Average Error: 7.7 → 0.5
Time: 22.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.025967367497455730113268706012740099348 \cdot 10^{54} \lor \neg \left(z \le 3.298414950520773924444494287766538320739 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{\frac{x}{\frac{1}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.025967367497455730113268706012740099348 \cdot 10^{54} \lor \neg \left(z \le 3.298414950520773924444494287766538320739 \cdot 10^{-6}\right):\\
\;\;\;\;\frac{\cosh x \cdot y}{\frac{x}{\frac{1}{z}}}\\

\mathbf{else}:\\
\;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\

\end{array}
double f(double x, double y, double z) {
        double r324633 = x;
        double r324634 = cosh(r324633);
        double r324635 = y;
        double r324636 = r324635 / r324633;
        double r324637 = r324634 * r324636;
        double r324638 = z;
        double r324639 = r324637 / r324638;
        return r324639;
}
↓
double f(double x, double y, double z) {
        double r324640 = z;
        double r324641 = -1.0259673674974557e+54;
        bool r324642 = r324640 <= r324641;
        double r324643 = 3.298414950520774e-06;
        bool r324644 = r324640 <= r324643;
        double r324645 = !r324644;
        bool r324646 = r324642 || r324645;
        double r324647 = x;
        double r324648 = cosh(r324647);
        double r324649 = y;
        double r324650 = r324648 * r324649;
        double r324651 = 1.0;
        double r324652 = r324651 / r324640;
        double r324653 = r324647 / r324652;
        double r324654 = r324650 / r324653;
        double r324655 = r324649 / r324640;
        double r324656 = r324655 / r324647;
        double r324657 = r324648 * r324656;
        double r324658 = r324646 ? r324654 : r324657;
        return r324658;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.7
Target0.5
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.0259673674974557e+54 or 3.298414950520774e-06 < z

    1. Initial program 12.2

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*12.6

      \[\leadsto \color{blue}{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\]

    4. Simplified10.7

      \[\leadsto \frac{\cosh x}{\color{blue}{\frac{x}{\frac{y}{z}}}}\]
    5. Using strategy rm

    6. Applied div-inv10.8

      \[\leadsto \frac{\cosh x}{\frac{x}{\color{blue}{y \cdot \frac{1}{z}}}}\]

    7. Applied *-un-lft-identity10.8

      \[\leadsto \frac{\cosh x}{\frac{\color{blue}{1 \cdot x}}{y \cdot \frac{1}{z}}}\]

    8. Applied times-frac0.9

      \[\leadsto \frac{\cosh x}{\color{blue}{\frac{1}{y} \cdot \frac{x}{\frac{1}{z}}}}\]

    9. Applied associate-/r*0.5

      \[\leadsto \color{blue}{\frac{\frac{\cosh x}{\frac{1}{y}}}{\frac{x}{\frac{1}{z}}}}\]

    10. Simplified0.5

      \[\leadsto \frac{\color{blue}{\cosh x \cdot y}}{\frac{x}{\frac{1}{z}}}\]

    if -1.0259673674974557e+54 < z < 3.298414950520774e-06

    1. Initial program 0.6

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*0.7

      \[\leadsto \color{blue}{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\]

    4. Simplified0.6

      \[\leadsto \frac{\cosh x}{\color{blue}{\frac{x}{\frac{y}{z}}}}\]
    5. Using strategy rm

    6. Applied div-inv0.6

      \[\leadsto \color{blue}{\cosh x \cdot \frac{1}{\frac{x}{\frac{y}{z}}}}\]

    7. Simplified0.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.025967367497455730113268706012740099348 \cdot 10^{54} \lor \neg \left(z \le 3.298414950520773924444494287766538320739 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{\frac{x}{\frac{1}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))