Average Error: 7.8 → 5.0
Time: 4.6s
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.84154847940333438 \cdot 10^{268}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\frac{t \cdot z}{a}\right)\\ \mathbf{elif}\;x \cdot y \le 5.0132949514285239 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{\frac{a}{x \cdot y - z \cdot t}}\\ \mathbf{elif}\;x \cdot y \le 2.10280870876133442 \cdot 10^{131}:\\ \;\;\;\;\left(\frac{x \cdot y}{a} - \frac{z}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\frac{t \cdot z}{a}\right)\\ \end{array}\]
\frac{x \cdot y - z \cdot t}{a}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -1.84154847940333438 \cdot 10^{268}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\frac{t \cdot z}{a}\right)\\

\mathbf{elif}\;x \cdot y \le 5.0132949514285239 \cdot 10^{-62}:\\
\;\;\;\;\frac{1}{\frac{a}{x \cdot y - z \cdot t}}\\

\mathbf{elif}\;x \cdot y \le 2.10280870876133442 \cdot 10^{131}:\\
\;\;\;\;\left(\frac{x \cdot y}{a} - \frac{z}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\frac{t \cdot z}{a}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r933934 = x;
        double r933935 = y;
        double r933936 = r933934 * r933935;
        double r933937 = z;
        double r933938 = t;
        double r933939 = r933937 * r933938;
        double r933940 = r933936 - r933939;
        double r933941 = a;
        double r933942 = r933940 / r933941;
        return r933942;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r933943 = x;
        double r933944 = y;
        double r933945 = r933943 * r933944;
        double r933946 = -1.8415484794033344e+268;
        bool r933947 = r933945 <= r933946;
        double r933948 = 1.0;
        double r933949 = r933943 / r933948;
        double r933950 = a;
        double r933951 = r933944 / r933950;
        double r933952 = t;
        double r933953 = z;
        double r933954 = r933952 * r933953;
        double r933955 = r933954 / r933950;
        double r933956 = -r933955;
        double r933957 = fma(r933949, r933951, r933956);
        double r933958 = 5.013294951428524e-62;
        bool r933959 = r933945 <= r933958;
        double r933960 = r933953 * r933952;
        double r933961 = r933945 - r933960;
        double r933962 = r933950 / r933961;
        double r933963 = r933948 / r933962;
        double r933964 = 2.1028087087613344e+131;
        bool r933965 = r933945 <= r933964;
        double r933966 = r933945 / r933950;
        double r933967 = cbrt(r933950);
        double r933968 = r933953 / r933967;
        double r933969 = r933967 * r933967;
        double r933970 = r933952 / r933969;
        double r933971 = r933968 * r933970;
        double r933972 = r933966 - r933971;
        double r933973 = -r933968;
        double r933974 = r933973 + r933968;
        double r933975 = r933970 * r933974;
        double r933976 = r933972 + r933975;
        double r933977 = r933965 ? r933976 : r933957;
        double r933978 = r933959 ? r933963 : r933977;
        double r933979 = r933947 ? r933957 : r933978;
        return r933979;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.8
Target6.3
Herbie5.0
\[\begin{array}{l} \mathbf{if}\;z \lt -2.46868496869954822 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.30983112197837121 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x y) < -1.8415484794033344e+268 or 2.1028087087613344e+131 < (* x y)

    1. Initial program 28.0

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

    3. Applied div-sub28.0

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

    4. Simplified28.0

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

    6. Applied *-un-lft-identity28.0

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

    7. Applied times-frac8.0

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

    8. Applied fma-neg8.0

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

    if -1.8415484794033344e+268 < (* x y) < 5.013294951428524e-62

    1. Initial program 4.6

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

    3. Applied clear-num4.9

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

    if 5.013294951428524e-62 < (* x y) < 2.1028087087613344e+131

    1. Initial program 3.1

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

    3. Applied div-sub3.1

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

    4. Simplified3.1

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

    6. Applied add-cube-cbrt3.4

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

    7. Applied times-frac2.8

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

    8. Applied add-sqr-sqrt31.9

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

    9. Applied prod-diff31.9

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

    10. Simplified2.8

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

    11. Simplified2.8

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

  4. Final simplification5.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.84154847940333438 \cdot 10^{268}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\frac{t \cdot z}{a}\right)\\ \mathbf{elif}\;x \cdot y \le 5.0132949514285239 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{\frac{a}{x \cdot y - z \cdot t}}\\ \mathbf{elif}\;x \cdot y \le 2.10280870876133442 \cdot 10^{131}:\\ \;\;\;\;\left(\frac{x \cdot y}{a} - \frac{z}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1}, \frac{y}{a}, -\frac{t \cdot z}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(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))