Average Error: 7.7 → 0.4
Time: 28.1s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -75754556562732115991855045745747951616 \lor \neg \left(z \le 189132.753253453527577221393585205078125\right):\\ \;\;\;\;\frac{\frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{\mathsf{fma}\left(e^{x}, e^{x}, \mathsf{expm1}\left(-2 \cdot x\right)\right)} \cdot y}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -75754556562732115991855045745747951616 \lor \neg \left(z \le 189132.753253453527577221393585205078125\right):\\
\;\;\;\;\frac{\frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{\mathsf{fma}\left(e^{x}, e^{x}, \mathsf{expm1}\left(-2 \cdot x\right)\right)} \cdot y}{z \cdot \left(2 \cdot x\right)}\\

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

\end{array}
double f(double x, double y, double z) {
        double r384387 = x;
        double r384388 = cosh(r384387);
        double r384389 = y;
        double r384390 = r384389 / r384387;
        double r384391 = r384388 * r384390;
        double r384392 = z;
        double r384393 = r384391 / r384392;
        return r384393;
}
↓
double f(double x, double y, double z) {
        double r384394 = z;
        double r384395 = -7.575455656273212e+37;
        bool r384396 = r384394 <= r384395;
        double r384397 = 189132.75325345353;
        bool r384398 = r384394 <= r384397;
        double r384399 = !r384398;
        bool r384400 = r384396 || r384399;
        double r384401 = x;
        double r384402 = exp(r384401);
        double r384403 = 3.0;
        double r384404 = pow(r384402, r384403);
        double r384405 = -r384401;
        double r384406 = exp(r384405);
        double r384407 = pow(r384406, r384403);
        double r384408 = r384404 + r384407;
        double r384409 = -2.0;
        double r384410 = r384409 * r384401;
        double r384411 = expm1(r384410);
        double r384412 = fma(r384402, r384402, r384411);
        double r384413 = r384408 / r384412;
        double r384414 = y;
        double r384415 = r384413 * r384414;
        double r384416 = 2.0;
        double r384417 = r384416 * r384401;
        double r384418 = r384394 * r384417;
        double r384419 = r384415 / r384418;
        double r384420 = r384402 + r384406;
        double r384421 = r384420 * r384414;
        double r384422 = r384421 / r384394;
        double r384423 = r384422 / r384417;
        double r384424 = r384400 ? r384419 : r384423;
        return r384424;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.7
Target0.5
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 < -7.575455656273212e+37 or 189132.75325345353 < z

    1. Initial program 12.5

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

    3. Applied cosh-def12.5

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

    4. Applied frac-times12.5

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

    5. Applied associate-/l/0.3

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

    7. Applied flip3-+0.5

      \[\leadsto \frac{\color{blue}{\frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{e^{x} \cdot e^{x} + \left(e^{-x} \cdot e^{-x} - e^{x} \cdot e^{-x}\right)}} \cdot y}{z \cdot \left(2 \cdot x\right)}\]

    8. Simplified0.5

      \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(e^{x}, e^{x}, \mathsf{expm1}\left(-2 \cdot x\right)\right)}} \cdot y}{z \cdot \left(2 \cdot x\right)}\]

    if -7.575455656273212e+37 < z < 189132.75325345353

    1. Initial program 0.4

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

    3. Applied cosh-def0.4

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

    4. Applied frac-times0.4

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

    5. Applied associate-/l/17.4

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

    7. Applied associate-/r*0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -75754556562732115991855045745747951616 \lor \neg \left(z \le 189132.753253453527577221393585205078125\right):\\ \;\;\;\;\frac{\frac{{\left(e^{x}\right)}^{3} + {\left(e^{-x}\right)}^{3}}{\mathsf{fma}\left(e^{x}, e^{x}, \mathsf{expm1}\left(-2 \cdot x\right)\right)} \cdot y}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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