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

Average Error: 7.9 → 0.4

Time: 7.0s

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 r1467832 = x;
        double r1467833 = y;
        double r1467834 = r1467832 * r1467833;
        double r1467835 = z;
        double r1467836 = t;
        double r1467837 = r1467835 * r1467836;
        double r1467838 = r1467834 - r1467837;
        double r1467839 = a;
        double r1467840 = r1467838 / r1467839;
        return r1467840;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r1467841 = x;
        double r1467842 = y;
        double r1467843 = r1467841 * r1467842;
        double r1467844 = z;
        double r1467845 = t;
        double r1467846 = r1467844 * r1467845;
        double r1467847 = r1467843 - r1467846;
        double r1467848 = -1.337162358091865e+282;
        bool r1467849 = r1467847 <= r1467848;
        double r1467850 = a;
        double r1467851 = r1467842 / r1467850;
        double r1467852 = r1467841 * r1467851;
        double r1467853 = cbrt(r1467850);
        double r1467854 = 3.0;
        double r1467855 = pow(r1467853, r1467854);
        double r1467856 = r1467845 / r1467855;
        double r1467857 = r1467844 * r1467856;
        double r1467858 = r1467852 - r1467857;
        double r1467859 = -2.5522379957491985e-254;
        bool r1467860 = r1467847 <= r1467859;
        double r1467861 = r1467843 / r1467850;
        double r1467862 = r1467845 * r1467844;
        double r1467863 = r1467862 / r1467850;
        double r1467864 = r1467861 - r1467863;
        double r1467865 = 0.0;
        bool r1467866 = r1467847 <= r1467865;
        double r1467867 = r1467853 * r1467853;
        double r1467868 = r1467844 / r1467867;
        double r1467869 = cbrt(r1467853);
        double r1467870 = r1467869 * r1467869;
        double r1467871 = r1467870 * r1467869;
        double r1467872 = r1467845 / r1467871;
        double r1467873 = r1467868 * r1467872;
        double r1467874 = r1467852 - r1467873;
        double r1467875 = 6.974835817633435e+251;
        bool r1467876 = r1467847 <= r1467875;
        double r1467877 = 1.0;
        double r1467878 = r1467877 / r1467850;
        double r1467879 = r1467878 * r1467847;
        double r1467880 = r1467876 ? r1467879 : r1467858;
        double r1467881 = r1467866 ? r1467874 : r1467880;
        double r1467882 = r1467860 ? r1467864 : r1467881;
        double r1467883 = r1467849 ? r1467858 : r1467882;
        return r1467883;
}

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))