Average Error: 7.5 → 0.8
Time: 5.9s
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t = -\infty \lor \neg \left(x \cdot y - z \cdot t \le 1.38915788899152083 \cdot 10^{298}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -z \cdot \frac{t}{a}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot y - t \cdot z\right)\\ \end{array}\]
\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 1.38915788899152083 \cdot 10^{298}\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 r838565 = x;
        double r838566 = y;
        double r838567 = r838565 * r838566;
        double r838568 = z;
        double r838569 = t;
        double r838570 = r838568 * r838569;
        double r838571 = r838567 - r838570;
        double r838572 = a;
        double r838573 = r838571 / r838572;
        return r838573;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r838574 = x;
        double r838575 = y;
        double r838576 = r838574 * r838575;
        double r838577 = z;
        double r838578 = t;
        double r838579 = r838577 * r838578;
        double r838580 = r838576 - r838579;
        double r838581 = -inf.0;
        bool r838582 = r838580 <= r838581;
        double r838583 = 1.3891578889915208e+298;
        bool r838584 = r838580 <= r838583;
        double r838585 = !r838584;
        bool r838586 = r838582 || r838585;
        double r838587 = a;
        double r838588 = cbrt(r838587);
        double r838589 = r838588 * r838588;
        double r838590 = r838574 / r838589;
        double r838591 = r838575 / r838588;
        double r838592 = r838578 / r838587;
        double r838593 = r838577 * r838592;
        double r838594 = -r838593;
        double r838595 = fma(r838590, r838591, r838594);
        double r838596 = r838578 / r838589;
        double r838597 = r838577 / r838588;
        double r838598 = -r838597;
        double r838599 = r838598 + r838597;
        double r838600 = r838596 * r838599;
        double r838601 = r838595 + r838600;
        double r838602 = 1.0;
        double r838603 = r838602 / r838587;
        double r838604 = r838578 * r838577;
        double r838605 = r838576 - r838604;
        double r838606 = r838603 * r838605;
        double r838607 = r838586 ? r838601 : r838606;
        return r838607;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.5
Target6.1
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 1.3891578889915208e+298 < (- (* x y) (* z t))

    1. Initial program 60.9

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

    3. Applied div-sub60.9

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

    4. Simplified60.9

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

    6. Applied add-cube-cbrt60.9

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

    7. Applied times-frac33.9

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

    8. Applied add-cube-cbrt34.0

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

    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*1.3

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

    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)) < 1.3891578889915208e+298

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t = -\infty \lor \neg \left(x \cdot y - z \cdot t \le 1.38915788899152083 \cdot 10^{298}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -z \cdot \frac{t}{a}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(x \cdot y - t \cdot z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))