Average Error: 7.9 → 0.4
Time: 13.1s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -813868602.68375670909881591796875 \lor \neg \left(z \le 2.464261935710089975344020916546644554505 \cdot 10^{44}\right):\\ \;\;\;\;\left(\cosh x \cdot y\right) \cdot \frac{1}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{z \cdot \frac{x}{y}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -813868602.68375670909881591796875 \lor \neg \left(z \le 2.464261935710089975344020916546644554505 \cdot 10^{44}\right):\\
\;\;\;\;\left(\cosh x \cdot y\right) \cdot \frac{1}{z \cdot x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r342819 = x;
        double r342820 = cosh(r342819);
        double r342821 = y;
        double r342822 = r342821 / r342819;
        double r342823 = r342820 * r342822;
        double r342824 = z;
        double r342825 = r342823 / r342824;
        return r342825;
}
↓
double f(double x, double y, double z) {
        double r342826 = z;
        double r342827 = -813868602.6837567;
        bool r342828 = r342826 <= r342827;
        double r342829 = 2.46426193571009e+44;
        bool r342830 = r342826 <= r342829;
        double r342831 = !r342830;
        bool r342832 = r342828 || r342831;
        double r342833 = x;
        double r342834 = cosh(r342833);
        double r342835 = y;
        double r342836 = r342834 * r342835;
        double r342837 = 1.0;
        double r342838 = r342826 * r342833;
        double r342839 = r342837 / r342838;
        double r342840 = r342836 * r342839;
        double r342841 = 0.5;
        double r342842 = exp(r342833);
        double r342843 = -r342833;
        double r342844 = exp(r342843);
        double r342845 = r342842 + r342844;
        double r342846 = r342841 * r342845;
        double r342847 = r342833 / r342835;
        double r342848 = r342826 * r342847;
        double r342849 = r342846 / r342848;
        double r342850 = r342832 ? r342840 : r342849;
        return r342850;
}

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

Original7.9
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 < -813868602.6837567 or 2.46426193571009e+44 < z

    1. Initial program 13.0

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

    3. Applied associate-*r/13.0

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

    4. Applied associate-/l/0.3

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

    6. Applied div-inv0.4

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

    if -813868602.6837567 < z < 2.46426193571009e+44

    1. Initial program 0.5

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

    3. Applied associate-*r/0.5

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

    4. Applied associate-/l/17.1

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

    6. Applied div-inv18.9

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

    7. Taylor expanded around inf 17.1

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

    8. Simplified17.1

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

    10. Applied *-un-lft-identity17.1

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

    11. Applied times-frac0.5

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

    12. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -813868602.68375670909881591796875 \lor \neg \left(z \le 2.464261935710089975344020916546644554505 \cdot 10^{44}\right):\\ \;\;\;\;\left(\cosh x \cdot y\right) \cdot \frac{1}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{z \cdot \frac{x}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +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 1.03853053593515302e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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