Average Error: 7.8 → 0.7
Time: 16.6s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.56082919711251703348519476677496253443 \cdot 10^{-15} \lor \neg \left(z \le 4.489894380771518759583953311198839017656 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{\frac{\left(e^{-x} + e^{x}\right) \cdot y}{x \cdot z}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.56082919711251703348519476677496253443 \cdot 10^{-15} \lor \neg \left(z \le 4.489894380771518759583953311198839017656 \cdot 10^{-96}\right):\\
\;\;\;\;\frac{\frac{\left(e^{-x} + e^{x}\right) \cdot y}{x \cdot z}}{2}\\

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

\end{array}
double f(double x, double y, double z) {
        double r489733 = x;
        double r489734 = cosh(r489733);
        double r489735 = y;
        double r489736 = r489735 / r489733;
        double r489737 = r489734 * r489736;
        double r489738 = z;
        double r489739 = r489737 / r489738;
        return r489739;
}
↓
double f(double x, double y, double z) {
        double r489740 = z;
        double r489741 = -2.560829197112517e-15;
        bool r489742 = r489740 <= r489741;
        double r489743 = 4.489894380771519e-96;
        bool r489744 = r489740 <= r489743;
        double r489745 = !r489744;
        bool r489746 = r489742 || r489745;
        double r489747 = x;
        double r489748 = -r489747;
        double r489749 = exp(r489748);
        double r489750 = exp(r489747);
        double r489751 = r489749 + r489750;
        double r489752 = y;
        double r489753 = r489751 * r489752;
        double r489754 = r489747 * r489740;
        double r489755 = r489753 / r489754;
        double r489756 = 2.0;
        double r489757 = r489755 / r489756;
        double r489758 = r489752 / r489740;
        double r489759 = r489758 / r489747;
        double r489760 = cosh(r489747);
        double r489761 = r489759 * r489760;
        double r489762 = r489746 ? r489757 : r489761;
        return r489762;
}

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

Original7.8
Target0.5
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.560829197112517e-15 or 4.489894380771519e-96 < z

    1. Initial program 10.5

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

    2. Simplified0.9

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

    4. Applied cosh-def0.9

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

    5. Applied associate-*r/0.9

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

    6. Simplified0.9

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

    if -2.560829197112517e-15 < z < 4.489894380771519e-96

    1. Initial program 0.3

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

    2. Simplified23.8

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

    4. Applied *-un-lft-identity23.8

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

    5. Applied associate-*l*23.8

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

    6. Simplified0.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.56082919711251703348519476677496253443 \cdot 10^{-15} \lor \neg \left(z \le 4.489894380771518759583953311198839017656 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{\frac{\left(e^{-x} + e^{x}\right) \cdot y}{x \cdot z}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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))