;\;\;\;z \cdot \left(x \cdot y - x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq -6.567089915928027 \cdot 10^{+56}:\\ \;\;\;\;x + \left(x \cdot \left(y \cdot z\right) - x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(x \cdot y - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021015 
(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))))