Average Error: 7.2 → 0.4
Time: 4.0s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -3.58400891098028717 \cdot 10^{108}:\\ \;\;\;\;\frac{1}{\frac{z}{\cosh x \cdot y} \cdot x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.5273947602591855 \cdot 10^{302}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{1} \cdot \frac{y}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -3.58400891098028717 \cdot 10^{108}:\\
\;\;\;\;\frac{1}{\frac{z}{\cosh x \cdot y} \cdot x}\\

\mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.5273947602591855 \cdot 10^{302}:\\
\;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\cosh x}{1} \cdot \frac{y}{x \cdot z}\\

\end{array}
double f(double x, double y, double z) {
        double r395321 = x;
        double r395322 = cosh(r395321);
        double r395323 = y;
        double r395324 = r395323 / r395321;
        double r395325 = r395322 * r395324;
        double r395326 = z;
        double r395327 = r395325 / r395326;
        return r395327;
}
↓
double f(double x, double y, double z) {
        double r395328 = x;
        double r395329 = cosh(r395328);
        double r395330 = y;
        double r395331 = r395330 / r395328;
        double r395332 = r395329 * r395331;
        double r395333 = z;
        double r395334 = r395332 / r395333;
        double r395335 = -3.584008910980287e+108;
        bool r395336 = r395334 <= r395335;
        double r395337 = 1.0;
        double r395338 = r395329 * r395330;
        double r395339 = r395333 / r395338;
        double r395340 = r395339 * r395328;
        double r395341 = r395337 / r395340;
        double r395342 = 1.5273947602591855e+302;
        bool r395343 = r395334 <= r395342;
        double r395344 = r395329 / r395337;
        double r395345 = r395328 * r395333;
        double r395346 = r395330 / r395345;
        double r395347 = r395344 * r395346;
        double r395348 = r395343 ? r395334 : r395347;
        double r395349 = r395336 ? r395341 : r395348;
        return r395349;
}

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

Original7.2
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 (/ (* (cosh x) (/ y x)) z) < -3.584008910980287e+108

    1. Initial program 19.0

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

    3. Applied div-inv19.1

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

    4. Applied associate-*r*19.1

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

    6. Applied clear-num19.1

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

    7. Simplified0.6

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

    if -3.584008910980287e+108 < (/ (* (cosh x) (/ y x)) z) < 1.5273947602591855e+302

    1. Initial program 0.2

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

    3. Applied div-inv0.3

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

    4. Applied associate-*r*0.3

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

    6. Applied associate-*l*0.3

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

    7. Simplified0.2

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

    if 1.5273947602591855e+302 < (/ (* (cosh x) (/ y x)) z)

    1. Initial program 60.5

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

    3. Applied div-inv60.5

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

    4. Applied associate-*r*60.5

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

    6. Applied clear-num60.5

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

    7. Simplified0.8

      \[\leadsto \frac{1}{\color{blue}{\frac{z}{\cosh x \cdot y} \cdot x}}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt0.8

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{z}{\cosh x \cdot y} \cdot x}\]

    10. Applied times-frac1.0

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{z}{\cosh x \cdot y}} \cdot \frac{\sqrt[3]{1}}{x}}\]

    11. Simplified1.0

      \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z}} \cdot \frac{\sqrt[3]{1}}{x}\]

    12. Simplified1.0

      \[\leadsto \frac{\cosh x \cdot y}{z} \cdot \color{blue}{\frac{1}{x}}\]
    13. Using strategy rm

    14. Applied *-un-lft-identity1.0

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

    15. Applied times-frac1.0

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

    16. Applied associate-*l*0.8

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

    17. Simplified2.2

      \[\leadsto \frac{\cosh x}{1} \cdot \color{blue}{\frac{y}{x \cdot z}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -3.58400891098028717 \cdot 10^{108}:\\ \;\;\;\;\frac{1}{\frac{z}{\cosh x \cdot y} \cdot x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.5273947602591855 \cdot 10^{302}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{1} \cdot \frac{y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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))