Average Error: 7.6 → 0.4
Time: 19.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -2.917391625482233487164607746012211025076 \cdot 10^{296} \lor \neg \left(\cosh x \cdot \frac{y}{x} \le 1.441100974532180611209834362198683583791 \cdot 10^{193}\right):\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(\left(e^{x} + e^{-x}\right) \cdot y\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{2 \cdot x}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -2.917391625482233487164607746012211025076 \cdot 10^{296} \lor \neg \left(\cosh x \cdot \frac{y}{x} \le 1.441100974532180611209834362198683583791 \cdot 10^{193}\right):\\
\;\;\;\;\frac{\frac{1}{2} \cdot \left(\left(e^{x} + e^{-x}\right) \cdot y\right)}{z \cdot x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r437405 = x;
        double r437406 = cosh(r437405);
        double r437407 = y;
        double r437408 = r437407 / r437405;
        double r437409 = r437406 * r437408;
        double r437410 = z;
        double r437411 = r437409 / r437410;
        return r437411;
}
↓
double f(double x, double y, double z) {
        double r437412 = x;
        double r437413 = cosh(r437412);
        double r437414 = y;
        double r437415 = r437414 / r437412;
        double r437416 = r437413 * r437415;
        double r437417 = -2.9173916254822335e+296;
        bool r437418 = r437416 <= r437417;
        double r437419 = 1.4411009745321806e+193;
        bool r437420 = r437416 <= r437419;
        double r437421 = !r437420;
        bool r437422 = r437418 || r437421;
        double r437423 = 0.5;
        double r437424 = exp(r437412);
        double r437425 = -r437412;
        double r437426 = exp(r437425);
        double r437427 = r437424 + r437426;
        double r437428 = r437427 * r437414;
        double r437429 = r437423 * r437428;
        double r437430 = z;
        double r437431 = r437430 * r437412;
        double r437432 = r437429 / r437431;
        double r437433 = 2.0;
        double r437434 = r437433 * r437412;
        double r437435 = r437428 / r437434;
        double r437436 = r437435 / r437430;
        double r437437 = r437422 ? r437432 : r437436;
        return r437437;
}

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

Original7.6
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 (* (cosh x) (/ y x)) < -2.9173916254822335e+296 or 1.4411009745321806e+193 < (* (cosh x) (/ y x))

    1. Initial program 37.4

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

    3. Applied div-inv37.5

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

    5. Applied associate-*r/37.5

      \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x}} \cdot \frac{1}{z}\]

    6. Applied associate-*l/0.9

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

    7. Simplified0.8

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

    8. Taylor expanded around inf 0.9

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

    9. Simplified0.9

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

    if -2.9173916254822335e+296 < (* (cosh x) (/ y x)) < 1.4411009745321806e+193

    1. Initial program 0.2

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

    3. Applied cosh-def0.3

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

    4. Applied frac-times0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -2.917391625482233487164607746012211025076 \cdot 10^{296} \lor \neg \left(\cosh x \cdot \frac{y}{x} \le 1.441100974532180611209834362198683583791 \cdot 10^{193}\right):\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(\left(e^{x} + e^{-x}\right) \cdot y\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{2 \cdot x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))