Average Error: 7.1 → 2.2
Time: 7.4s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1067161694993487.5:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 2713.6534281564946:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{1}{\frac{y - t}{2}}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1067161694993487.5:\\
\;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\

\mathbf{elif}\;z \le 2713.6534281564946:\\
\;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{z} \cdot \frac{1}{\frac{y - t}{2}}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r1262573 = x;
        double r1262574 = 2.0;
        double r1262575 = r1262573 * r1262574;
        double r1262576 = y;
        double r1262577 = z;
        double r1262578 = r1262576 * r1262577;
        double r1262579 = t;
        double r1262580 = r1262579 * r1262577;
        double r1262581 = r1262578 - r1262580;
        double r1262582 = r1262575 / r1262581;
        return r1262582;
}
↓
double f(double x, double y, double z, double t) {
        double r1262583 = z;
        double r1262584 = -1067161694993487.5;
        bool r1262585 = r1262583 <= r1262584;
        double r1262586 = x;
        double r1262587 = r1262586 / r1262583;
        double r1262588 = y;
        double r1262589 = t;
        double r1262590 = r1262588 - r1262589;
        double r1262591 = 2.0;
        double r1262592 = r1262590 / r1262591;
        double r1262593 = r1262587 / r1262592;
        double r1262594 = 2713.6534281564946;
        bool r1262595 = r1262583 <= r1262594;
        double r1262596 = r1262583 * r1262590;
        double r1262597 = r1262596 / r1262591;
        double r1262598 = r1262586 / r1262597;
        double r1262599 = 1.0;
        double r1262600 = r1262599 / r1262592;
        double r1262601 = r1262587 * r1262600;
        double r1262602 = r1262595 ? r1262598 : r1262601;
        double r1262603 = r1262585 ? r1262593 : r1262602;
        return r1262603;
}

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

Original7.1
Target2.3
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1067161694993487.5

    1. Initial program 12.0

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

    2. Simplified9.5

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

    4. Applied *-un-lft-identity9.5

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

    5. Applied times-frac9.5

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

    6. Applied associate-/r*2.1

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

    7. Simplified2.1

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

    if -1067161694993487.5 < z < 2713.6534281564946

    1. Initial program 2.4

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

    2. Simplified2.4

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

    if 2713.6534281564946 < z

    1. Initial program 10.7

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

    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 associate-/r*1.9

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

    7. Simplified1.9

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

    9. Applied div-inv2.0

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{1}{\frac{y - t}{2}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1067161694993487.5:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y - t}{2}}\\ \mathbf{elif}\;z \le 2713.6534281564946:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(y - t\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{1}{\frac{y - t}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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))))