Average Error: 7.8 → 5.0
Time: 11.0s
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.84154847940333438 \cdot 10^{268}:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t \cdot z}{a}\\ \mathbf{elif}\;x \cdot y \le 5.0132949514285239 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{\frac{a}{x \cdot y - z \cdot t}}\\ \mathbf{elif}\;x \cdot y \le 2.10280870876133442 \cdot 10^{131}:\\ \;\;\;\;\left(\frac{x \cdot y}{a} - \frac{z}{\sqrt[3]{a}} \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) + \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\left(-\frac{z}{\sqrt[3]{a}}\right) + \frac{z}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a} - \frac{t \cdot z}{a}\\ \end{array}\]
\frac{x \cdot y - z \cdot t}{a}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -1.84154847940333438 \cdot 10^{268}:\\
\;\;\;\;x \cdot \frac{y}{a} - \frac{t \cdot z}{a}\\

\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}:\\
\;\;\;\;x \cdot \frac{y}{a} - \frac{t \cdot z}{a}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r4416 = x;
        double r4417 = y;
        double r4418 = r4416 * r4417;
        double r4419 = z;
        double r4420 = t;
        double r4421 = r4419 * r4420;
        double r4422 = r4418 - r4421;
        double r4423 = a;
        double r4424 = r4422 / r4423;
        return r4424;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r4425 = x;
        double r4426 = y;
        double r4427 = r4425 * r4426;
        double r4428 = -1.8415484794033344e+268;
        bool r4429 = r4427 <= r4428;
        double r4430 = a;
        double r4431 = r4426 / r4430;
        double r4432 = r4425 * r4431;
        double r4433 = t;
        double r4434 = z;
        double r4435 = r4433 * r4434;
        double r4436 = r4435 / r4430;
        double r4437 = r4432 - r4436;
        double r4438 = 5.013294951428524e-62;
        bool r4439 = r4427 <= r4438;
        double r4440 = 1.0;
        double r4441 = r4434 * r4433;
        double r4442 = r4427 - r4441;
        double r4443 = r4430 / r4442;
        double r4444 = r4440 / r4443;
        double r4445 = 2.1028087087613344e+131;
        bool r4446 = r4427 <= r4445;
        double r4447 = r4427 / r4430;
        double r4448 = cbrt(r4430);
        double r4449 = r4434 / r4448;
        double r4450 = r4448 * r4448;
        double r4451 = r4433 / r4450;
        double r4452 = r4449 * r4451;
        double r4453 = r4447 - r4452;
        double r4454 = -r4449;
        double r4455 = r4454 + r4449;
        double r4456 = r4451 * r4455;
        double r4457 = r4453 + r4456;
        double r4458 = r4446 ? r4457 : r4437;
        double r4459 = r4439 ? r4444 : r4458;
        double r4460 = r4429 ? r4437 : r4459;
        return r4460;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified8.0

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

    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

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

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))