Average Error: 7.5 → 0.4
Time: 17.7s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.259547158282228448649525588294824107293 \cdot 10^{-30}:\\ \;\;\;\;y \cdot \frac{\frac{\frac{\frac{1}{2}}{e^{x}} + e^{x} \cdot \frac{1}{2}}{z}}{x}\\ \mathbf{elif}\;z \le 1634804152264766393417728:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{2}}{e^{x}} + e^{x} \cdot \frac{1}{2}}{z} \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \cosh x}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.259547158282228448649525588294824107293 \cdot 10^{-30}:\\
\;\;\;\;y \cdot \frac{\frac{\frac{\frac{1}{2}}{e^{x}} + e^{x} \cdot \frac{1}{2}}{z}}{x}\\

\mathbf{elif}\;z \le 1634804152264766393417728:\\
\;\;\;\;\frac{\frac{\frac{\frac{1}{2}}{e^{x}} + e^{x} \cdot \frac{1}{2}}{z} \cdot y}{x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r9832391 = x;
        double r9832392 = cosh(r9832391);
        double r9832393 = y;
        double r9832394 = r9832393 / r9832391;
        double r9832395 = r9832392 * r9832394;
        double r9832396 = z;
        double r9832397 = r9832395 / r9832396;
        return r9832397;
}
↓
double f(double x, double y, double z) {
        double r9832398 = z;
        double r9832399 = -1.2595471582822284e-30;
        bool r9832400 = r9832398 <= r9832399;
        double r9832401 = y;
        double r9832402 = 0.5;
        double r9832403 = x;
        double r9832404 = exp(r9832403);
        double r9832405 = r9832402 / r9832404;
        double r9832406 = r9832404 * r9832402;
        double r9832407 = r9832405 + r9832406;
        double r9832408 = r9832407 / r9832398;
        double r9832409 = r9832408 / r9832403;
        double r9832410 = r9832401 * r9832409;
        double r9832411 = 1.6348041522647664e+24;
        bool r9832412 = r9832398 <= r9832411;
        double r9832413 = r9832408 * r9832401;
        double r9832414 = r9832413 / r9832403;
        double r9832415 = cosh(r9832403);
        double r9832416 = r9832401 * r9832415;
        double r9832417 = r9832403 * r9832398;
        double r9832418 = r9832416 / r9832417;
        double r9832419 = r9832412 ? r9832414 : r9832418;
        double r9832420 = r9832400 ? r9832410 : r9832419;
        return r9832420;
}

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

Original7.5
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 3 regimes

  2. if z < -1.2595471582822284e-30

    1. Initial program 10.6

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified9.6

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

    5. Applied *-un-lft-identity9.6

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

    6. Applied times-frac0.5

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

    7. Simplified0.5

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

    if -1.2595471582822284e-30 < z < 1.6348041522647664e+24

    1. Initial program 0.3

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

    2. Taylor expanded around inf 19.8

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

    3. Simplified0.5

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

    if 1.6348041522647664e+24 < z

    1. Initial program 12.2

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

    3. Applied associate-*r/12.2

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

    4. Applied associate-/l/0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.259547158282228448649525588294824107293 \cdot 10^{-30}:\\ \;\;\;\;y \cdot \frac{\frac{\frac{\frac{1}{2}}{e^{x}} + e^{x} \cdot \frac{1}{2}}{z}}{x}\\ \mathbf{elif}\;z \le 1634804152264766393417728:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{2}}{e^{x}} + e^{x} \cdot \frac{1}{2}}{z} \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \cosh x}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))