Average Error: 7.5 → 0.7
Time: 17.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -13294780528925495035856397041561239552:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;y \le 1.449820321413413963499843591018471786151 \cdot 10^{72}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \left(\frac{\frac{1}{2}}{e^{x}} + e^{x} \cdot \frac{1}{2}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -13294780528925495035856397041561239552:\\
\;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\

\mathbf{elif}\;y \le 1.449820321413413963499843591018471786151 \cdot 10^{72}:\\
\;\;\;\;\frac{\frac{y}{x} \cdot \left(\frac{\frac{1}{2}}{e^{x}} + e^{x} \cdot \frac{1}{2}\right)}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r27162409 = x;
        double r27162410 = cosh(r27162409);
        double r27162411 = y;
        double r27162412 = r27162411 / r27162409;
        double r27162413 = r27162410 * r27162412;
        double r27162414 = z;
        double r27162415 = r27162413 / r27162414;
        return r27162415;
}
↓
double f(double x, double y, double z) {
        double r27162416 = y;
        double r27162417 = -1.3294780528925495e+37;
        bool r27162418 = r27162416 <= r27162417;
        double r27162419 = x;
        double r27162420 = cosh(r27162419);
        double r27162421 = r27162420 * r27162416;
        double r27162422 = z;
        double r27162423 = r27162419 * r27162422;
        double r27162424 = r27162421 / r27162423;
        double r27162425 = 1.449820321413414e+72;
        bool r27162426 = r27162416 <= r27162425;
        double r27162427 = r27162416 / r27162419;
        double r27162428 = 0.5;
        double r27162429 = exp(r27162419);
        double r27162430 = r27162428 / r27162429;
        double r27162431 = r27162429 * r27162428;
        double r27162432 = r27162430 + r27162431;
        double r27162433 = r27162427 * r27162432;
        double r27162434 = r27162433 / r27162422;
        double r27162435 = r27162426 ? r27162434 : r27162424;
        double r27162436 = r27162418 ? r27162424 : r27162435;
        return r27162436;
}

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
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.3294780528925495e+37 or 1.449820321413414e+72 < y

    1. Initial program 26.9

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

    3. Applied associate-*r/26.9

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

    4. Applied associate-/l/0.4

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

    if -1.3294780528925495e+37 < y < 1.449820321413414e+72

    1. Initial program 0.8

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

    3. Applied associate-*r/0.8

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

    4. Applied associate-/l/9.5

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

    6. Applied associate-/r*9.2

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

    7. Taylor expanded around inf 9.5

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

    8. Simplified0.8

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -13294780528925495035856397041561239552:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;y \le 1.449820321413413963499843591018471786151 \cdot 10^{72}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \left(\frac{\frac{1}{2}}{e^{x}} + e^{x} \cdot \frac{1}{2}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

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))