Average Error: 7.5 → 0.4
Time: 4.7s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -1.3555263400315947 \cdot 10^{223}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 2.79890311216328603 \cdot 10^{202}:\\ \;\;\;\;\frac{e^{\log \left(\cosh x\right)} \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{x} + e^{-x}\right) \cdot y\right) \cdot \frac{1}{z \cdot \left(2 \cdot x\right)}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -1.3555263400315947 \cdot 10^{223}:\\
\;\;\;\;\frac{y}{z} \cdot \frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\

\mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 2.79890311216328603 \cdot 10^{202}:\\
\;\;\;\;\frac{e^{\log \left(\cosh x\right)} \cdot \frac{y}{x}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r786303 = x;
        double r786304 = cosh(r786303);
        double r786305 = y;
        double r786306 = r786305 / r786303;
        double r786307 = r786304 * r786306;
        double r786308 = z;
        double r786309 = r786307 / r786308;
        return r786309;
}
↓
double f(double x, double y, double z) {
        double r786310 = x;
        double r786311 = cosh(r786310);
        double r786312 = y;
        double r786313 = r786312 / r786310;
        double r786314 = r786311 * r786313;
        double r786315 = -1.3555263400315947e+223;
        bool r786316 = r786314 <= r786315;
        double r786317 = z;
        double r786318 = r786312 / r786317;
        double r786319 = exp(r786310);
        double r786320 = 0.5;
        double r786321 = r786320 / r786319;
        double r786322 = fma(r786319, r786320, r786321);
        double r786323 = r786322 / r786310;
        double r786324 = r786318 * r786323;
        double r786325 = 2.798903112163286e+202;
        bool r786326 = r786314 <= r786325;
        double r786327 = log(r786311);
        double r786328 = exp(r786327);
        double r786329 = r786328 * r786313;
        double r786330 = r786329 / r786317;
        double r786331 = -r786310;
        double r786332 = exp(r786331);
        double r786333 = r786319 + r786332;
        double r786334 = r786333 * r786312;
        double r786335 = 1.0;
        double r786336 = 2.0;
        double r786337 = r786336 * r786310;
        double r786338 = r786317 * r786337;
        double r786339 = r786335 / r786338;
        double r786340 = r786334 * r786339;
        double r786341 = r786326 ? r786330 : r786340;
        double r786342 = r786316 ? r786324 : r786341;
        return r786342;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.5
Target0.4
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)) < -1.3555263400315947e+223

    1. Initial program 33.2

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

    2. Taylor expanded around inf 0.8

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

    3. Simplified0.9

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

    if -1.3555263400315947e+223 < (* (cosh x) (/ y x)) < 2.798903112163286e+202

    1. Initial program 0.2

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

    3. Applied add-exp-log0.3

      \[\leadsto \frac{\color{blue}{e^{\log \left(\cosh x\right)}} \cdot \frac{y}{x}}{z}\]

    if 2.798903112163286e+202 < (* (cosh x) (/ y x))

    1. Initial program 28.7

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

    3. Applied cosh-def28.7

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

    4. Applied frac-times28.7

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

    5. Applied associate-/l/0.5

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

    7. Applied div-inv0.7

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -1.3555263400315947 \cdot 10^{223}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 2.79890311216328603 \cdot 10^{202}:\\ \;\;\;\;\frac{e^{\log \left(\cosh x\right)} \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{x} + e^{-x}\right) \cdot y\right) \cdot \frac{1}{z \cdot \left(2 \cdot x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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))