Average Error: 7.7 → 0.5
Time: 12.1s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -2132121890186478992287162732707315712:\\ \;\;\;\;\frac{1}{\frac{2 \cdot x}{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 2.012757038928455801214359719345400453991 \cdot 10^{-70}:\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -2132121890186478992287162732707315712:\\
\;\;\;\;\frac{1}{\frac{2 \cdot x}{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}}\\

\mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 2.012757038928455801214359719345400453991 \cdot 10^{-70}:\\
\;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}\\

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

\end{array}
double f(double x, double y, double z) {
        double r397821 = x;
        double r397822 = cosh(r397821);
        double r397823 = y;
        double r397824 = r397823 / r397821;
        double r397825 = r397822 * r397824;
        double r397826 = z;
        double r397827 = r397825 / r397826;
        return r397827;
}
↓
double f(double x, double y, double z) {
        double r397828 = x;
        double r397829 = cosh(r397828);
        double r397830 = y;
        double r397831 = r397830 / r397828;
        double r397832 = r397829 * r397831;
        double r397833 = z;
        double r397834 = r397832 / r397833;
        double r397835 = -2.132121890186479e+36;
        bool r397836 = r397834 <= r397835;
        double r397837 = 1.0;
        double r397838 = 2.0;
        double r397839 = r397838 * r397828;
        double r397840 = exp(r397828);
        double r397841 = -r397828;
        double r397842 = exp(r397841);
        double r397843 = r397840 + r397842;
        double r397844 = r397843 * r397830;
        double r397845 = r397844 / r397833;
        double r397846 = r397839 / r397845;
        double r397847 = r397837 / r397846;
        double r397848 = 2.0127570389284558e-70;
        bool r397849 = r397834 <= r397848;
        double r397850 = r397833 * r397839;
        double r397851 = r397844 / r397850;
        double r397852 = r397845 / r397839;
        double r397853 = r397849 ? r397851 : r397852;
        double r397854 = r397836 ? r397847 : r397853;
        return r397854;
}

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
Target0.4
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 (/ (* (cosh x) (/ y x)) z) < -2.132121890186479e+36

    1. Initial program 14.0

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

    3. Applied cosh-def14.0

      \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{y}{x}}{z}\]

    4. Applied frac-times14.0

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

    5. Applied associate-/l/12.5

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

    7. Applied associate-/r*0.3

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

    9. Applied clear-num0.4

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

    if -2.132121890186479e+36 < (/ (* (cosh x) (/ y x)) z) < 2.0127570389284558e-70

    1. Initial program 0.2

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

    3. Applied cosh-def0.2

      \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{y}{x}}{z}\]

    4. Applied frac-times0.2

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

    5. Applied associate-/l/0.4

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

    if 2.0127570389284558e-70 < (/ (* (cosh x) (/ y x)) z)

    1. Initial program 11.0

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

    3. Applied cosh-def11.0

      \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{y}{x}}{z}\]

    4. Applied frac-times11.1

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

    5. Applied associate-/l/10.7

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

    7. Applied associate-/r*0.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -2132121890186478992287162732707315712:\\ \;\;\;\;\frac{1}{\frac{2 \cdot x}{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 2.012757038928455801214359719345400453991 \cdot 10^{-70}:\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(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 1.03853053593515302e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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