Average Error: 7.7 → 0.5
Time: 15.4s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7333611518295001906581357141952796033024:\\ \;\;\;\;\cosh x \cdot \left(\frac{\frac{1}{z}}{x} \cdot y\right)\\ \mathbf{elif}\;z \le 3.874474762317707999930582517660829469561 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{y}{\frac{x}{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7333611518295001906581357141952796033024:\\
\;\;\;\;\cosh x \cdot \left(\frac{\frac{1}{z}}{x} \cdot y\right)\\

\mathbf{elif}\;z \le 3.874474762317707999930582517660829469561 \cdot 10^{-25}:\\
\;\;\;\;\frac{\frac{y}{\frac{x}{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r377780 = x;
        double r377781 = cosh(r377780);
        double r377782 = y;
        double r377783 = r377782 / r377780;
        double r377784 = r377781 * r377783;
        double r377785 = z;
        double r377786 = r377784 / r377785;
        return r377786;
}
↓
double f(double x, double y, double z) {
        double r377787 = z;
        double r377788 = -7.333611518295002e+39;
        bool r377789 = r377787 <= r377788;
        double r377790 = x;
        double r377791 = cosh(r377790);
        double r377792 = 1.0;
        double r377793 = r377792 / r377787;
        double r377794 = r377793 / r377790;
        double r377795 = y;
        double r377796 = r377794 * r377795;
        double r377797 = r377791 * r377796;
        double r377798 = 3.874474762317708e-25;
        bool r377799 = r377787 <= r377798;
        double r377800 = 0.5;
        double r377801 = exp(r377790);
        double r377802 = r377800 / r377801;
        double r377803 = fma(r377800, r377801, r377802);
        double r377804 = r377790 / r377803;
        double r377805 = r377795 / r377804;
        double r377806 = r377805 / r377787;
        double r377807 = r377791 * r377795;
        double r377808 = r377790 * r377787;
        double r377809 = r377807 / r377808;
        double r377810 = r377799 ? r377806 : r377809;
        double r377811 = r377789 ? r377797 : r377810;
        return r377811;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.7
Target0.4
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 3 regimes

  2. if z < -7.333611518295002e+39

    1. Initial program 12.7

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

    3. Applied associate-/l*13.1

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

    4. Simplified12.8

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

    6. Applied div-inv12.8

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

    7. Simplified11.7

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

    9. Applied *-un-lft-identity11.7

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

    10. Applied div-inv11.8

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

    11. Applied times-frac0.4

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

    12. Simplified0.4

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

    if -7.333611518295002e+39 < z < 3.874474762317708e-25

    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. Simplified0.5

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

    5. Taylor expanded around inf 19.7

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

    6. Simplified0.6

      \[\leadsto \color{blue}{\frac{\frac{y}{\frac{x}{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}{z}}\]

    if 3.874474762317708e-25 < z

    1. Initial program 11.3

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

    3. Applied associate-*r/11.3

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

    4. Applied associate-/l/0.3

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7333611518295001906581357141952796033024:\\ \;\;\;\;\cosh x \cdot \left(\frac{\frac{1}{z}}{x} \cdot y\right)\\ \mathbf{elif}\;z \le 3.874474762317707999930582517660829469561 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{y}{\frac{x}{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"

  :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))