Average Error: 7.5 → 0.4
Time: 19.6s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.378113244966585428960316749794424299186 \cdot 10^{-37} \lor \neg \left(z \le 180493624317508243685376\right):\\ \;\;\;\;\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.378113244966585428960316749794424299186 \cdot 10^{-37} \lor \neg \left(z \le 180493624317508243685376\right):\\
\;\;\;\;\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{z \cdot x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r359686 = x;
        double r359687 = cosh(r359686);
        double r359688 = y;
        double r359689 = r359688 / r359686;
        double r359690 = r359687 * r359689;
        double r359691 = z;
        double r359692 = r359690 / r359691;
        return r359692;
}
↓
double f(double x, double y, double z) {
        double r359693 = z;
        double r359694 = -3.3781132449665854e-37;
        bool r359695 = r359693 <= r359694;
        double r359696 = 1.8049362431750824e+23;
        bool r359697 = r359693 <= r359696;
        double r359698 = !r359697;
        bool r359699 = r359695 || r359698;
        double r359700 = y;
        double r359701 = 0.5;
        double r359702 = x;
        double r359703 = exp(r359702);
        double r359704 = -r359702;
        double r359705 = exp(r359704);
        double r359706 = r359703 + r359705;
        double r359707 = r359701 * r359706;
        double r359708 = r359700 * r359707;
        double r359709 = r359693 * r359702;
        double r359710 = r359708 / r359709;
        double r359711 = r359700 / r359693;
        double r359712 = r359707 / r359702;
        double r359713 = r359711 * r359712;
        double r359714 = r359699 ? r359710 : r359713;
        return r359714;
}

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

Original7.5
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.3781132449665854e-37 or 1.8049362431750824e+23 < z

    1. Initial program 11.3

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

    if -3.3781132449665854e-37 < z < 1.8049362431750824e+23

    1. Initial program 0.4

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]

    2. Taylor expanded around inf 19.6

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

    3. Simplified19.6

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

    5. Applied times-frac0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.378113244966585428960316749794424299186 \cdot 10^{-37} \lor \neg \left(z \le 180493624317508243685376\right):\\ \;\;\;\;\frac{y \cdot \left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))