Average Error: 7.3 → 0.9
Time: 5.0s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.8438321628099627 \cdot 10^{84} \lor \neg \left(z \le 1.4502045605748618 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\cosh x}}{z} \cdot \left(\sqrt{\cosh x} \cdot \frac{y}{x}\right)\\ \end{array}\]
\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{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(2 \cdot x\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 r594211 = x;
        double r594212 = cosh(r594211);
        double r594213 = y;
        double r594214 = r594213 / r594211;
        double r594215 = r594212 * r594214;
        double r594216 = z;
        double r594217 = r594215 / r594216;
        return r594217;
}
↓
double f(double x, double y, double z) {
        double r594218 = z;
        double r594219 = -4.843832162809963e+84;
        bool r594220 = r594218 <= r594219;
        double r594221 = 1.4502045605748618e-55;
        bool r594222 = r594218 <= r594221;
        double r594223 = !r594222;
        bool r594224 = r594220 || r594223;
        double r594225 = x;
        double r594226 = exp(r594225);
        double r594227 = -r594225;
        double r594228 = exp(r594227);
        double r594229 = r594226 + r594228;
        double r594230 = y;
        double r594231 = r594229 * r594230;
        double r594232 = 2.0;
        double r594233 = r594232 * r594225;
        double r594234 = r594218 * r594233;
        double r594235 = r594231 / r594234;
        double r594236 = cosh(r594225);
        double r594237 = sqrt(r594236);
        double r594238 = r594237 / r594218;
        double r594239 = r594230 / r594225;
        double r594240 = r594237 * r594239;
        double r594241 = r594238 * r594240;
        double r594242 = r594224 ? r594235 : r594241;
        return r594242;
}

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 cosh-def11.3

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

    4. Applied frac-times11.3

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

    5. Applied associate-/l/0.5

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

    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

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

Reproduce

herbie shell --seed 2020035 
(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))