Average Error: 7.9 → 0.4
Time: 12.7s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.53262895416257164 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right) \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;z \le 2.20785778145389367 \cdot 10^{-49}:\\ \;\;\;\;\frac{\cosh x}{z \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x \cdot z}\right)\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.53262895416257164 \cdot 10^{-20}:\\
\;\;\;\;\left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right) \cdot \frac{y}{x \cdot z}\\

\mathbf{elif}\;z \le 2.20785778145389367 \cdot 10^{-49}:\\
\;\;\;\;\frac{\cosh x}{z \cdot \frac{x}{y}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x \cdot z}\right)\\

\end{array}
double f(double x, double y, double z) {
        double r553664 = x;
        double r553665 = cosh(r553664);
        double r553666 = y;
        double r553667 = r553666 / r553664;
        double r553668 = r553665 * r553667;
        double r553669 = z;
        double r553670 = r553668 / r553669;
        return r553670;
}
↓
double f(double x, double y, double z) {
        double r553671 = z;
        double r553672 = -1.5326289541625716e-20;
        bool r553673 = r553671 <= r553672;
        double r553674 = 0.5;
        double r553675 = x;
        double r553676 = exp(r553675);
        double r553677 = -r553675;
        double r553678 = exp(r553677);
        double r553679 = r553676 + r553678;
        double r553680 = r553674 * r553679;
        double r553681 = y;
        double r553682 = r553675 * r553671;
        double r553683 = r553681 / r553682;
        double r553684 = r553680 * r553683;
        double r553685 = 2.2078577814538937e-49;
        bool r553686 = r553671 <= r553685;
        double r553687 = cosh(r553675);
        double r553688 = r553675 / r553681;
        double r553689 = r553671 * r553688;
        double r553690 = r553687 / r553689;
        double r553691 = sqrt(r553687);
        double r553692 = r553691 * r553683;
        double r553693 = r553691 * r553692;
        double r553694 = r553686 ? r553690 : r553693;
        double r553695 = r553673 ? r553684 : r553694;
        return r553695;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.9
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 z < -1.5326289541625716e-20

    1. Initial program 11.9

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

    3. Applied *-un-lft-identity11.9

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

    4. Applied times-frac11.8

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

    5. Simplified11.8

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

    6. Simplified0.3

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

    7. Taylor expanded around inf 0.3

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

    if -1.5326289541625716e-20 < z < 2.2078577814538937e-49

    1. Initial program 0.3

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

    3. Applied associate-/l*0.4

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

    4. Simplified0.4

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

    if 2.2078577814538937e-49 < z

    1. Initial program 10.3

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

    3. Applied *-un-lft-identity10.3

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

    4. Applied times-frac10.3

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

    5. Simplified10.3

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

    6. Simplified0.6

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

    8. Applied add-sqr-sqrt0.6

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

    9. Applied associate-*l*0.6

      \[\leadsto \color{blue}{\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x \cdot z}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.53262895416257164 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right) \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;z \le 2.20785778145389367 \cdot 10^{-49}:\\ \;\;\;\;\frac{\cosh x}{z \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\cosh x} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x \cdot z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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))