Average Error: 6.9 → 2.7
Time: 7.0s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.031032483723507254 \lor \neg \left(x \le 3.6948490856919295 \cdot 10^{-142}\right):\\ \;\;\;\;\frac{1}{z} \cdot \frac{x}{\frac{y - t}{2}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.031032483723507254 \lor \neg \left(x \le 3.6948490856919295 \cdot 10^{-142}\right):\\
\;\;\;\;\frac{1}{z} \cdot \frac{x}{\frac{y - t}{2}}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r562662 = x;
        double r562663 = 2.0;
        double r562664 = r562662 * r562663;
        double r562665 = y;
        double r562666 = z;
        double r562667 = r562665 * r562666;
        double r562668 = t;
        double r562669 = r562668 * r562666;
        double r562670 = r562667 - r562669;
        double r562671 = r562664 / r562670;
        return r562671;
}
↓
double f(double x, double y, double z, double t) {
        double r562672 = x;
        double r562673 = -0.031032483723507254;
        bool r562674 = r562672 <= r562673;
        double r562675 = 3.6948490856919295e-142;
        bool r562676 = r562672 <= r562675;
        double r562677 = !r562676;
        bool r562678 = r562674 || r562677;
        double r562679 = 1.0;
        double r562680 = z;
        double r562681 = r562679 / r562680;
        double r562682 = y;
        double r562683 = t;
        double r562684 = r562682 - r562683;
        double r562685 = 2.0;
        double r562686 = r562684 / r562685;
        double r562687 = r562672 / r562686;
        double r562688 = r562681 * r562687;
        double r562689 = r562685 / r562684;
        double r562690 = r562689 / r562680;
        double r562691 = r562672 * r562690;
        double r562692 = r562678 ? r562688 : r562691;
        return r562692;
}

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

Original6.9
Target2.1
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.031032483723507254 or 3.6948490856919295e-142 < x

    1. Initial program 9.5

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

    2. Simplified8.5

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

    4. Applied *-un-lft-identity8.5

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

    5. Applied times-frac8.5

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

    6. Applied *-un-lft-identity8.5

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

    7. Applied times-frac3.0

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

    8. Simplified3.0

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

    if -0.031032483723507254 < x < 3.6948490856919295e-142

    1. Initial program 3.3

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

    2. Simplified2.0

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

    4. Applied div-inv2.3

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

    5. Simplified2.3

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.031032483723507254 \lor \neg \left(x \le 3.6948490856919295 \cdot 10^{-142}\right):\\ \;\;\;\;\frac{1}{z} \cdot \frac{x}{\frac{y - t}{2}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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))))