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

Average Error: 3.1 → 0.1

Time: 4.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq -4.490989771135522 \cdot 10^{+167}:\\ \;\;\;\;x + \left(x \cdot z\right) \cdot \left(y - 1\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq 2.320352310306588 \cdot 10^{+301}:\\ \;\;\;\;x + x \cdot \left(y \cdot z - z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sqrt[3]{y - 1} \cdot \left(\left(x \cdot z\right) \cdot \left(\sqrt[3]{y - 1} \cdot \sqrt[3]{y - 1}\right)\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x (- 1.0 (* (- 1.0 y) z))) -4.490989771135522e+167)
   (+ x (* (* x z) (- y 1.0)))
   (if (<= (* x (- 1.0 (* (- 1.0 y) z))) 2.320352310306588e+301)
     (+ x (* x (- (* y z) z)))
     (+
      x
      (*
       (cbrt (- y 1.0))
       (* (* x z) (* (cbrt (- y 1.0)) (cbrt (- 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 tmp;
	if ((x * (1.0 - ((1.0 - y) * z))) <= -4.490989771135522e+167) {
		tmp = x + ((x * z) * (y - 1.0));
	} else if ((x * (1.0 - ((1.0 - y) * z))) <= 2.320352310306588e+301) {
		tmp = x + (x * ((y * z) - z));
	} else {
		tmp = x + (cbrt(y - 1.0) * ((x * z) * (cbrt(y - 1.0) * cbrt(y - 1.0))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	3.1
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 x (-.f64 1 (*.f64 (-.f64 1 y) z))) < -4.49098977113552204e167

   1. Initial program 9.2

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

   3. Applied sub-neg_binary64_18485 9.2

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

   4. Applied distribute-rgt-in_binary64_18442 9.2

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

   5. Simplified 9.2

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

   6. Simplified 9.2

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

   8. Applied *-un-lft-identity_binary64_18492 9.2

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

   9. Applied distribute-rgt-out--_binary64_18446 9.2

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

   10. Applied associate-*r*_binary64_18432 0.1

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

   if -4.49098977113552204e167 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z))) < 2.32035231030658789e301

   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

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

   if 2.32035231030658789e301 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z)))

   1. Initial program 51.5

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

   3. Applied sub-neg_binary64_18485 51.5

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

   4. Applied distribute-rgt-in_binary64_18442 51.5

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

   5. Simplified 51.5

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

   6. Simplified 51.5

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

   8. Applied *-un-lft-identity_binary64_18492 51.5

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

   9. Applied distribute-rgt-out--_binary64_18446 51.5

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

   10. Applied associate-*r*_binary64_18432 0.2

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

   12. Applied add-cube-cbrt_binary64_18527 1.2

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

   13. Applied associate-*r*_binary64_18432 1.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq -4.490989771135522 \cdot 10^{+167}:\\ \;\;\;\;x + \left(x \cdot z\right) \cdot \left(y - 1\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq 2.320352310306588 \cdot 10^{+301}:\\ \;\;\;\;x + x \cdot \left(y \cdot z - z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \sqrt[3]{y - 1} \cdot \left(\left(x \cdot z\right) \cdot \left(\sqrt[3]{y - 1} \cdot \sqrt[3]{y - 1}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020344 
(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))))