Average Error: 7.3 → 0.9
Time: 4.3s
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t = -\infty \lor \neg \left(x \cdot y - z \cdot t \le 6.2962358828876013 \cdot 10^{295}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\frac{z}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a} - \left(t \cdot z\right) \cdot \frac{1}{a}\\ \end{array}\]
\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 6.2962358828876013 \cdot 10^{295}\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{x \cdot y}{a} - \left(t \cdot z\right) \cdot \frac{1}{a}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r876287 = x;
        double r876288 = y;
        double r876289 = r876287 * r876288;
        double r876290 = z;
        double r876291 = t;
        double r876292 = r876290 * r876291;
        double r876293 = r876289 - r876292;
        double r876294 = a;
        double r876295 = r876293 / r876294;
        return r876295;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r876296 = x;
        double r876297 = y;
        double r876298 = r876296 * r876297;
        double r876299 = z;
        double r876300 = t;
        double r876301 = r876299 * r876300;
        double r876302 = r876298 - r876301;
        double r876303 = -inf.0;
        bool r876304 = r876302 <= r876303;
        double r876305 = 6.296235882887601e+295;
        bool r876306 = r876302 <= r876305;
        double r876307 = !r876306;
        bool r876308 = r876304 || r876307;
        double r876309 = a;
        double r876310 = cbrt(r876309);
        double r876311 = r876310 * r876310;
        double r876312 = r876296 / r876311;
        double r876313 = r876297 / r876310;
        double r876314 = r876299 / r876310;
        double r876315 = r876300 / r876311;
        double r876316 = r876314 * r876315;
        double r876317 = -r876316;
        double r876318 = fma(r876312, r876313, r876317);
        double r876319 = -r876314;
        double r876320 = r876319 + r876314;
        double r876321 = r876315 * r876320;
        double r876322 = r876318 + r876321;
        double r876323 = r876298 / r876309;
        double r876324 = r876300 * r876299;
        double r876325 = 1.0;
        double r876326 = r876325 / r876309;
        double r876327 = r876324 * r876326;
        double r876328 = r876323 - r876327;
        double r876329 = r876308 ? r876322 : r876328;
        return r876329;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.3
Target5.6
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;z \lt -2.46868496869954822 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.30983112197837121 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* x y) (* z t)) < -inf.0 or 6.296235882887601e+295 < (- (* x y) (* z t))

    1. Initial program 59.9

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

    3. Applied div-sub59.9

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

    4. Simplified59.9

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

    6. Applied add-cube-cbrt60.0

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

    7. Applied times-frac31.8

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

    8. Applied add-cube-cbrt31.8

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

    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)) < 6.296235882887601e+295

    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 div-inv0.9

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot t = -\infty \lor \neg \left(x \cdot y - z \cdot t \le 6.2962358828876013 \cdot 10^{295}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt[3]{a} \cdot \sqrt[3]{a}}, \frac{y}{\sqrt[3]{a}}, -\frac{z}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a} - \left(t \cdot z\right) \cdot \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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))