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

Average Error: 7.9 → 0.4

Time: 9.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \le -1.337162358091865 \cdot 10^{282}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{{\left(\sqrt[3]{a}\right)}^{3}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \le -2.5522379957491985 \cdot 10^{-254}:\\ \;\;\;\;\frac{x \cdot y}{a} - \frac{t \cdot z}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \le 0.0:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{t}{\left(\sqrt[3]{\sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}\right) \cdot \sqrt[3]{\sqrt[3]{a}}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \le 6.9748358176334352 \cdot 10^{251}:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot y - z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{{\left(\sqrt[3]{a}\right)}^{3}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r1402439 = x;
        double r1402440 = y;
        double r1402441 = r1402439 * r1402440;
        double r1402442 = z;
        double r1402443 = t;
        double r1402444 = r1402442 * r1402443;
        double r1402445 = r1402441 - r1402444;
        double r1402446 = a;
        double r1402447 = r1402445 / r1402446;
        return r1402447;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r1402448 = x;
        double r1402449 = y;
        double r1402450 = r1402448 * r1402449;
        double r1402451 = z;
        double r1402452 = t;
        double r1402453 = r1402451 * r1402452;
        double r1402454 = r1402450 - r1402453;
        double r1402455 = -1.337162358091865e+282;
        bool r1402456 = r1402454 <= r1402455;
        double r1402457 = a;
        double r1402458 = r1402449 / r1402457;
        double r1402459 = r1402448 * r1402458;
        double r1402460 = cbrt(r1402457);
        double r1402461 = 3.0;
        double r1402462 = pow(r1402460, r1402461);
        double r1402463 = r1402452 / r1402462;
        double r1402464 = r1402451 * r1402463;
        double r1402465 = r1402459 - r1402464;
        double r1402466 = -2.5522379957491985e-254;
        bool r1402467 = r1402454 <= r1402466;
        double r1402468 = r1402450 / r1402457;
        double r1402469 = r1402452 * r1402451;
        double r1402470 = r1402469 / r1402457;
        double r1402471 = r1402468 - r1402470;
        double r1402472 = 0.0;
        bool r1402473 = r1402454 <= r1402472;
        double r1402474 = r1402460 * r1402460;
        double r1402475 = r1402451 / r1402474;
        double r1402476 = cbrt(r1402460);
        double r1402477 = r1402476 * r1402476;
        double r1402478 = r1402477 * r1402476;
        double r1402479 = r1402452 / r1402478;
        double r1402480 = r1402475 * r1402479;
        double r1402481 = r1402459 - r1402480;
        double r1402482 = 6.974835817633435e+251;
        bool r1402483 = r1402454 <= r1402482;
        double r1402484 = 1.0;
        double r1402485 = r1402484 / r1402457;
        double r1402486 = r1402485 * r1402454;
        double r1402487 = r1402483 ? r1402486 : r1402465;
        double r1402488 = r1402473 ? r1402481 : r1402487;
        double r1402489 = r1402467 ? r1402471 : r1402488;
        double r1402490 = r1402456 ? r1402465 : r1402489;
        return r1402490;
}

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	6.0
Herbie	0.4

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

Derivation

1. Split input into 4 regimes

2. if (- (* x y) (* z t)) < -1.337162358091865e+282 or 6.974835817633435e+251 < (- (* x y) (* z t))

   1. Initial program 45.2

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

   3. Applied div-sub 45.2

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

   5. Applied add-cube-cbrt 45.4

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

   6. Applied times-frac 23.6

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

   8. Applied *-un-lft-identity 23.6

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

   9. Applied times-frac 0.8

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

   10. Simplified 0.8

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

   12. Applied div-inv 0.8

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

   13. Applied associate-*l* 0.8

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

   14. Simplified 0.8

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

   if -1.337162358091865e+282 < (- (* x y) (* z t)) < -2.5522379957491985e-254

   1. Initial program 0.2

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

   3. Applied div-sub 0.2

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

   5. Applied add-cube-cbrt 0.8

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

   6. Applied times-frac 4.0

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

   7. Taylor expanded around 0 0.2

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

   if -2.5522379957491985e-254 < (- (* x y) (* z t)) < 0.0

   1. Initial program 18.0

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

   3. Applied div-sub 18.0

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

   5. Applied add-cube-cbrt 18.1

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

   6. Applied times-frac 9.8

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

   8. Applied *-un-lft-identity 9.8

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

   9. Applied times-frac 0.6

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

   10. Simplified 0.6

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

   12. Applied add-cube-cbrt 0.7

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

   if 0.0 < (- (* x y) (* z t)) < 6.974835817633435e+251

   1. Initial program 0.3

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

   3. Applied div-sub 0.3

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

   5. Applied div-inv 0.4

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

   6. Applied div-inv 0.4

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

   7. Applied distribute-rgt-out-- 0.4

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(x \cdot y - z \cdot t\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t \le -1.337162358091865 \cdot 10^{282}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{{\left(\sqrt[3]{a}\right)}^{3}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \le -2.5522379957491985 \cdot 10^{-254}:\\ \;\;\;\;\frac{x \cdot y}{a} - \frac{t \cdot z}{a}\\ \mathbf{elif}\;x \cdot y - z \cdot t \le 0.0:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{t}{\left(\sqrt[3]{\sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}\right) \cdot \sqrt[3]{\sqrt[3]{a}}}\\ \mathbf{elif}\;x \cdot y - z \cdot t \le 6.9748358176334352 \cdot 10^{251}:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot y - z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - z \cdot \frac{t}{{\left(\sqrt[3]{a}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))