Average Error: 7.3 → 0.4
Time: 11.6s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2548401331254628.0:\\ \;\;\;\;\frac{y}{\frac{x \cdot z}{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}}\\ \mathbf{elif}\;z \le 4.164739628396679 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(y, e^{x}, \frac{y}{e^{x}}\right)}{x}}{2}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{x \cdot z}{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2548401331254628.0:\\
\;\;\;\;\frac{y}{\frac{x \cdot z}{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}}\\

\mathbf{elif}\;z \le 4.164739628396679 \cdot 10^{+36}:\\
\;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(y, e^{x}, \frac{y}{e^{x}}\right)}{x}}{2}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{y}{\frac{x \cdot z}{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}}\\

\end{array}
double f(double x, double y, double z) {
        double r8634977 = x;
        double r8634978 = cosh(r8634977);
        double r8634979 = y;
        double r8634980 = r8634979 / r8634977;
        double r8634981 = r8634978 * r8634980;
        double r8634982 = z;
        double r8634983 = r8634981 / r8634982;
        return r8634983;
}
↓
double f(double x, double y, double z) {
        double r8634984 = z;
        double r8634985 = -2548401331254628.0;
        bool r8634986 = r8634984 <= r8634985;
        double r8634987 = y;
        double r8634988 = x;
        double r8634989 = r8634988 * r8634984;
        double r8634990 = exp(r8634988);
        double r8634991 = 0.5;
        double r8634992 = r8634991 / r8634990;
        double r8634993 = fma(r8634990, r8634991, r8634992);
        double r8634994 = r8634989 / r8634993;
        double r8634995 = r8634987 / r8634994;
        double r8634996 = 4.164739628396679e+36;
        bool r8634997 = r8634984 <= r8634996;
        double r8634998 = r8634987 / r8634990;
        double r8634999 = fma(r8634987, r8634990, r8634998);
        double r8635000 = r8634999 / r8634988;
        double r8635001 = 2.0;
        double r8635002 = r8635000 / r8635001;
        double r8635003 = r8635002 / r8634984;
        double r8635004 = r8634997 ? r8635003 : r8634995;
        double r8635005 = r8634986 ? r8634995 : r8635004;
        return r8635005;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.3
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2548401331254628.0 or 4.164739628396679e+36 < z

    1. Initial program 11.6

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

    3. Applied div-inv11.7

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

    4. Applied associate-*r*11.7

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

    5. Taylor expanded around inf 0.3

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

    6. Simplified0.3

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

    if -2548401331254628.0 < z < 4.164739628396679e+36

    1. Initial program 0.6

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

    3. Applied div-inv0.8

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

    4. Applied associate-*r*0.8

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

    6. Applied cosh-def0.8

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

    7. Applied associate-*l/0.8

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

    8. Applied associate-*l/0.8

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

    9. Simplified0.6

      \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(y, e^{x}, \frac{1 \cdot y}{e^{x}}\right)}{x}}}{2}}{z}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2548401331254628.0:\\ \;\;\;\;\frac{y}{\frac{x \cdot z}{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}}\\ \mathbf{elif}\;z \le 4.164739628396679 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(y, e^{x}, \frac{y}{e^{x}}\right)}{x}}{2}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{x \cdot z}{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +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))