Average Error: 7.8 → 0.7
Time: 18.1s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.56082919711251703348519476677496253443 \cdot 10^{-15} \lor \neg \left(z \le 4.489894380771518759583953311198839017656 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{y}{\frac{\frac{x}{\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}}}{\sqrt[3]{\cosh x}} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{x}{\cosh x}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.56082919711251703348519476677496253443 \cdot 10^{-15} \lor \neg \left(z \le 4.489894380771518759583953311198839017656 \cdot 10^{-96}\right):\\
\;\;\;\;\frac{y}{\frac{\frac{x}{\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}}}{\sqrt[3]{\cosh x}} \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{y}{z}}{\frac{x}{\cosh x}}\\

\end{array}
double f(double x, double y, double z) {
        double r468680 = x;
        double r468681 = cosh(r468680);
        double r468682 = y;
        double r468683 = r468682 / r468680;
        double r468684 = r468681 * r468683;
        double r468685 = z;
        double r468686 = r468684 / r468685;
        return r468686;
}
↓
double f(double x, double y, double z) {
        double r468687 = z;
        double r468688 = -2.560829197112517e-15;
        bool r468689 = r468687 <= r468688;
        double r468690 = 4.489894380771519e-96;
        bool r468691 = r468687 <= r468690;
        double r468692 = !r468691;
        bool r468693 = r468689 || r468692;
        double r468694 = y;
        double r468695 = x;
        double r468696 = cosh(r468695);
        double r468697 = cbrt(r468696);
        double r468698 = r468697 * r468697;
        double r468699 = r468695 / r468698;
        double r468700 = r468699 / r468697;
        double r468701 = r468700 * r468687;
        double r468702 = r468694 / r468701;
        double r468703 = r468694 / r468687;
        double r468704 = r468695 / r468696;
        double r468705 = r468703 / r468704;
        double r468706 = r468693 ? r468702 : r468705;
        return r468706;
}

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

Original7.8
Target0.5
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.560829197112517e-15 or 4.489894380771519e-96 < z

    1. Initial program 10.5

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

    2. Simplified10.5

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

    4. Applied div-inv10.6

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

    5. Applied associate-/l*0.9

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

    6. Simplified0.8

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

    8. Applied add-cube-cbrt0.9

      \[\leadsto \frac{y}{z \cdot \frac{x}{\color{blue}{\left(\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}\right) \cdot \sqrt[3]{\cosh x}}}}\]

    9. Applied associate-/r*0.8

      \[\leadsto \frac{y}{z \cdot \color{blue}{\frac{\frac{x}{\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}}}{\sqrt[3]{\cosh x}}}}\]

    if -2.560829197112517e-15 < z < 4.489894380771519e-96

    1. Initial program 0.3

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

    2. Simplified0.3

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

    4. Applied div-inv0.4

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

    5. Applied associate-/l*23.9

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

    6. Simplified23.8

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

    8. Applied associate-/r*0.3

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{\frac{x}{\cosh x}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.56082919711251703348519476677496253443 \cdot 10^{-15} \lor \neg \left(z \le 4.489894380771518759583953311198839017656 \cdot 10^{-96}\right):\\ \;\;\;\;\frac{y}{\frac{\frac{x}{\sqrt[3]{\cosh x} \cdot \sqrt[3]{\cosh x}}}{\sqrt[3]{\cosh x}} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{x}{\cosh x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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))