Average Error: 7.5 → 0.4
Time: 23.8s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.499151431129345690915331400810285626293 \cdot 10^{-22} \lor \neg \left(z \le 3.123371243102039260157787505142381731931 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{y}{\frac{z \cdot x}{\left(e^{x} + e^{-x}\right) \cdot \frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{e^{x} + e^{-x}}}}{\frac{x}{\frac{1}{2}}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.499151431129345690915331400810285626293 \cdot 10^{-22} \lor \neg \left(z \le 3.123371243102039260157787505142381731931 \cdot 10^{-43}\right):\\
\;\;\;\;\frac{y}{\frac{z \cdot x}{\left(e^{x} + e^{-x}\right) \cdot \frac{1}{2}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{y}{\frac{z}{e^{x} + e^{-x}}}}{\frac{x}{\frac{1}{2}}}\\

\end{array}
double f(double x, double y, double z) {
        double r340401 = x;
        double r340402 = cosh(r340401);
        double r340403 = y;
        double r340404 = r340403 / r340401;
        double r340405 = r340402 * r340404;
        double r340406 = z;
        double r340407 = r340405 / r340406;
        return r340407;
}
↓
double f(double x, double y, double z) {
        double r340408 = z;
        double r340409 = -7.499151431129346e-22;
        bool r340410 = r340408 <= r340409;
        double r340411 = 3.1233712431020393e-43;
        bool r340412 = r340408 <= r340411;
        double r340413 = !r340412;
        bool r340414 = r340410 || r340413;
        double r340415 = y;
        double r340416 = x;
        double r340417 = r340408 * r340416;
        double r340418 = exp(r340416);
        double r340419 = -r340416;
        double r340420 = exp(r340419);
        double r340421 = r340418 + r340420;
        double r340422 = 0.5;
        double r340423 = r340421 * r340422;
        double r340424 = r340417 / r340423;
        double r340425 = r340415 / r340424;
        double r340426 = r340408 / r340421;
        double r340427 = r340415 / r340426;
        double r340428 = r340416 / r340422;
        double r340429 = r340427 / r340428;
        double r340430 = r340414 ? r340425 : r340429;
        return r340430;
}

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 2 regimes

  2. if z < -7.499151431129346e-22 or 3.1233712431020393e-43 < z

    1. Initial program 10.7

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

    if -7.499151431129346e-22 < z < 3.1233712431020393e-43

    1. Initial program 0.3

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

    2. Taylor expanded around inf 22.1

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

    3. Simplified22.1

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

    5. Applied times-frac22.1

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

    6. Applied associate-/r*0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.499151431129345690915331400810285626293 \cdot 10^{-22} \lor \neg \left(z \le 3.123371243102039260157787505142381731931 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{y}{\frac{z \cdot x}{\left(e^{x} + e^{-x}\right) \cdot \frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{e^{x} + e^{-x}}}}{\frac{x}{\frac{1}{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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