Average Error: 6.6 → 5.0
Time: 12.2s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 1.252162427360217230393195741755373415123 \cdot 10^{-291} \lor \neg \left(t \le 1.175763829828214372931466074322922795802 \cdot 10^{77}\right):\\ \;\;\;\;\frac{x \cdot 2}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}} \cdot \frac{\frac{1}{z}}{\sqrt[3]{y - t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;t \le 1.252162427360217230393195741755373415123 \cdot 10^{-291} \lor \neg \left(t \le 1.175763829828214372931466074322922795802 \cdot 10^{77}\right):\\
\;\;\;\;\frac{x \cdot 2}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}} \cdot \frac{\frac{1}{z}}{\sqrt[3]{y - t}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r351394 = x;
        double r351395 = 2.0;
        double r351396 = r351394 * r351395;
        double r351397 = y;
        double r351398 = z;
        double r351399 = r351397 * r351398;
        double r351400 = t;
        double r351401 = r351400 * r351398;
        double r351402 = r351399 - r351401;
        double r351403 = r351396 / r351402;
        return r351403;
}
↓
double f(double x, double y, double z, double t) {
        double r351404 = t;
        double r351405 = 1.2521624273602172e-291;
        bool r351406 = r351404 <= r351405;
        double r351407 = 1.1757638298282144e+77;
        bool r351408 = r351404 <= r351407;
        double r351409 = !r351408;
        bool r351410 = r351406 || r351409;
        double r351411 = x;
        double r351412 = 2.0;
        double r351413 = r351411 * r351412;
        double r351414 = y;
        double r351415 = r351414 - r351404;
        double r351416 = cbrt(r351415);
        double r351417 = r351416 * r351416;
        double r351418 = r351413 / r351417;
        double r351419 = 1.0;
        double r351420 = z;
        double r351421 = r351419 / r351420;
        double r351422 = r351421 / r351416;
        double r351423 = r351418 * r351422;
        double r351424 = r351413 / r351420;
        double r351425 = r351424 / r351415;
        double r351426 = r351410 ? r351423 : r351425;
        return r351426;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.6
Target2.0
Herbie5.0
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061113708240820439530037456 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.045027827330126029709547581125571222799 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < 1.2521624273602172e-291 or 1.1757638298282144e+77 < t

    1. Initial program 7.5

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified6.2

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}}\]
    3. Using strategy rm

    4. Applied associate-/r*6.3

      \[\leadsto \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt7.0

      \[\leadsto \frac{\frac{x \cdot 2}{z}}{\color{blue}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}}\]

    7. Applied div-inv7.0

      \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right) \cdot \frac{1}{z}}}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}\]

    8. Applied times-frac5.0

      \[\leadsto \color{blue}{\frac{x \cdot 2}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}} \cdot \frac{\frac{1}{z}}{\sqrt[3]{y - t}}}\]

    if 1.2521624273602172e-291 < t < 1.1757638298282144e+77

    1. Initial program 4.3

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

    2. Simplified4.3

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}}\]
    3. Using strategy rm

    4. Applied associate-/r*5.0

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

  4. Final simplification5.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.252162427360217230393195741755373415123 \cdot 10^{-291} \lor \neg \left(t \le 1.175763829828214372931466074322922795802 \cdot 10^{77}\right):\\ \;\;\;\;\frac{x \cdot 2}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}} \cdot \frac{\frac{1}{z}}{\sqrt[3]{y - t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2) (if (< (/ (* x 2) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2)))

  (/ (* x 2) (- (* y z) (* t z))))