Average Error: 8.0 → 0.4
Time: 3.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -5.2383203256810599 \cdot 10^{302}:\\ \;\;\;\;\frac{\cosh x}{\frac{z \cdot x}{y}}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 6.08538018889700869 \cdot 10^{170}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -5.2383203256810599 \cdot 10^{302}:\\
\;\;\;\;\frac{\cosh x}{\frac{z \cdot x}{y}}\\

\mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 6.08538018889700869 \cdot 10^{170}:\\
\;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r479451 = x;
        double r479452 = cosh(r479451);
        double r479453 = y;
        double r479454 = r479453 / r479451;
        double r479455 = r479452 * r479454;
        double r479456 = z;
        double r479457 = r479455 / r479456;
        return r479457;
}
↓
double f(double x, double y, double z) {
        double r479458 = x;
        double r479459 = cosh(r479458);
        double r479460 = y;
        double r479461 = r479460 / r479458;
        double r479462 = r479459 * r479461;
        double r479463 = -5.23832032568106e+302;
        bool r479464 = r479462 <= r479463;
        double r479465 = z;
        double r479466 = r479465 * r479458;
        double r479467 = r479466 / r479460;
        double r479468 = r479459 / r479467;
        double r479469 = 6.085380188897009e+170;
        bool r479470 = r479462 <= r479469;
        double r479471 = r479462 / r479465;
        double r479472 = r479459 * r479460;
        double r479473 = r479472 / r479465;
        double r479474 = r479473 / r479458;
        double r479475 = r479470 ? r479471 : r479474;
        double r479476 = r479464 ? r479468 : r479475;
        return r479476;
}

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

Original8.0
Target0.5
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* (cosh x) (/ y x)) < -5.23832032568106e+302

    1. Initial program 61.0

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

    3. Applied associate-*r/61.0

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

    4. Applied associate-/l/0.7

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

    6. Applied associate-/l*0.8

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

    if -5.23832032568106e+302 < (* (cosh x) (/ y x)) < 6.085380188897009e+170

    1. Initial program 0.3

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

    if 6.085380188897009e+170 < (* (cosh x) (/ y x))

    1. Initial program 24.2

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

    3. Applied associate-*r/24.2

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

    4. Applied associate-/l/1.5

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

    6. Applied associate-/r*1.1

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -5.2383203256810599 \cdot 10^{302}:\\ \;\;\;\;\frac{\cosh x}{\frac{z \cdot x}{y}}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 6.08538018889700869 \cdot 10^{170}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +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 1.0385305359351529e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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