Average Error: 7.0 → 2.5
Time: 8.5s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot 2 \le -8882633821861867 \lor \neg \left(x \cdot 2 \le 7.5458448607072867 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\frac{x}{\frac{y - t}{2}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot 2 \le -8882633821861867 \lor \neg \left(x \cdot 2 \le 7.5458448607072867 \cdot 10^{-72}\right):\\
\;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\frac{x}{\frac{y - t}{2}}}{z}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r515089 = x;
        double r515090 = 2.0;
        double r515091 = r515089 * r515090;
        double r515092 = y;
        double r515093 = z;
        double r515094 = r515092 * r515093;
        double r515095 = t;
        double r515096 = r515095 * r515093;
        double r515097 = r515094 - r515096;
        double r515098 = r515091 / r515097;
        return r515098;
}
↓
double f(double x, double y, double z, double t) {
        double r515099 = x;
        double r515100 = 2.0;
        double r515101 = r515099 * r515100;
        double r515102 = -8882633821861867.0;
        bool r515103 = r515101 <= r515102;
        double r515104 = 7.545844860707287e-72;
        bool r515105 = r515101 <= r515104;
        double r515106 = !r515105;
        bool r515107 = r515103 || r515106;
        double r515108 = 1.0;
        double r515109 = cbrt(r515108);
        double r515110 = r515109 * r515109;
        double r515111 = r515110 / r515108;
        double r515112 = y;
        double r515113 = t;
        double r515114 = r515112 - r515113;
        double r515115 = r515114 / r515100;
        double r515116 = r515099 / r515115;
        double r515117 = z;
        double r515118 = r515116 / r515117;
        double r515119 = r515111 * r515118;
        double r515120 = r515099 / r515117;
        double r515121 = r515120 / r515115;
        double r515122 = r515107 ? r515119 : r515121;
        return r515122;
}

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
Target2.1
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* x 2.0) < -8882633821861867.0 or 7.545844860707287e-72 < (* x 2.0)

    1. Initial program 10.7

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

    2. Simplified9.8

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

    4. Applied *-un-lft-identity9.8

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

    5. Applied times-frac9.7

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

    6. Applied *-un-lft-identity9.7

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

    7. Applied times-frac3.0

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

    8. Simplified3.0

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

    10. Applied *-un-lft-identity3.0

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

    11. Applied add-cube-cbrt3.0

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

    12. Applied times-frac3.0

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

    13. Applied associate-*l*3.0

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

    14. Simplified2.9

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

    if -8882633821861867.0 < (* x 2.0) < 7.545844860707287e-72

    1. Initial program 3.2

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

    2. Simplified2.0

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

    4. Applied *-un-lft-identity2.0

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

    5. Applied times-frac2.0

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

    6. Applied associate-/r*2.1

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

    7. Simplified2.1

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

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \le -8882633821861867 \lor \neg \left(x \cdot 2 \le 7.5458448607072867 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\frac{x}{\frac{y - t}{2}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))))