Average Error: 7.9 → 0.5
Time: 20.1s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -9.318083108674960798059641283288945178294 \cdot 10^{-34}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;z \le 2.232206736370805648104041633838868572772 \cdot 10^{46}:\\ \;\;\;\;\frac{\frac{\frac{\frac{y}{e^{x}} + y \cdot e^{x}}{x}}{2}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -9.318083108674960798059641283288945178294 \cdot 10^{-34}:\\
\;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\

\mathbf{elif}\;z \le 2.232206736370805648104041633838868572772 \cdot 10^{46}:\\
\;\;\;\;\frac{\frac{\frac{\frac{y}{e^{x}} + y \cdot e^{x}}{x}}{2}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r17104443 = x;
        double r17104444 = cosh(r17104443);
        double r17104445 = y;
        double r17104446 = r17104445 / r17104443;
        double r17104447 = r17104444 * r17104446;
        double r17104448 = z;
        double r17104449 = r17104447 / r17104448;
        return r17104449;
}
↓
double f(double x, double y, double z) {
        double r17104450 = z;
        double r17104451 = -9.31808310867496e-34;
        bool r17104452 = r17104450 <= r17104451;
        double r17104453 = x;
        double r17104454 = cosh(r17104453);
        double r17104455 = y;
        double r17104456 = r17104454 * r17104455;
        double r17104457 = r17104453 * r17104450;
        double r17104458 = r17104456 / r17104457;
        double r17104459 = 2.2322067363708056e+46;
        bool r17104460 = r17104450 <= r17104459;
        double r17104461 = exp(r17104453);
        double r17104462 = r17104455 / r17104461;
        double r17104463 = r17104455 * r17104461;
        double r17104464 = r17104462 + r17104463;
        double r17104465 = r17104464 / r17104453;
        double r17104466 = 2.0;
        double r17104467 = r17104465 / r17104466;
        double r17104468 = r17104467 / r17104450;
        double r17104469 = r17104460 ? r17104468 : r17104458;
        double r17104470 = r17104452 ? r17104458 : r17104469;
        return r17104470;
}

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

Derivation

  1. Split input into 2 regimes

  2. if z < -9.31808310867496e-34 or 2.2322067363708056e+46 < z

    1. Initial program 12.1

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

    3. Applied associate-*r/12.1

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

    4. Applied associate-/l/0.4

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

    if -9.31808310867496e-34 < z < 2.2322067363708056e+46

    1. Initial program 0.7

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

    3. Applied cosh-def0.7

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

    4. Applied associate-*l/0.7

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

    5. Simplified0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.318083108674960798059641283288945178294 \cdot 10^{-34}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;z \le 2.232206736370805648104041633838868572772 \cdot 10^{46}:\\ \;\;\;\;\frac{\frac{\frac{\frac{y}{e^{x}} + y \cdot e^{x}}{x}}{2}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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))