Average Error: 7.3 → 0.9
Time: 4.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.8438321628099627 \cdot 10^{84} \lor \neg \left(z \le 1.4502045605748618 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{y \cdot \left(e^{x} + e^{-x}\right)}{x \cdot \left(z \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\cosh x}}{z} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x}\right)\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -4.8438321628099627 \cdot 10^{84} \lor \neg \left(z \le 1.4502045605748618 \cdot 10^{-55}\right):\\
\;\;\;\;\frac{y \cdot \left(e^{x} + e^{-x}\right)}{x \cdot \left(z \cdot 2\right)}\\

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

\end{array}
double f(double x, double y, double z) {
        double r438206 = x;
        double r438207 = cosh(r438206);
        double r438208 = y;
        double r438209 = r438208 / r438206;
        double r438210 = r438207 * r438209;
        double r438211 = z;
        double r438212 = r438210 / r438211;
        return r438212;
}
↓
double f(double x, double y, double z) {
        double r438213 = z;
        double r438214 = -4.843832162809963e+84;
        bool r438215 = r438213 <= r438214;
        double r438216 = 1.4502045605748618e-55;
        bool r438217 = r438213 <= r438216;
        double r438218 = !r438217;
        bool r438219 = r438215 || r438218;
        double r438220 = y;
        double r438221 = x;
        double r438222 = exp(r438221);
        double r438223 = -r438221;
        double r438224 = exp(r438223);
        double r438225 = r438222 + r438224;
        double r438226 = r438220 * r438225;
        double r438227 = 2.0;
        double r438228 = r438213 * r438227;
        double r438229 = r438221 * r438228;
        double r438230 = r438226 / r438229;
        double r438231 = cosh(r438221);
        double r438232 = sqrt(r438231);
        double r438233 = r438232 / r438213;
        double r438234 = r438220 / r438221;
        double r438235 = r438232 * r438234;
        double r438236 = r438233 * r438235;
        double r438237 = r438219 ? r438230 : r438236;
        return r438237;
}

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

Original7.3
Target0.4
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.843832162809963e+84 or 1.4502045605748618e-55 < z

    1. Initial program 11.3

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

    3. Applied associate-/l*11.7

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

    5. Applied associate-/r/11.3

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

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

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

    7. Applied times-frac11.4

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

    8. Simplified10.5

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

    10. Applied associate-*r/10.4

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

    12. Applied cosh-def10.4

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

    13. Applied frac-times10.5

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

    14. Applied associate-/l/0.6

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

    if -4.843832162809963e+84 < z < 1.4502045605748618e-55

    1. Initial program 1.3

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

    3. Applied associate-/l*1.4

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

    5. Applied div-inv1.5

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

    6. Applied add-sqr-sqrt1.5

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

    7. Applied times-frac1.5

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

    8. Simplified1.5

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.8438321628099627 \cdot 10^{84} \lor \neg \left(z \le 1.4502045605748618 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{y \cdot \left(e^{x} + e^{-x}\right)}{x \cdot \left(z \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\cosh x}}{z} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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))