Average Error: 6.8 → 3.1
Time: 3.2s
Precision: 64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.2537193603035936 \cdot 10^{90}:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{\frac{y - t}{2}}{x}}\\ \mathbf{elif}\;z \le -2.1648081773488171 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{\frac{\frac{z \cdot \left(y - t\right)}{2}}{x}}\\ \mathbf{elif}\;z \le 2.3904605014014527 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{\frac{y - t}{2}}{x}}\\ \mathbf{elif}\;z \le 1.827128106419149 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{2}{z \cdot \left(y - t\right)}}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{\frac{y - t}{2}}{x}}\\ \end{array}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.2537193603035936 \cdot 10^{90}:\\
\;\;\;\;\frac{\frac{1}{z}}{\frac{\frac{y - t}{2}}{x}}\\

\mathbf{elif}\;z \le -2.1648081773488171 \cdot 10^{-305}:\\
\;\;\;\;\frac{1}{\frac{\frac{z \cdot \left(y - t\right)}{2}}{x}}\\

\mathbf{elif}\;z \le 2.3904605014014527 \cdot 10^{-230}:\\
\;\;\;\;\frac{\frac{1}{z}}{\frac{\frac{y - t}{2}}{x}}\\

\mathbf{elif}\;z \le 1.827128106419149 \cdot 10^{-51}:\\
\;\;\;\;\frac{\frac{2}{z \cdot \left(y - t\right)}}{\frac{1}{x}}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r559527 = x;
        double r559528 = 2.0;
        double r559529 = r559527 * r559528;
        double r559530 = y;
        double r559531 = z;
        double r559532 = r559530 * r559531;
        double r559533 = t;
        double r559534 = r559533 * r559531;
        double r559535 = r559532 - r559534;
        double r559536 = r559529 / r559535;
        return r559536;
}
↓
double f(double x, double y, double z, double t) {
        double r559537 = z;
        double r559538 = -3.2537193603035936e+90;
        bool r559539 = r559537 <= r559538;
        double r559540 = 1.0;
        double r559541 = r559540 / r559537;
        double r559542 = y;
        double r559543 = t;
        double r559544 = r559542 - r559543;
        double r559545 = 2.0;
        double r559546 = r559544 / r559545;
        double r559547 = x;
        double r559548 = r559546 / r559547;
        double r559549 = r559541 / r559548;
        double r559550 = -2.164808177348817e-305;
        bool r559551 = r559537 <= r559550;
        double r559552 = r559537 * r559544;
        double r559553 = r559552 / r559545;
        double r559554 = r559553 / r559547;
        double r559555 = r559540 / r559554;
        double r559556 = 2.3904605014014527e-230;
        bool r559557 = r559537 <= r559556;
        double r559558 = 1.827128106419149e-51;
        bool r559559 = r559537 <= r559558;
        double r559560 = r559545 / r559552;
        double r559561 = r559540 / r559547;
        double r559562 = r559560 / r559561;
        double r559563 = r559559 ? r559562 : r559549;
        double r559564 = r559557 ? r559549 : r559563;
        double r559565 = r559551 ? r559555 : r559564;
        double r559566 = r559539 ? r559549 : r559565;
        return r559566;
}

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
Target2.1
Herbie3.1
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -3.2537193603035936e+90 or -2.164808177348817e-305 < z < 2.3904605014014527e-230 or 1.827128106419149e-51 < z

    1. Initial program 10.2

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

    2. Simplified8.2

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

    4. Applied clear-num8.5

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

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

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

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

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

    8. Applied times-frac8.5

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

    9. Applied times-frac3.8

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

    10. Applied associate-/r*3.4

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

    11. Simplified3.4

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

    if -3.2537193603035936e+90 < z < -2.164808177348817e-305

    1. Initial program 2.8

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

    2. Simplified2.7

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

    4. Applied clear-num3.1

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

    if 2.3904605014014527e-230 < z < 1.827128106419149e-51

    1. Initial program 1.8

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

    2. Simplified1.7

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

    4. Applied clear-num2.0

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

    6. Applied div-inv2.0

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

    7. Applied associate-/r*2.2

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

    8. Simplified2.2

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.2537193603035936 \cdot 10^{90}:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{\frac{y - t}{2}}{x}}\\ \mathbf{elif}\;z \le -2.1648081773488171 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{\frac{\frac{z \cdot \left(y - t\right)}{2}}{x}}\\ \mathbf{elif}\;z \le 2.3904605014014527 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{\frac{y - t}{2}}{x}}\\ \mathbf{elif}\;z \le 1.827128106419149 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{2}{z \cdot \left(y - t\right)}}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{\frac{y - t}{2}}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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))))