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

Average Error: 7.7 → 1.1

Time: 11.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \le -1.99518096941667601 \cdot 10^{305} \lor \neg \left(x \cdot y - z \cdot t \le 7.95286389093006834 \cdot 10^{292}\right):\\ \;\;\;\;\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}} - t \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{x \cdot y - z \cdot t}}\\ \end{array}\]

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)))) <= -1.995180969416676e+305) || !(((double) (((double) (x * y)) - ((double) (z * t)))) <= 7.952863890930068e+292))) {
		VAR = ((double) (((double) (((double) (x / ((double) (((double) cbrt(a)) * ((double) cbrt(a)))))) * ((double) (y / ((double) cbrt(a)))))) - ((double) (t * ((double) (z / a))))));
	} else {
		VAR = ((double) (1.0 / ((double) (a / ((double) (((double) (x * y)) - ((double) (z * t))))))));
	}
	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	7.7
Target	5.9
Herbie	1.1

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

Derivation

1. Split input into 2 regimes

2. if (- (* x y) (* z t)) < -1.99518096941667601e305 or 7.95286389093006834e292 < (- (* x y) (* z t))

   1. Initial program 59.2

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

   3. Applied div-sub 59.2

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

   4. Simplified 59.2

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

   6. Applied add-cube-cbrt 59.2

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

   7. Applied times-frac 31.1

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

   9. Applied *-un-lft-identity 31.1

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

   10. Applied times-frac 0.8

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

   11. Simplified 0.8

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

   if -1.99518096941667601e305 < (- (* x y) (* z t)) < 7.95286389093006834e292

   1. Initial program 0.8

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

   3. Applied clear-num 1.1

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \le -1.99518096941667601 \cdot 10^{305} \lor \neg \left(x \cdot y - z \cdot t \le 7.95286389093006834 \cdot 10^{292}\right):\\ \;\;\;\;\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}} - t \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{x \cdot y - z \cdot t}}\\ \end{array}\]

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))