Data.Colour.Matrix:inverse from colour-2.3.3, B

Average Error: 7.8 → 1.6

Time: 6.8s

Precision: binary64

Math

\[\frac{x \cdot y - z \cdot t}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} = -inf.0:\\ \;\;\;\;x \cdot \frac{y}{a} - \sqrt{z \cdot \frac{t}{a}} \cdot \sqrt{z \cdot \frac{t}{a}}\\ \mathbf{elif}\;\frac{x \cdot y - z \cdot t}{a} \le 7.9357084036002786 \cdot 10^{291}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{y}{a}} \cdot \left(x \cdot \left(\sqrt[3]{\frac{y}{a}} \cdot \sqrt[3]{\frac{y}{a}}\right)\right) - z \cdot \frac{t}{a}\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (((double) (x * y)) - ((double) (z * t)))) / a));
}

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((((double) (((double) (((double) (x * y)) - ((double) (z * t)))) / a)) <= -inf.0)) {
		VAR = ((double) (((double) (x * ((double) (y / a)))) - ((double) (((double) sqrt(((double) (z * ((double) (t / a)))))) * ((double) sqrt(((double) (z * ((double) (t / a))))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (((double) (x * y)) - ((double) (z * t)))) / a)) <= 7.935708403600279e+291)) {
			VAR_1 = ((double) (((double) (((double) (x * y)) - ((double) (z * t)))) / a));
		} else {
			VAR_1 = ((double) (((double) (((double) cbrt(((double) (y / a)))) * ((double) (x * ((double) (((double) cbrt(((double) (y / a)))) * ((double) cbrt(((double) (y / a)))))))))) - ((double) (z * ((double) (t / a))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.8
Target	6.3
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;z \lt -2.46868496869954822 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.30983112197837121 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ (- (* x y) (* z t)) a) < -inf.0

   1. Initial program 64.0

      \[\frac{x \cdot y - z \cdot t}{a}\]
   2. Using strategy rm

   3. Applied div-sub 64.0

      \[\leadsto \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}}\]

   4. Simplified 31.6

      \[\leadsto \color{blue}{x \cdot \frac{y}{a}} - \frac{z \cdot t}{a}\]

   5. Simplified 0.3

      \[\leadsto x \cdot \frac{y}{a} - \color{blue}{z \cdot \frac{t}{a}}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt 9.4

      \[\leadsto x \cdot \frac{y}{a} - \color{blue}{\sqrt{z \cdot \frac{t}{a}} \cdot \sqrt{z \cdot \frac{t}{a}}}\]

   if -inf.0 < (/ (- (* x y) (* z t)) a) < 7.9357084036002786e291

   1. Initial program 0.9

      \[\frac{x \cdot y - z \cdot t}{a}\]

   if 7.9357084036002786e291 < (/ (- (* x y) (* z t)) a)

   1. Initial program 53.2

      \[\frac{x \cdot y - z \cdot t}{a}\]
   2. Using strategy rm

   3. Applied div-sub 53.2

      \[\leadsto \color{blue}{\frac{x \cdot y}{a} - \frac{z \cdot t}{a}}\]

   4. Simplified 30.3

      \[\leadsto \color{blue}{x \cdot \frac{y}{a}} - \frac{z \cdot t}{a}\]

   5. Simplified 4.2

      \[\leadsto x \cdot \frac{y}{a} - \color{blue}{z \cdot \frac{t}{a}}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 4.7

      \[\leadsto x \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{y}{a}} \cdot \sqrt[3]{\frac{y}{a}}\right) \cdot \sqrt[3]{\frac{y}{a}}\right)} - z \cdot \frac{t}{a}\]

   8. Applied associate-*r* 4.7

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} = -inf.0:\\ \;\;\;\;x \cdot \frac{y}{a} - \sqrt{z \cdot \frac{t}{a}} \cdot \sqrt{z \cdot \frac{t}{a}}\\ \mathbf{elif}\;\frac{x \cdot y - z \cdot t}{a} \le 7.9357084036002786 \cdot 10^{291}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{y}{a}} \cdot \left(x \cdot \left(\sqrt[3]{\frac{y}{a}} \cdot \sqrt[3]{\frac{y}{a}}\right)\right) - z \cdot \frac{t}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

  (/ (- (* x y) (* z t)) a))