Average Error: 7.6 → 0.7
Time: 4.1s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -6.47943049490942291 \cdot 10^{-17} \lor \neg \left(y \le 4072403334666.24561\right):\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \frac{y \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{x}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -6.47943049490942291 \cdot 10^{-17} \lor \neg \left(y \le 4072403334666.24561\right):\\
\;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r458393 = x;
        double r458394 = cosh(r458393);
        double r458395 = y;
        double r458396 = r458395 / r458393;
        double r458397 = r458394 * r458396;
        double r458398 = z;
        double r458399 = r458397 / r458398;
        return r458399;
}
↓
double f(double x, double y, double z) {
        double r458400 = y;
        double r458401 = -6.479430494909423e-17;
        bool r458402 = r458400 <= r458401;
        double r458403 = 4072403334666.2456;
        bool r458404 = r458400 <= r458403;
        double r458405 = !r458404;
        bool r458406 = r458402 || r458405;
        double r458407 = 0.5;
        double r458408 = x;
        double r458409 = r458408 * r458400;
        double r458410 = z;
        double r458411 = r458409 / r458410;
        double r458412 = r458407 * r458411;
        double r458413 = r458408 * r458410;
        double r458414 = r458400 / r458413;
        double r458415 = r458412 + r458414;
        double r458416 = -1.0;
        double r458417 = r458416 * r458408;
        double r458418 = exp(r458417);
        double r458419 = exp(r458408);
        double r458420 = r458418 + r458419;
        double r458421 = r458400 * r458420;
        double r458422 = r458421 / r458408;
        double r458423 = r458407 * r458422;
        double r458424 = r458423 / r458410;
        double r458425 = r458406 ? r458415 : r458424;
        return r458425;
}

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
Target0.5
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -6.479430494909423e-17 or 4072403334666.2456 < y

    1. Initial program 20.8

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

    2. Taylor expanded around 0 1.5

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]

    if -6.479430494909423e-17 < y < 4072403334666.2456

    1. Initial program 0.3

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.4

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

    5. Applied div-inv0.4

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

    6. Applied times-frac0.5

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

    7. Simplified0.4

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

    9. Applied div-inv0.4

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

    10. Applied associate-*l*0.4

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

    11. Simplified0.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.47943049490942291 \cdot 10^{-17} \lor \neg \left(y \le 4072403334666.24561\right):\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \frac{y \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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))