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

Average Error: 7.2 → 1.0

Time: 19.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot x - z \cdot t = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{\sqrt[3]{a}}, \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a}}\right) + \mathsf{fma}\left(x, \frac{y}{a}, \frac{-\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \left(\frac{\sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{a}}} \cdot \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)\right)\\ \mathbf{elif}\;y \cdot x - z \cdot t \le 6.415733537003891 \cdot 10^{+276}:\\ \;\;\;\;\frac{1}{a} \cdot \left(y \cdot x - z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{\sqrt[3]{a}}, \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a}}\right) + \mathsf{fma}\left(x, \frac{y}{a}, \frac{-t}{\sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r38697934 = x;
        double r38697935 = y;
        double r38697936 = r38697934 * r38697935;
        double r38697937 = z;
        double r38697938 = t;
        double r38697939 = r38697937 * r38697938;
        double r38697940 = r38697936 - r38697939;
        double r38697941 = a;
        double r38697942 = r38697940 / r38697941;
        return r38697942;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r38697943 = y;
        double r38697944 = x;
        double r38697945 = r38697943 * r38697944;
        double r38697946 = z;
        double r38697947 = t;
        double r38697948 = r38697946 * r38697947;
        double r38697949 = r38697945 - r38697948;
        double r38697950 = -inf.0;
        bool r38697951 = r38697949 <= r38697950;
        double r38697952 = -r38697947;
        double r38697953 = a;
        double r38697954 = cbrt(r38697953);
        double r38697955 = r38697952 / r38697954;
        double r38697956 = r38697954 * r38697954;
        double r38697957 = r38697946 / r38697956;
        double r38697958 = r38697947 / r38697954;
        double r38697959 = r38697957 * r38697958;
        double r38697960 = fma(r38697955, r38697957, r38697959);
        double r38697961 = r38697943 / r38697953;
        double r38697962 = cbrt(r38697947);
        double r38697963 = r38697962 * r38697962;
        double r38697964 = -r38697963;
        double r38697965 = cbrt(r38697956);
        double r38697966 = r38697964 / r38697965;
        double r38697967 = cbrt(r38697954);
        double r38697968 = r38697962 / r38697967;
        double r38697969 = r38697968 * r38697957;
        double r38697970 = r38697966 * r38697969;
        double r38697971 = fma(r38697944, r38697961, r38697970);
        double r38697972 = r38697960 + r38697971;
        double r38697973 = 6.415733537003891e+276;
        bool r38697974 = r38697949 <= r38697973;
        double r38697975 = 1.0;
        double r38697976 = r38697975 / r38697953;
        double r38697977 = r38697976 * r38697949;
        double r38697978 = r38697955 * r38697957;
        double r38697979 = fma(r38697944, r38697961, r38697978);
        double r38697980 = r38697960 + r38697979;
        double r38697981 = r38697974 ? r38697977 : r38697980;
        double r38697982 = r38697951 ? r38697972 : r38697981;
        return r38697982;
}

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	1.0

\[\begin{array}{l} \mathbf{if}\;z \lt -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (- (* x y) (* z t)) < -inf.0

   1. Initial program 59.9

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

   3. Applied div-sub 59.9

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

   5. Applied add-cube-cbrt 59.9

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

   6. Applied times-frac 30.4

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

   7. Applied *-un-lft-identity 30.4

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

   8. Applied times-frac 0.8

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

   9. Applied prod-diff 0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\frac{t}{\sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \mathsf{fma}\left(-\frac{t}{\sqrt[3]{a}}, \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{t}{\sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 0.8

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

   12. Applied cbrt-prod 0.8

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

   13. Applied add-cube-cbrt 0.9

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

   14. Applied times-frac 0.9

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

   15. Applied associate-*l* 0.9

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

   if -inf.0 < (- (* x y) (* z t)) < 6.415733537003891e+276

   1. Initial program 0.9

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

   3. Applied div-sub 0.9

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

   5. Applied div-inv 0.9

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

   6. Applied div-inv 1.0

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

   7. Applied distribute-rgt-out-- 1.0

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

   if 6.415733537003891e+276 < (- (* x y) (* z t))

   1. Initial program 48.0

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

   3. Applied div-sub 48.0

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

   5. Applied add-cube-cbrt 48.1

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

   6. Applied times-frac 26.4

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

   7. Applied *-un-lft-identity 26.4

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

   8. Applied times-frac 0.8

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

   9. Applied prod-diff 0.8

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x - z \cdot t = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{\sqrt[3]{a}}, \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a}}\right) + \mathsf{fma}\left(x, \frac{y}{a}, \frac{-\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \left(\frac{\sqrt[3]{t}}{\sqrt[3]{\sqrt[3]{a}}} \cdot \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)\right)\\ \mathbf{elif}\;y \cdot x - z \cdot t \le 6.415733537003891 \cdot 10^{+276}:\\ \;\;\;\;\frac{1}{a} \cdot \left(y \cdot x - z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{\sqrt[3]{a}}, \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a}}\right) + \mathsf{fma}\left(x, \frac{y}{a}, \frac{-t}{\sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"

  :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))