Average Error: 8.1 → 0.6
Time: 17.1s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.641117804463787546198878896231329963893 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(e^{-x} + e^{x}\right) \cdot y}{\left(2 \cdot x\right) \cdot z}\\ \mathbf{elif}\;z \le 1.188638495701333717916059122397157453863 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{-x} + e^{x}\right) \cdot y}{\left(2 \cdot x\right) \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.641117804463787546198878896231329963893 \cdot 10^{-16}:\\
\;\;\;\;\frac{\left(e^{-x} + e^{x}\right) \cdot y}{\left(2 \cdot x\right) \cdot z}\\

\mathbf{elif}\;z \le 1.188638495701333717916059122397157453863 \cdot 10^{-69}:\\
\;\;\;\;\frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x} \cdot \frac{y}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r18949322 = x;
        double r18949323 = cosh(r18949322);
        double r18949324 = y;
        double r18949325 = r18949324 / r18949322;
        double r18949326 = r18949323 * r18949325;
        double r18949327 = z;
        double r18949328 = r18949326 / r18949327;
        return r18949328;
}
↓
double f(double x, double y, double z) {
        double r18949329 = z;
        double r18949330 = -4.641117804463788e-16;
        bool r18949331 = r18949329 <= r18949330;
        double r18949332 = x;
        double r18949333 = -r18949332;
        double r18949334 = exp(r18949333);
        double r18949335 = exp(r18949332);
        double r18949336 = r18949334 + r18949335;
        double r18949337 = y;
        double r18949338 = r18949336 * r18949337;
        double r18949339 = 2.0;
        double r18949340 = r18949339 * r18949332;
        double r18949341 = r18949340 * r18949329;
        double r18949342 = r18949338 / r18949341;
        double r18949343 = 1.1886384957013337e-69;
        bool r18949344 = r18949329 <= r18949343;
        double r18949345 = 0.5;
        double r18949346 = r18949345 / r18949335;
        double r18949347 = fma(r18949335, r18949345, r18949346);
        double r18949348 = r18949347 / r18949332;
        double r18949349 = r18949337 / r18949329;
        double r18949350 = r18949348 * r18949349;
        double r18949351 = r18949344 ? r18949350 : r18949342;
        double r18949352 = r18949331 ? r18949342 : r18949351;
        return r18949352;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original8.1
Target0.5
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.641117804463788e-16 or 1.1886384957013337e-69 < z

    1. Initial program 11.3

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

    3. Applied cosh-def11.3

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

    4. Applied frac-times11.3

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

    5. Applied associate-/l/0.6

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

    if -4.641117804463788e-16 < z < 1.1886384957013337e-69

    1. Initial program 0.3

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

    2. Taylor expanded around inf 23.6

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

    3. Simplified0.4

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.641117804463787546198878896231329963893 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(e^{-x} + e^{x}\right) \cdot y}{\left(2 \cdot x\right) \cdot z}\\ \mathbf{elif}\;z \le 1.188638495701333717916059122397157453863 \cdot 10^{-69}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{-x} + e^{x}\right) \cdot y}{\left(2 \cdot x\right) \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(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))