Average Error: 7.8 → 0.7
Time: 12.4s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -19.7335978520876729:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \le 7876022.030429827:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{x \cdot y}{z}, \frac{\frac{y}{x}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{y}{z}}{2 \cdot x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -19.7335978520876729:\\
\;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\

\mathbf{elif}\;y \le 7876022.030429827:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{x \cdot y}{z}, \frac{\frac{y}{x}}{z}\right)\\

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

\end{array}
double f(double x, double y, double z) {
        double r658475 = x;
        double r658476 = cosh(r658475);
        double r658477 = y;
        double r658478 = r658477 / r658475;
        double r658479 = r658476 * r658478;
        double r658480 = z;
        double r658481 = r658479 / r658480;
        return r658481;
}
↓
double f(double x, double y, double z) {
        double r658482 = y;
        double r658483 = -19.733597852087673;
        bool r658484 = r658482 <= r658483;
        double r658485 = x;
        double r658486 = cosh(r658485);
        double r658487 = z;
        double r658488 = r658485 * r658487;
        double r658489 = r658482 / r658488;
        double r658490 = r658486 * r658489;
        double r658491 = 7876022.030429827;
        bool r658492 = r658482 <= r658491;
        double r658493 = 0.5;
        double r658494 = r658485 * r658482;
        double r658495 = r658494 / r658487;
        double r658496 = r658482 / r658485;
        double r658497 = r658496 / r658487;
        double r658498 = fma(r658493, r658495, r658497);
        double r658499 = exp(r658485);
        double r658500 = -r658485;
        double r658501 = exp(r658500);
        double r658502 = r658499 + r658501;
        double r658503 = r658482 / r658487;
        double r658504 = r658502 * r658503;
        double r658505 = 2.0;
        double r658506 = r658505 * r658485;
        double r658507 = r658504 / r658506;
        double r658508 = r658492 ? r658498 : r658507;
        double r658509 = r658484 ? r658490 : r658508;
        return r658509;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.8
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -19.733597852087673

    1. Initial program 20.7

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

    3. Applied *-un-lft-identity20.7

      \[\leadsto \frac{\cosh x \cdot \frac{y}{x}}{\color{blue}{1 \cdot z}}\]

    4. Applied times-frac20.7

      \[\leadsto \color{blue}{\frac{\cosh x}{1} \cdot \frac{\frac{y}{x}}{z}}\]

    5. Simplified20.7

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

    6. Simplified0.2

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

    if -19.733597852087673 < y < 7876022.030429827

    1. Initial program 0.3

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

    2. Taylor expanded around 0 11.2

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

    3. Simplified0.9

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

    if 7876022.030429827 < y

    1. Initial program 22.1

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

    3. Applied *-un-lft-identity22.1

      \[\leadsto \frac{\cosh x \cdot \frac{y}{x}}{\color{blue}{1 \cdot z}}\]

    4. Applied times-frac21.9

      \[\leadsto \color{blue}{\frac{\cosh x}{1} \cdot \frac{\frac{y}{x}}{z}}\]

    5. Simplified21.9

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

    6. Simplified0.3

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

    8. Applied *-un-lft-identity0.3

      \[\leadsto \cosh x \cdot \frac{\color{blue}{1 \cdot y}}{x \cdot z}\]

    9. Applied times-frac0.3

      \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{y}{z}\right)}\]
    10. Using strategy rm

    11. Applied associate-*l/0.3

      \[\leadsto \cosh x \cdot \color{blue}{\frac{1 \cdot \frac{y}{z}}{x}}\]

    12. Applied cosh-def0.3

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

    13. Applied frac-times0.3

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

    14. Simplified0.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -19.7335978520876729:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;y \le 7876022.030429827:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{x \cdot y}{z}, \frac{\frac{y}{x}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{y}{z}}{2 \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +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))