Average Error: 7.5 → 0.8
Time: 6.2s
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t = -\infty \lor \neg \left(x \cdot y - z \cdot t \le 5.82727439177554069 \cdot 10^{289}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -z \cdot \frac{t}{a}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot y - t \cdot z\right)\\ \end{array}\]
\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 5.82727439177554069 \cdot 10^{289}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -z \cdot \frac{t}{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}{a} \cdot \left(x \cdot y - t \cdot z\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r989119 = x;
        double r989120 = y;
        double r989121 = r989119 * r989120;
        double r989122 = z;
        double r989123 = t;
        double r989124 = r989122 * r989123;
        double r989125 = r989121 - r989124;
        double r989126 = a;
        double r989127 = r989125 / r989126;
        return r989127;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r989128 = x;
        double r989129 = y;
        double r989130 = r989128 * r989129;
        double r989131 = z;
        double r989132 = t;
        double r989133 = r989131 * r989132;
        double r989134 = r989130 - r989133;
        double r989135 = -inf.0;
        bool r989136 = r989134 <= r989135;
        double r989137 = 5.827274391775541e+289;
        bool r989138 = r989134 <= r989137;
        double r989139 = !r989138;
        bool r989140 = r989136 || r989139;
        double r989141 = a;
        double r989142 = cbrt(r989141);
        double r989143 = r989142 * r989142;
        double r989144 = r989128 / r989143;
        double r989145 = r989129 / r989142;
        double r989146 = r989132 / r989141;
        double r989147 = r989131 * r989146;
        double r989148 = -r989147;
        double r989149 = fma(r989144, r989145, r989148);
        double r989150 = r989132 / r989143;
        double r989151 = r989131 / r989142;
        double r989152 = -r989151;
        double r989153 = r989152 + r989151;
        double r989154 = r989150 * r989153;
        double r989155 = r989149 + r989154;
        double r989156 = 1.0;
        double r989157 = r989156 / r989141;
        double r989158 = r989132 * r989131;
        double r989159 = r989130 - r989158;
        double r989160 = r989157 * r989159;
        double r989161 = r989140 ? r989155 : r989160;
        return r989161;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.5
Target6.0
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;z \lt -2.46868496869954822 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.30983112197837121 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* z t)) < -inf.0 or 5.827274391775541e+289 < (- (* x y) (* z t))

    1. Initial program 58.6

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

    3. Applied div-sub58.6

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

    4. Simplified58.6

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

    6. Applied add-cube-cbrt58.6

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

    7. Applied times-frac31.7

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

    8. Applied add-cube-cbrt31.7

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

    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

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

    13. Applied div-inv1.3

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

    14. Applied associate-*l*1.4

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

    15. Simplified0.8

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

    if -inf.0 < (- (* x y) (* z t)) < 5.827274391775541e+289

    1. Initial program 0.7

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

    3. Applied div-sub0.7

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

    4. Simplified0.7

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

    6. Applied div-inv0.8

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

    7. Applied div-inv0.8

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

    8. Applied distribute-rgt-out--0.8

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t = -\infty \lor \neg \left(x \cdot y - z \cdot t \le 5.82727439177554069 \cdot 10^{289}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -z \cdot \frac{t}{a}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot y - t \cdot z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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))