Average Error: 6.5 → 2.7
Time: 15.7s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.0322425658280421 \cdot 10^{101}:\\ \;\;\;\;\frac{\frac{\frac{x}{\sqrt[3]{y - t}}}{\sqrt[3]{y - t}} \cdot \frac{2}{z}}{\sqrt[3]{y - t}}\\ \mathbf{elif}\;z \le 1.6365850306645365 \cdot 10^{21}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z + \left(-t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.0322425658280421 \cdot 10^{101}:\\
\;\;\;\;\frac{\frac{\frac{x}{\sqrt[3]{y - t}}}{\sqrt[3]{y - t}} \cdot \frac{2}{z}}{\sqrt[3]{y - t}}\\

\mathbf{elif}\;z \le 1.6365850306645365 \cdot 10^{21}:\\
\;\;\;\;\frac{x \cdot 2}{y \cdot z + \left(-t\right) \cdot z}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r1281423 = x;
        double r1281424 = 2.0;
        double r1281425 = r1281423 * r1281424;
        double r1281426 = y;
        double r1281427 = z;
        double r1281428 = r1281426 * r1281427;
        double r1281429 = t;
        double r1281430 = r1281429 * r1281427;
        double r1281431 = r1281428 - r1281430;
        double r1281432 = r1281425 / r1281431;
        return r1281432;
}
↓
double f(double x, double y, double z, double t) {
        double r1281433 = z;
        double r1281434 = -2.032242565828042e+101;
        bool r1281435 = r1281433 <= r1281434;
        double r1281436 = x;
        double r1281437 = y;
        double r1281438 = t;
        double r1281439 = r1281437 - r1281438;
        double r1281440 = cbrt(r1281439);
        double r1281441 = r1281436 / r1281440;
        double r1281442 = r1281441 / r1281440;
        double r1281443 = 2.0;
        double r1281444 = r1281443 / r1281433;
        double r1281445 = r1281442 * r1281444;
        double r1281446 = r1281445 / r1281440;
        double r1281447 = 1.6365850306645365e+21;
        bool r1281448 = r1281433 <= r1281447;
        double r1281449 = r1281436 * r1281443;
        double r1281450 = r1281437 * r1281433;
        double r1281451 = -r1281438;
        double r1281452 = r1281451 * r1281433;
        double r1281453 = r1281450 + r1281452;
        double r1281454 = r1281449 / r1281453;
        double r1281455 = r1281449 / r1281433;
        double r1281456 = r1281455 / r1281439;
        double r1281457 = r1281448 ? r1281454 : r1281456;
        double r1281458 = r1281435 ? r1281446 : r1281457;
        return r1281458;
}

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

Original6.5
Target2.3
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.032242565828042e+101

    1. Initial program 11.9

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

    2. Simplified9.3

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

    4. Applied associate-/r*2.3

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

    6. Applied add-cube-cbrt2.9

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

    7. Applied *-un-lft-identity2.9

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

    8. Applied times-frac2.9

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

    9. Applied times-frac4.3

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

    10. Simplified4.3

      \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}} \cdot \frac{\frac{2}{z}}{\sqrt[3]{y - t}}\]
    11. Using strategy rm

    12. Applied associate-*r/3.1

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

    13. Simplified3.1

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

    if -2.032242565828042e+101 < z < 1.6365850306645365e+21

    1. Initial program 2.8

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

    2. Simplified2.7

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

    4. Applied sub-neg2.7

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

    5. Applied distribute-lft-in2.8

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

    6. Simplified2.8

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

    7. Simplified2.8

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

    if 1.6365850306645365e+21 < z

    1. Initial program 11.2

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

    2. Simplified8.6

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

    4. Applied associate-/r*2.0

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.0322425658280421 \cdot 10^{101}:\\ \;\;\;\;\frac{\frac{\frac{x}{\sqrt[3]{y - t}}}{\sqrt[3]{y - t}} \cdot \frac{2}{z}}{\sqrt[3]{y - t}}\\ \mathbf{elif}\;z \le 1.6365850306645365 \cdot 10^{21}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z + \left(-t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"

  :herbie-target
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))