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

Average Error: 3.6 → 0.2

Time: 4.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq -1.1353341014094774 \cdot 10^{+276} \lor \neg \left(x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq 8.254025059186815 \cdot 10^{-83}\right):\\ \;\;\;\;x + \left(x \cdot z\right) \cdot \left(y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(x \cdot \left(y \cdot z\right) - x \cdot z\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x (- 1.0 (* (- 1.0 y) z))) -1.1353341014094774e+276)
         (not (<= (* x (- 1.0 (* (- 1.0 y) z))) 8.254025059186815e-83)))
   (+ x (* (* x z) (- y 1.0)))
   (+ x (- (* x (* y z)) (* x z)))))

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))) <= -1.1353341014094774e+276) || !((x * (1.0 - ((1.0 - y) * z))) <= 8.254025059186815e-83)) {
		tmp = x + ((x * z) * (y - 1.0));
	} else {
		tmp = x + ((x * (y * z)) - (x * z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	3.6
Target	0.2
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z))) < -1.13533410140947738e276 or 8.254025059186815e-83 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z)))

   1. Initial program 8.5

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

   3. Applied sub-neg_binary64_23259 8.5

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

   4. Applied distribute-rgt-in_binary64_23216 8.5

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

   5. Simplified 8.5

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

   6. Simplified 8.5

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

   8. Applied *-un-lft-identity_binary64_23266 8.5

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

   9. Applied distribute-rgt-out--_binary64_23220 8.5

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

   10. Applied associate-*r*_binary64_23206 0.3

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

   if -1.13533410140947738e276 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z))) < 8.254025059186815e-83

   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_23259 0.1

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

   4. Applied distribute-rgt-in_binary64_23216 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

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

   8. Applied sub-neg_binary64_23259 0.1

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

   9. Applied distribute-rgt-in_binary64_23216 0.1

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

   10. Simplified 0.1

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

   11. Simplified 0.1

       \[\leadsto x + \left(x \cdot \left(y \cdot z\right) + \color{blue}{x \cdot \left(-z\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq -1.1353341014094774 \cdot 10^{+276} \lor \neg \left(x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq 8.254025059186815 \cdot 10^{-83}\right):\\ \;\;\;\;x + \left(x \cdot z\right) \cdot \left(y - 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(x \cdot \left(y \cdot z\right) - x \cdot z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020339 
(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))))