Average Error: 7.8 → 0.5
Time: 19.2s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.283612116277876309196541517459782371519 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{\frac{1}{e^{x}} + e^{x}}{\frac{z}{y}}}{x \cdot 2}\\ \mathbf{elif}\;y \le 1.743933658318041395875089693172887195128 \cdot 10^{-58}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(e^{x} + e^{-x}\right)}{\left(x \cdot 2\right) \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.283612116277876309196541517459782371519 \cdot 10^{-42}:\\
\;\;\;\;\frac{\frac{\frac{1}{e^{x}} + e^{x}}{\frac{z}{y}}}{x \cdot 2}\\

\mathbf{elif}\;y \le 1.743933658318041395875089693172887195128 \cdot 10^{-58}:\\
\;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r23344599 = x;
        double r23344600 = cosh(r23344599);
        double r23344601 = y;
        double r23344602 = r23344601 / r23344599;
        double r23344603 = r23344600 * r23344602;
        double r23344604 = z;
        double r23344605 = r23344603 / r23344604;
        return r23344605;
}
↓
double f(double x, double y, double z) {
        double r23344606 = y;
        double r23344607 = -2.2836121162778763e-42;
        bool r23344608 = r23344606 <= r23344607;
        double r23344609 = 1.0;
        double r23344610 = x;
        double r23344611 = exp(r23344610);
        double r23344612 = r23344609 / r23344611;
        double r23344613 = r23344612 + r23344611;
        double r23344614 = z;
        double r23344615 = r23344614 / r23344606;
        double r23344616 = r23344613 / r23344615;
        double r23344617 = 2.0;
        double r23344618 = r23344610 * r23344617;
        double r23344619 = r23344616 / r23344618;
        double r23344620 = 1.7439336583180414e-58;
        bool r23344621 = r23344606 <= r23344620;
        double r23344622 = 0.5;
        double r23344623 = r23344622 / r23344611;
        double r23344624 = fma(r23344622, r23344611, r23344623);
        double r23344625 = r23344624 / r23344610;
        double r23344626 = r23344606 * r23344625;
        double r23344627 = r23344626 / r23344614;
        double r23344628 = -r23344610;
        double r23344629 = exp(r23344628);
        double r23344630 = r23344611 + r23344629;
        double r23344631 = r23344606 * r23344630;
        double r23344632 = r23344618 * r23344614;
        double r23344633 = r23344631 / r23344632;
        double r23344634 = r23344621 ? r23344627 : r23344633;
        double r23344635 = r23344608 ? r23344619 : r23344634;
        return r23344635;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.8
Target0.4
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -2.2836121162778763e-42

    1. Initial program 17.7

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

    3. Applied cosh-def17.7

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

    4. Applied frac-times17.7

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

    5. Applied associate-/l/0.8

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

    7. Applied associate-/r*0.6

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

    8. Simplified0.7

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

    if -2.2836121162778763e-42 < y < 1.7439336583180414e-58

    1. Initial program 0.2

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

    2. Taylor expanded around inf 11.4

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

    3. Simplified11.9

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

    5. Applied add-cube-cbrt12.7

      \[\leadsto \frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x} \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

    6. Applied *-un-lft-identity12.7

      \[\leadsto \frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x} \cdot \frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

    7. Applied times-frac12.7

      \[\leadsto \frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right)}\]

    8. Applied associate-*r*10.1

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x} \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{y}{\sqrt[3]{z}}}\]

    9. Simplified10.1

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\sqrt[3]{z}}\]
    10. Using strategy rm

    11. Applied frac-times1.3

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}{x} \cdot y}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

    12. Simplified0.3

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

    if 1.7439336583180414e-58 < y

    1. Initial program 16.7

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

    3. Applied cosh-def16.7

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

    4. Applied frac-times16.7

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

    5. Applied associate-/l/0.6

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.283612116277876309196541517459782371519 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{\frac{1}{e^{x}} + e^{x}}{\frac{z}{y}}}{x \cdot 2}\\ \mathbf{elif}\;y \le 1.743933658318041395875089693172887195128 \cdot 10^{-58}:\\ \;\;\;\;\frac{y \cdot \frac{\mathsf{fma}\left(\frac{1}{2}, e^{x}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(e^{x} + e^{-x}\right)}{\left(x \cdot 2\right) \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +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))