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

Average Error: 3.3 → 0.1

Time: 6.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;z \cdot \left(x \cdot y - x\right)\\ \mathbf{elif}\;t_0 \leq 1.987519463954085 \cdot 10^{+180}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, z, 1\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot z, x\right) - x \cdot z\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* (- 1.0 y) z)))))
   (if (<= t_0 (- INFINITY))
     (* z (- (* x y) x))
     (if (<= t_0 1.987519463954085e+180)
       (* x (- (fma y z 1.0) z))
       (- (fma y (* x z) x) (* 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 t_0 = x * (1.0 - ((1.0 - y) * z));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = z * ((x * y) - x);
	} else if (t_0 <= 1.987519463954085e+180) {
		tmp = x * (fma(y, z, 1.0) - z);
	} else {
		tmp = fma(y, (x * z), x) - (x * z);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	3.3
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))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded in y around 0 0.3

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

   4. Taylor expanded in z around inf 0.3

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

   if -inf.0 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z))) < 1.98751946395408493e180

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   if 1.98751946395408493e180 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z)))

   1. Initial program 11.4

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

   2. Simplified 11.4

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

   3. Taylor expanded in y around 0 0.1

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

   4. Applied fma-def_binary64 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq -\infty:\\ \;\;\;\;z \cdot \left(x \cdot y - x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq 1.987519463954085 \cdot 10^{+180}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(y, z, 1\right) - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x \cdot z, x\right) - x \cdot z\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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))))