Average Error: 7.7 → 0.6
Time: 11.8s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.40953097935408513 \cdot 10^{35} \lor \neg \left(z \le 5.82511916007843312 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right) \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.40953097935408513 \cdot 10^{35} \lor \neg \left(z \le 5.82511916007843312 \cdot 10^{-85}\right):\\
\;\;\;\;\frac{\left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right) \cdot y}{z \cdot x}\\

\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 r573605 = x;
        double r573606 = cosh(r573605);
        double r573607 = y;
        double r573608 = r573607 / r573605;
        double r573609 = r573606 * r573608;
        double r573610 = z;
        double r573611 = r573609 / r573610;
        return r573611;
}
↓
double f(double x, double y, double z) {
        double r573612 = z;
        double r573613 = -1.4095309793540851e+35;
        bool r573614 = r573612 <= r573613;
        double r573615 = 5.825119160078433e-85;
        bool r573616 = r573612 <= r573615;
        double r573617 = !r573616;
        bool r573618 = r573614 || r573617;
        double r573619 = 0.5;
        double r573620 = x;
        double r573621 = exp(r573620);
        double r573622 = -r573620;
        double r573623 = exp(r573622);
        double r573624 = r573621 + r573623;
        double r573625 = r573619 * r573624;
        double r573626 = y;
        double r573627 = r573625 * r573626;
        double r573628 = r573612 * r573620;
        double r573629 = r573627 / r573628;
        double r573630 = r573624 * r573626;
        double r573631 = r573630 / r573612;
        double r573632 = 2.0;
        double r573633 = r573632 * r573620;
        double r573634 = r573631 / r573633;
        double r573635 = r573618 ? r573629 : r573634;
        return r573635;
}

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

Original7.7
Target0.4
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.4095309793540851e+35 or 5.825119160078433e-85 < z

    1. Initial program 11.2

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

    3. Applied div-inv11.2

      \[\leadsto \color{blue}{\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}}\]

    4. Taylor expanded around inf 0.7

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

    5. Simplified0.7

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

    if -1.4095309793540851e+35 < z < 5.825119160078433e-85

    1. Initial program 0.7

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

    3. Applied div-inv0.8

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

    5. Applied cosh-def0.8

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

    6. Applied frac-times0.8

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

    7. Applied associate-*l/0.5

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

    8. Simplified0.4

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.40953097935408513 \cdot 10^{35} \lor \neg \left(z \le 5.82511916007843312 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)\right) \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))