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

Average Error: 7.5 → 0.8

Time: 5.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;\left(x \cdot \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 7.116535952425097 \cdot 10^{+286}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - \sqrt[3]{\frac{t}{a}} \cdot \left(z \cdot \left(\sqrt[3]{\frac{t}{a}} \cdot \sqrt[3]{\frac{t}{a}}\right)\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a) {
	return (((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)))) <= ((double) -(((double) INFINITY))))) {
		VAR = ((double) (((double) (((double) (x * ((double) (((double) cbrt(y)) * (((double) cbrt(y)) / ((double) (((double) cbrt(a)) * ((double) cbrt(a))))))))) * (((double) cbrt(y)) / ((double) cbrt(a))))) - ((double) (z * (t / a)))));
	} else {
		double VAR_1;
		if ((((double) (((double) (x * y)) - ((double) (z * t)))) <= 7.116535952425097e+286)) {
			VAR_1 = (((double) (((double) (x * y)) - ((double) (z * t)))) / a);
		} else {
			VAR_1 = ((double) (((double) (x * (y / a))) - ((double) (((double) cbrt((t / a))) * ((double) (z * ((double) (((double) cbrt((t / a))) * ((double) cbrt((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	6.2
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;z < -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z < 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

1. Split input into 3 regimes

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

   1. Initial program Error: 64.0 bits

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

   3. Applied div-sub Error: 64.0 bits

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

   4. Simplified Error: 31.0 bits

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

   5. Simplified Error: 0.3 bits

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

   7. Applied add-cube-cbrt Error: 0.9 bits

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

   8. Applied add-cube-cbrt Error: 1.0 bits

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

   9. Applied times-frac Error: 1.0 bits

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

   10. Applied associate-*r* Error: 1.0 bits

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

   11. Simplified Error: 0.9 bits

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

   if -inf.0 < (- (* x y) (* z t)) < 7.1165359524250968e286

   1. Initial program Error: 0.8 bits

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

   if 7.1165359524250968e286 < (- (* x y) (* z t))

   1. Initial program Error: 52.6 bits

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

   3. Applied div-sub Error: 52.6 bits

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

   4. Simplified Error: 28.8 bits

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

   5. Simplified Error: 0.2 bits

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

   7. Applied add-cube-cbrt Error: 0.8 bits

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

   8. Applied associate-*r* Error: 0.8 bits

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

4. Final simplification Error: 0.8 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \leq -\infty:\\ \;\;\;\;\left(x \cdot \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}} - z \cdot \frac{t}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \leq 7.116535952425097 \cdot 10^{+286}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - \sqrt[3]{\frac{t}{a}} \cdot \left(z \cdot \left(\sqrt[3]{\frac{t}{a}} \cdot \sqrt[3]{\frac{t}{a}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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))