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

Average Error: 7.2 → 0.8

Time: 3.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t = -\infty \lor \neg \left(x \cdot y - z \cdot t \le 1.289984888446294143698575098275272478903 \cdot 10^{281}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - \left(\sqrt[3]{t \cdot \frac{z}{a}} \cdot \sqrt[3]{t \cdot \frac{z}{a}}\right) \cdot \sqrt[3]{t \cdot \frac{z}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a} - \left(t \cdot z\right) \cdot \frac{1}{a}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r878923 = x;
        double r878924 = y;
        double r878925 = r878923 * r878924;
        double r878926 = z;
        double r878927 = t;
        double r878928 = r878926 * r878927;
        double r878929 = r878925 - r878928;
        double r878930 = a;
        double r878931 = r878929 / r878930;
        return r878931;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r878932 = x;
        double r878933 = y;
        double r878934 = r878932 * r878933;
        double r878935 = z;
        double r878936 = t;
        double r878937 = r878935 * r878936;
        double r878938 = r878934 - r878937;
        double r878939 = -inf.0;
        bool r878940 = r878938 <= r878939;
        double r878941 = 1.2899848884462941e+281;
        bool r878942 = r878938 <= r878941;
        double r878943 = !r878942;
        bool r878944 = r878940 || r878943;
        double r878945 = a;
        double r878946 = r878933 / r878945;
        double r878947 = r878932 * r878946;
        double r878948 = r878935 / r878945;
        double r878949 = r878936 * r878948;
        double r878950 = cbrt(r878949);
        double r878951 = r878950 * r878950;
        double r878952 = r878951 * r878950;
        double r878953 = r878947 - r878952;
        double r878954 = r878934 / r878945;
        double r878955 = r878936 * r878935;
        double r878956 = 1.0;
        double r878957 = r878956 / r878945;
        double r878958 = r878955 * r878957;
        double r878959 = r878954 - r878958;
        double r878960 = r878944 ? r878953 : r878959;
        return r878960;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.2
Target	5.7
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;z \lt -2.468684968699548224247694913169778644284 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.309831121978371209578784129518242708809 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (* x y) (* z t)) < -inf.0 or 1.2899848884462941e+281 < (- (* x y) (* z t))

   1. Initial program 55.6

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

   3. Applied div-sub 55.6

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

   4. Simplified 55.6

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

   6. Applied *-un-lft-identity 55.6

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

   7. Applied times-frac 29.8

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

   8. Simplified 29.8

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

   10. Applied *-un-lft-identity 29.8

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

   11. Applied times-frac 0.3

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

   12. Simplified 0.3

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

   14. Applied add-cube-cbrt 0.8

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

   if -inf.0 < (- (* x y) (* z t)) < 1.2899848884462941e+281

   1. Initial program 0.7

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

   3. Applied div-sub 0.7

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

   4. Simplified 0.7

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

   6. Applied *-un-lft-identity 0.7

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

   7. Applied times-frac 5.4

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

   8. Simplified 5.4

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

   10. Applied div-inv 5.5

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

   11. Applied associate-*r* 0.8

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t = -\infty \lor \neg \left(x \cdot y - z \cdot t \le 1.289984888446294143698575098275272478903 \cdot 10^{281}\right):\\ \;\;\;\;x \cdot \frac{y}{a} - \left(\sqrt[3]{t \cdot \frac{z}{a}} \cdot \sqrt[3]{t \cdot \frac{z}{a}}\right) \cdot \sqrt[3]{t \cdot \frac{z}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a} - \left(t \cdot z\right) \cdot \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(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))