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

Average Error: 7.9 → 1.8

Time: 38.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \le -8.8508885940779006 \cdot 10^{184} \lor \neg \left(x \cdot y - z \cdot t \le 2.0890422248165597 \cdot 10^{209}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{x \cdot y - z \cdot t}{\sqrt[3]{a}}\\ \end{array}\]

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)))) <= -8.8508885940779e+184) || !(((double) (((double) (x * y)) - ((double) (z * t)))) <= 2.0890422248165597e+209))) {
		VAR = ((double) (((double) (x * ((double) (y / a)))) - ((double) (((double) (t / ((double) (((double) cbrt(a)) * ((double) cbrt(a)))))) * ((double) (z / ((double) cbrt(a))))))));
	} else {
		VAR = ((double) (((double) (1.0 / ((double) (((double) cbrt(a)) * ((double) cbrt(a)))))) * ((double) (((double) (((double) (x * y)) - ((double) (z * t)))) / ((double) cbrt(a))))));
	}
	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	7.9
Target	6.3
Herbie	1.8

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

Derivation

1. Split input into 2 regimes

2. if (- (* x y) (* z t)) < -8.8508885940779006e184 or 2.0890422248165597e209 < (- (* x y) (* z t))

   1. Initial program 28.0

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

   3. Applied div-sub 28.0

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

   4. Simplified 28.0

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

   6. Applied add-cube-cbrt 28.4

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

   7. Applied times-frac 15.6

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

   9. Applied *-un-lft-identity 15.6

      \[\leadsto \frac{x \cdot y}{\color{blue}{1 \cdot a}} - \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\]

   10. Applied times-frac 1.7

       \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y}{a}} - \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\]

   11. Simplified 1.7

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

   if -8.8508885940779006e184 < (- (* x y) (* z t)) < 2.0890422248165597e209

   1. Initial program 0.9

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

   3. Applied add-cube-cbrt 1.9

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

   4. Applied *-un-lft-identity 1.9

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y - z \cdot t\right)}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   5. Applied times-frac 1.9

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{x \cdot y - z \cdot t}{\sqrt[3]{a}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \le -8.8508885940779006 \cdot 10^{184} \lor \neg \left(x \cdot y - z \cdot t \le 2.0890422248165597 \cdot 10^{209}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{x \cdot y - z \cdot t}{\sqrt[3]{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(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))