Average Error: 7.0 → 1.9
Time: 1.5m
Precision: 64
\[\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.744943014796802 \cdot 10^{-186}:\\ \;\;\;\;\frac{\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt{2.0}}{\sqrt[3]{y - t}}}{\sqrt[3]{y - t}} \cdot \frac{\frac{\sqrt{2.0}}{\frac{z}{\sqrt[3]{x}}}}{\sqrt[3]{y - t}}\\ \mathbf{elif}\;z \le 1.8116285421753165 \cdot 10^{-82}:\\ \;\;\;\;\frac{2.0 \cdot x}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt{2.0}}{\sqrt[3]{y - t}}}{\sqrt[3]{y - t}} \cdot \frac{\frac{\sqrt{2.0}}{\frac{z}{\sqrt[3]{x}}}}{\sqrt[3]{y - t}}\\ \end{array}\]
\frac{x \cdot 2.0}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.744943014796802 \cdot 10^{-186}:\\
\;\;\;\;\frac{\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt{2.0}}{\sqrt[3]{y - t}}}{\sqrt[3]{y - t}} \cdot \frac{\frac{\sqrt{2.0}}{\frac{z}{\sqrt[3]{x}}}}{\sqrt[3]{y - t}}\\

\mathbf{elif}\;z \le 1.8116285421753165 \cdot 10^{-82}:\\
\;\;\;\;\frac{2.0 \cdot x}{\left(y - t\right) \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt{2.0}}{\sqrt[3]{y - t}}}{\sqrt[3]{y - t}} \cdot \frac{\frac{\sqrt{2.0}}{\frac{z}{\sqrt[3]{x}}}}{\sqrt[3]{y - t}}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r21300847 = x;
        double r21300848 = 2.0;
        double r21300849 = r21300847 * r21300848;
        double r21300850 = y;
        double r21300851 = z;
        double r21300852 = r21300850 * r21300851;
        double r21300853 = t;
        double r21300854 = r21300853 * r21300851;
        double r21300855 = r21300852 - r21300854;
        double r21300856 = r21300849 / r21300855;
        return r21300856;
}
↓
double f(double x, double y, double z, double t) {
        double r21300857 = z;
        double r21300858 = -5.744943014796802e-186;
        bool r21300859 = r21300857 <= r21300858;
        double r21300860 = x;
        double r21300861 = cbrt(r21300860);
        double r21300862 = r21300861 * r21300861;
        double r21300863 = 2.0;
        double r21300864 = sqrt(r21300863);
        double r21300865 = r21300862 * r21300864;
        double r21300866 = y;
        double r21300867 = t;
        double r21300868 = r21300866 - r21300867;
        double r21300869 = cbrt(r21300868);
        double r21300870 = r21300865 / r21300869;
        double r21300871 = r21300870 / r21300869;
        double r21300872 = r21300857 / r21300861;
        double r21300873 = r21300864 / r21300872;
        double r21300874 = r21300873 / r21300869;
        double r21300875 = r21300871 * r21300874;
        double r21300876 = 1.8116285421753165e-82;
        bool r21300877 = r21300857 <= r21300876;
        double r21300878 = r21300863 * r21300860;
        double r21300879 = r21300868 * r21300857;
        double r21300880 = r21300878 / r21300879;
        double r21300881 = r21300877 ? r21300880 : r21300875;
        double r21300882 = r21300859 ? r21300875 : r21300881;
        return r21300882;
}

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

Original7.0
Target2.0
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2.0\\ \mathbf{elif}\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z} \lt 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2.0}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2.0\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.744943014796802e-186 or 1.8116285421753165e-82 < z

    1. Initial program 8.2

      \[\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\]

    2. Simplified3.4

      \[\leadsto \color{blue}{\frac{\frac{2.0}{\frac{z}{x}}}{y - t}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt4.0

      \[\leadsto \frac{\frac{2.0}{\frac{z}{x}}}{\color{blue}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}}\]

    5. Applied add-cube-cbrt4.2

      \[\leadsto \frac{\frac{2.0}{\frac{z}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}\]

    6. Applied *-un-lft-identity4.2

      \[\leadsto \frac{\frac{2.0}{\frac{\color{blue}{1 \cdot z}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}\]

    7. Applied times-frac4.2

      \[\leadsto \frac{\frac{2.0}{\color{blue}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{z}{\sqrt[3]{x}}}}}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}\]

    8. Applied add-sqr-sqrt4.2

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{2.0} \cdot \sqrt{2.0}}}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{z}{\sqrt[3]{x}}}}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}\]

    9. Applied times-frac4.0

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{2.0}}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\sqrt{2.0}}{\frac{z}{\sqrt[3]{x}}}}}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}\]

    10. Applied times-frac1.4

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2.0}}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}} \cdot \frac{\frac{\sqrt{2.0}}{\frac{z}{\sqrt[3]{x}}}}{\sqrt[3]{y - t}}}\]

    11. Simplified1.4

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{2.0} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}{\sqrt[3]{y - t}}}{\sqrt[3]{y - t}}} \cdot \frac{\frac{\sqrt{2.0}}{\frac{z}{\sqrt[3]{x}}}}{\sqrt[3]{y - t}}\]

    if -5.744943014796802e-186 < z < 1.8116285421753165e-82

    1. Initial program 3.3

      \[\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\]

    2. Simplified15.2

      \[\leadsto \color{blue}{\frac{\frac{2.0}{\frac{z}{x}}}{y - t}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity15.2

      \[\leadsto \frac{\frac{2.0}{\frac{z}{x}}}{\color{blue}{1 \cdot \left(y - t\right)}}\]

    5. Applied associate-/r/15.1

      \[\leadsto \frac{\color{blue}{\frac{2.0}{z} \cdot x}}{1 \cdot \left(y - t\right)}\]

    6. Applied times-frac13.1

      \[\leadsto \color{blue}{\frac{\frac{2.0}{z}}{1} \cdot \frac{x}{y - t}}\]

    7. Simplified13.1

      \[\leadsto \color{blue}{\frac{2.0}{z}} \cdot \frac{x}{y - t}\]
    8. Using strategy rm

    9. Applied frac-times3.3

      \[\leadsto \color{blue}{\frac{2.0 \cdot x}{z \cdot \left(y - t\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.744943014796802 \cdot 10^{-186}:\\ \;\;\;\;\frac{\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt{2.0}}{\sqrt[3]{y - t}}}{\sqrt[3]{y - t}} \cdot \frac{\frac{\sqrt{2.0}}{\frac{z}{\sqrt[3]{x}}}}{\sqrt[3]{y - t}}\\ \mathbf{elif}\;z \le 1.8116285421753165 \cdot 10^{-82}:\\ \;\;\;\;\frac{2.0 \cdot x}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt{2.0}}{\sqrt[3]{y - t}}}{\sqrt[3]{y - t}} \cdot \frac{\frac{\sqrt{2.0}}{\frac{z}{\sqrt[3]{x}}}}{\sqrt[3]{y - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +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))))