Average Error: 7.7 → 1.1
Time: 4.3s
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t = -\infty \lor \neg \left(x \cdot y - z \cdot t \le 1.79932804528860272 \cdot 10^{298}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\frac{z}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{x \cdot y}} - \frac{1}{\frac{a}{t \cdot z}}\\ \end{array}\]
\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 1.79932804528860272 \cdot 10^{298}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{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}:\\
\;\;\;\;\frac{1}{\frac{a}{x \cdot y}} - \frac{1}{\frac{a}{t \cdot z}}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r729942 = x;
        double r729943 = y;
        double r729944 = r729942 * r729943;
        double r729945 = z;
        double r729946 = t;
        double r729947 = r729945 * r729946;
        double r729948 = r729944 - r729947;
        double r729949 = a;
        double r729950 = r729948 / r729949;
        return r729950;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r729951 = x;
        double r729952 = y;
        double r729953 = r729951 * r729952;
        double r729954 = z;
        double r729955 = t;
        double r729956 = r729954 * r729955;
        double r729957 = r729953 - r729956;
        double r729958 = -inf.0;
        bool r729959 = r729957 <= r729958;
        double r729960 = 1.7993280452886027e+298;
        bool r729961 = r729957 <= r729960;
        double r729962 = !r729961;
        bool r729963 = r729959 || r729962;
        double r729964 = a;
        double r729965 = cbrt(r729964);
        double r729966 = r729965 * r729965;
        double r729967 = r729951 / r729966;
        double r729968 = r729952 / r729965;
        double r729969 = r729954 / r729965;
        double r729970 = r729955 / r729966;
        double r729971 = r729969 * r729970;
        double r729972 = -r729971;
        double r729973 = fma(r729967, r729968, r729972);
        double r729974 = -r729969;
        double r729975 = r729974 + r729969;
        double r729976 = r729970 * r729975;
        double r729977 = r729973 + r729976;
        double r729978 = 1.0;
        double r729979 = r729964 / r729953;
        double r729980 = r729978 / r729979;
        double r729981 = r729955 * r729954;
        double r729982 = r729964 / r729981;
        double r729983 = r729978 / r729982;
        double r729984 = r729980 - r729983;
        double r729985 = r729963 ? r729977 : r729984;
        return r729985;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.7
Target6.2
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;z \lt -2.46868496869954822 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.30983112197837121 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* z t)) < -inf.0 or 1.7993280452886027e+298 < (- (* x y) (* z t))

    1. Initial program 60.7

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

    3. Applied div-sub60.7

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

    4. Simplified60.7

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

    6. Applied add-cube-cbrt60.8

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

    7. Applied times-frac33.7

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

    8. Applied add-cube-cbrt33.8

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

    9. Applied times-frac1.3

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

    10. Applied prod-diff1.3

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

    11. Simplified1.3

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

    if -inf.0 < (- (* x y) (* z t)) < 1.7993280452886027e+298

    1. Initial program 0.8

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

    3. Applied div-sub0.8

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

    4. Simplified0.8

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

    6. Applied clear-num1.0

      \[\leadsto \frac{x \cdot y}{a} - \color{blue}{\frac{1}{\frac{a}{t \cdot z}}}\]
    7. Using strategy rm

    8. Applied clear-num1.1

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t = -\infty \lor \neg \left(x \cdot y - z \cdot t \le 1.79932804528860272 \cdot 10^{298}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\frac{z}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a}{x \cdot y}} - \frac{1}{\frac{a}{t \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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))