Average Error: 6.8 → 2.1
Time: 10.1s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -49355.0736941491268:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 1.74137454924600484 \cdot 10^{79}:\\ \;\;\;\;\frac{x}{\frac{1 \cdot \left(z \cdot \left(y - t\right)\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{x}}} \cdot \frac{\sqrt[3]{x}}{\frac{y - t}{2}}\\ \end{array}\]
\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{1 \cdot \left(z \cdot \left(y - t\right)\right)}{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 r751650 = x;
        double r751651 = 2.0;
        double r751652 = r751650 * r751651;
        double r751653 = y;
        double r751654 = z;
        double r751655 = r751653 * r751654;
        double r751656 = t;
        double r751657 = r751656 * r751654;
        double r751658 = r751655 - r751657;
        double r751659 = r751652 / r751658;
        return r751659;
}
↓
double f(double x, double y, double z, double t) {
        double r751660 = z;
        double r751661 = -49355.07369414913;
        bool r751662 = r751660 <= r751661;
        double r751663 = x;
        double r751664 = r751663 / r751660;
        double r751665 = y;
        double r751666 = t;
        double r751667 = r751665 - r751666;
        double r751668 = 2.0;
        double r751669 = r751667 / r751668;
        double r751670 = r751664 / r751669;
        double r751671 = 1.741374549246005e+79;
        bool r751672 = r751660 <= r751671;
        double r751673 = 1.0;
        double r751674 = r751660 * r751667;
        double r751675 = r751673 * r751674;
        double r751676 = r751675 / r751668;
        double r751677 = r751663 / r751676;
        double r751678 = cbrt(r751663);
        double r751679 = r751660 / r751678;
        double r751680 = r751678 / r751679;
        double r751681 = r751678 / r751669;
        double r751682 = r751680 * r751681;
        double r751683 = r751672 ? r751677 : r751682;
        double r751684 = r751662 ? r751670 : r751683;
        return r751684;
}

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 *-un-lft-identity2.4

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

    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

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(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))))