Average Error: 6.8 → 2.1
Time: 10.0s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -49355.0736941491268:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 1.74137454924600484 \cdot 10^{79}:\\ \;\;\;\;\frac{x}{\frac{{\left(z \cdot \left(y - t\right)\right)}^{1}}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -49355.0736941491268:\\
\;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\

\mathbf{elif}\;z \le 1.74137454924600484 \cdot 10^{79}:\\
\;\;\;\;\frac{x}{\frac{{\left(z \cdot \left(y - t\right)\right)}^{1}}{2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r550826 = x;
        double r550827 = 2.0;
        double r550828 = r550826 * r550827;
        double r550829 = y;
        double r550830 = z;
        double r550831 = r550829 * r550830;
        double r550832 = t;
        double r550833 = r550832 * r550830;
        double r550834 = r550831 - r550833;
        double r550835 = r550828 / r550834;
        return r550835;
}
↓
double f(double x, double y, double z, double t) {
        double r550836 = z;
        double r550837 = -49355.07369414913;
        bool r550838 = r550836 <= r550837;
        double r550839 = x;
        double r550840 = r550839 / r550836;
        double r550841 = y;
        double r550842 = t;
        double r550843 = r550841 - r550842;
        double r550844 = 2.0;
        double r550845 = r550843 / r550844;
        double r550846 = r550840 / r550845;
        double r550847 = 1.741374549246005e+79;
        bool r550848 = r550836 <= r550847;
        double r550849 = r550836 * r550843;
        double r550850 = 1.0;
        double r550851 = pow(r550849, r550850);
        double r550852 = r550851 / r550844;
        double r550853 = r550839 / r550852;
        double r550854 = cbrt(r550839);
        double r550855 = r550836 / r550854;
        double r550856 = r550854 / r550855;
        double r550857 = r550854 / r550845;
        double r550858 = r550856 * r550857;
        double r550859 = r550848 ? r550853 : r550858;
        double r550860 = r550838 ? r550846 : r550859;
        return r550860;
}

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

Original6.8
Target2.2
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -49355.07369414913

    1. Initial program 11.2

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

    2. Simplified9.2

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

    4. Applied *-un-lft-identity9.2

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

    5. Applied times-frac9.2

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

    6. Applied associate-/r*1.7

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

    7. Simplified1.7

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

    if -49355.07369414913 < z < 1.741374549246005e+79

    1. Initial program 2.4

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

    2. Simplified2.4

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

    4. Applied pow12.4

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

    5. Applied pow12.4

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

    6. Applied pow-prod-down2.4

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

    if 1.741374549246005e+79 < z

    1. Initial program 12.7

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

    2. Simplified10.3

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

    4. Applied *-un-lft-identity10.3

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

    5. Applied times-frac10.2

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

    6. Applied add-cube-cbrt10.6

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

    7. Applied times-frac1.9

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

    8. Simplified1.9

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -49355.0736941491268:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 1.74137454924600484 \cdot 10^{79}:\\ \;\;\;\;\frac{x}{\frac{{\left(z \cdot \left(y - t\right)\right)}^{1}}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))) 1.0450278273301259e-269) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2)))

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