Average Error: 7.5 → 0.4
Time: 4.2s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -4.359963571427226706906654116630111705967 \cdot 10^{288} \lor \neg \left(\cosh x \cdot \frac{y}{x} \le 4.818059594116706179187911079448660909001 \cdot 10^{190}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{x}}{2 \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -4.359963571427226706906654116630111705967 \cdot 10^{288} \lor \neg \left(\cosh x \cdot \frac{y}{x} \le 4.818059594116706179187911079448660909001 \cdot 10^{190}\right):\\
\;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{x}}{2 \cdot z}\\

\end{array}
double f(double x, double y, double z) {
        double r517403 = x;
        double r517404 = cosh(r517403);
        double r517405 = y;
        double r517406 = r517405 / r517403;
        double r517407 = r517404 * r517406;
        double r517408 = z;
        double r517409 = r517407 / r517408;
        return r517409;
}
↓
double f(double x, double y, double z) {
        double r517410 = x;
        double r517411 = cosh(r517410);
        double r517412 = y;
        double r517413 = r517412 / r517410;
        double r517414 = r517411 * r517413;
        double r517415 = -4.3599635714272267e+288;
        bool r517416 = r517414 <= r517415;
        double r517417 = 4.818059594116706e+190;
        bool r517418 = r517414 <= r517417;
        double r517419 = !r517418;
        bool r517420 = r517416 || r517419;
        double r517421 = z;
        double r517422 = r517410 * r517421;
        double r517423 = r517412 / r517422;
        double r517424 = r517411 * r517423;
        double r517425 = exp(r517410);
        double r517426 = -r517410;
        double r517427 = exp(r517426);
        double r517428 = r517425 + r517427;
        double r517429 = r517428 * r517412;
        double r517430 = r517429 / r517410;
        double r517431 = 2.0;
        double r517432 = r517431 * r517421;
        double r517433 = r517430 / r517432;
        double r517434 = r517420 ? r517424 : r517433;
        return r517434;
}

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

Original7.5
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935152855236908684227749499669 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (cosh x) (/ y x)) < -4.3599635714272267e+288 or 4.818059594116706e+190 < (* (cosh x) (/ y x))

    1. Initial program 35.5

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

    3. Applied *-un-lft-identity35.5

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

    4. Applied times-frac35.5

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

    5. Simplified35.5

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

    6. Simplified0.7

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

    if -4.3599635714272267e+288 < (* (cosh x) (/ y x)) < 4.818059594116706e+190

    1. Initial program 0.3

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

    3. Applied *-un-lft-identity0.3

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

    4. Applied times-frac0.3

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

    5. Simplified0.3

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

    6. Simplified8.8

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

    8. Applied add-cube-cbrt9.7

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

    9. Applied times-frac2.8

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

    10. Applied associate-*r*2.8

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

    12. Applied cosh-def2.8

      \[\leadsto \left(\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{x}\right) \cdot \frac{\sqrt[3]{y}}{z}\]

    13. Applied associate-*l/2.8

      \[\leadsto \color{blue}{\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{x}}{2}} \cdot \frac{\sqrt[3]{y}}{z}\]

    14. Applied frac-times1.4

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x} + e^{-x}\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{x}\right) \cdot \sqrt[3]{y}}{2 \cdot z}}\]

    15. Simplified0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -4.359963571427226706906654116630111705967 \cdot 10^{288} \lor \neg \left(\cosh x \cdot \frac{y}{x} \le 4.818059594116706179187911079448660909001 \cdot 10^{190}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{x}}{2 \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(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))