Average Error: 6.5 → 3.1
Time: 13.5s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.5348954012425702 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{y - t}}{z}\\ \mathbf{elif}\;z \le 3.0801408755089634 \cdot 10^{-122}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z + \left(-t \cdot z\right)}\\ \mathbf{elif}\;z \le 9.9894340906327121 \cdot 10^{98}:\\ \;\;\;\;\frac{x \cdot 2}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}} \cdot \frac{\frac{1}{z}}{\sqrt[3]{y - t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.5348954012425702 \cdot 10^{-168}:\\
\;\;\;\;\frac{\frac{x \cdot 2}{y - t}}{z}\\

\mathbf{elif}\;z \le 3.0801408755089634 \cdot 10^{-122}:\\
\;\;\;\;\frac{x \cdot 2}{y \cdot z + \left(-t \cdot z\right)}\\

\mathbf{elif}\;z \le 9.9894340906327121 \cdot 10^{98}:\\
\;\;\;\;\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 r379811 = x;
        double r379812 = 2.0;
        double r379813 = r379811 * r379812;
        double r379814 = y;
        double r379815 = z;
        double r379816 = r379814 * r379815;
        double r379817 = t;
        double r379818 = r379817 * r379815;
        double r379819 = r379816 - r379818;
        double r379820 = r379813 / r379819;
        return r379820;
}
↓
double f(double x, double y, double z, double t) {
        double r379821 = z;
        double r379822 = -5.53489540124257e-168;
        bool r379823 = r379821 <= r379822;
        double r379824 = x;
        double r379825 = 2.0;
        double r379826 = r379824 * r379825;
        double r379827 = y;
        double r379828 = t;
        double r379829 = r379827 - r379828;
        double r379830 = r379826 / r379829;
        double r379831 = r379830 / r379821;
        double r379832 = 3.0801408755089634e-122;
        bool r379833 = r379821 <= r379832;
        double r379834 = r379827 * r379821;
        double r379835 = r379828 * r379821;
        double r379836 = -r379835;
        double r379837 = r379834 + r379836;
        double r379838 = r379826 / r379837;
        double r379839 = 9.989434090632712e+98;
        bool r379840 = r379821 <= r379839;
        double r379841 = cbrt(r379829);
        double r379842 = r379841 * r379841;
        double r379843 = r379826 / r379842;
        double r379844 = 1.0;
        double r379845 = r379844 / r379821;
        double r379846 = r379845 / r379841;
        double r379847 = r379843 * r379846;
        double r379848 = r379826 / r379821;
        double r379849 = r379848 / r379829;
        double r379850 = r379840 ? r379847 : r379849;
        double r379851 = r379833 ? r379838 : r379850;
        double r379852 = r379823 ? r379831 : r379851;
        return r379852;
}

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
Herbie3.1
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if z < -5.53489540124257e-168

    1. Initial program 6.6

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

    2. Simplified5.3

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

    4. Applied associate-/r*3.7

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

    6. Applied div-inv3.8

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z} \cdot \frac{1}{y - t}}\]
    7. Using strategy rm

    8. Applied associate-*l/3.6

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

    9. Simplified3.5

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

    if -5.53489540124257e-168 < z < 3.0801408755089634e-122

    1. Initial program 4.2

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

    2. Simplified4.2

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

    4. Applied sub-neg4.2

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

    5. Applied distribute-rgt-in4.2

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

    6. Simplified4.2

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

    if 3.0801408755089634e-122 < z < 9.989434090632712e+98

    1. Initial program 2.3

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

    2. Simplified2.3

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

    4. Applied associate-/r*3.3

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

    6. Applied add-cube-cbrt4.1

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

    7. Applied div-inv4.1

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

    8. Applied times-frac1.7

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

    if 9.989434090632712e+98 < z

    1. Initial program 13.5

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

    2. Simplified9.9

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

    4. Applied associate-/r*2.2

      \[\leadsto \color{blue}{\frac{\frac{x \cdot 2}{z}}{y - t}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.5348954012425702 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{y - t}}{z}\\ \mathbf{elif}\;z \le 3.0801408755089634 \cdot 10^{-122}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z + \left(-t \cdot z\right)}\\ \mathbf{elif}\;z \le 9.9894340906327121 \cdot 10^{98}:\\ \;\;\;\;\frac{x \cdot 2}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}} \cdot \frac{\frac{1}{z}}{\sqrt[3]{y - t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(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))))