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

Average Error: 7.5 → 0.7

Time: 5.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} = -inf.0:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{y}{a}\right) - z \cdot \frac{t}{a}\\ \mathbf{elif}\;\frac{x \cdot y - z \cdot t}{a} \le 1.25326139336449662 \cdot 10^{304}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}} - z \cdot \frac{t}{a}\\ \end{array}\]

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) (((double) (((double) cbrt(x)) * ((double) cbrt(x)))) * ((double) (((double) cbrt(x)) * ((double) (y / a)))))) - ((double) (z * ((double) (t / a))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (((double) (x * y)) - ((double) (z * t)))) / a)) <= 1.2532613933644966e+304)) {
			VAR_1 = ((double) (((double) (((double) (x * y)) - ((double) (z * t)))) / a));
		} else {
			VAR_1 = ((double) (((double) (((double) (x * ((double) (((double) cbrt(y)) * ((double) (((double) cbrt(y)) / ((double) (((double) cbrt(a)) * ((double) cbrt(a)))))))))) * ((double) (((double) cbrt(y)) / ((double) cbrt(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	7.5
Target	6.1
Herbie	0.7

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

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 33.6

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

   5. Simplified 0.3

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

   7. Applied add-cube-cbrt 0.8

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

   8. Applied associate-*l* 0.8

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

   9. Simplified 0.8

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

   if -inf.0 < (/ (- (* x y) (* z t)) a) < 1.25326139336449662e304

   1. Initial program 0.7

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

   if 1.25326139336449662e304 < (/ (- (* x y) (* z t)) a)

   1. Initial program 62.1

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

   3. Applied div-sub 62.1

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

   4. Simplified 34.0

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

   5. Simplified 0.8

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

   7. Applied add-cube-cbrt 1.3

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

   8. Applied add-cube-cbrt 1.4

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

   9. Applied times-frac 1.4

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

   10. Applied associate-*r* 1.4

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

   11. Simplified 1.4

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y - z \cdot t}{a} = -inf.0:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{y}{a}\right) - z \cdot \frac{t}{a}\\ \mathbf{elif}\;\frac{x \cdot y - z \cdot t}{a} \le 1.25326139336449662 \cdot 10^{304}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}} - z \cdot \frac{t}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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))