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

Average Error: 7.8 → 1.0

Time: 6.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \le -3.97470660796454297 \cdot 10^{291}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \left(\sqrt[3]{\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}} \cdot \sqrt[3]{\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}}\right) \cdot \sqrt[3]{\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \le 7.8325114634375287 \cdot 10^{275}:\\ \;\;\;\;\frac{1}{\frac{a}{x \cdot y - z \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y - \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\\ \end{array}\]

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) (x * y)) - ((double) (z * t)))) <= -3.974706607964543e+291)) {
		VAR = ((double) (((double) (x / ((double) (a / y)))) - ((double) (((double) (((double) cbrt(((double) (((double) (t / ((double) (((double) cbrt(a)) * ((double) cbrt(a)))))) * ((double) (z / ((double) cbrt(a)))))))) * ((double) cbrt(((double) (((double) (t / ((double) (((double) cbrt(a)) * ((double) cbrt(a)))))) * ((double) (z / ((double) cbrt(a)))))))))) * ((double) cbrt(((double) (((double) (t / ((double) (((double) cbrt(a)) * ((double) cbrt(a)))))) * ((double) (z / ((double) cbrt(a))))))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (x * y)) - ((double) (z * t)))) <= 7.832511463437529e+275)) {
			VAR_1 = ((double) (1.0 / ((double) (a / ((double) (((double) (x * y)) - ((double) (z * t))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (x / a)) * y)) - ((double) (((double) (t / ((double) (((double) cbrt(a)) * ((double) cbrt(a)))))) * ((double) (z / ((double) cbrt(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	6.1
Herbie	1.0

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

Derivation

1. Split input into 3 regimes

2. if (- (* x y) (* z t)) < -3.97470660796454297e291

   1. Initial program 53.7

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

   3. Applied div-sub 53.7

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

   4. Simplified 53.7

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

   6. Applied add-cube-cbrt 53.8

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

   7. Applied times-frac 26.7

      \[\leadsto \frac{x \cdot y}{a} - \color{blue}{\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}}\]
   8. Using strategy rm

   9. Applied associate-/l* 0.9

      \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 1.1

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

   if -3.97470660796454297e291 < (- (* x y) (* z t)) < 7.8325114634375287e275

   1. Initial program 0.8

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

   3. Applied clear-num 1.0

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{x \cdot y - z \cdot t}}}\]

   if 7.8325114634375287e275 < (- (* x y) (* z t))

   1. Initial program 49.7

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

   3. Applied div-sub 49.7

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

   4. Simplified 49.7

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

   6. Applied add-cube-cbrt 49.9

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

   7. Applied times-frac 26.6

      \[\leadsto \frac{x \cdot y}{a} - \color{blue}{\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}}\]
   8. Using strategy rm

   9. Applied associate-/l* 0.8

      \[\leadsto \color{blue}{\frac{x}{\frac{a}{y}}} - \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\]
   10. Using strategy rm

   11. Applied associate-/r/ 0.8

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \le -3.97470660796454297 \cdot 10^{291}:\\ \;\;\;\;\frac{x}{\frac{a}{y}} - \left(\sqrt[3]{\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}} \cdot \sqrt[3]{\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}}\right) \cdot \sqrt[3]{\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \le 7.8325114634375287 \cdot 10^{275}:\\ \;\;\;\;\frac{1}{\frac{a}{x \cdot y - z \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a} \cdot y - \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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))