Average Error: 7.9 → 0.4
Time: 21.3s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.394362921332223351644270586304115137557 \cdot 10^{-33}:\\ \;\;\;\;\frac{y}{z \cdot \frac{x}{\cosh x}}\\ \mathbf{elif}\;z \le 2.035540255879025646360089979447494101805 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{\frac{x}{\cosh x}} \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{x}{\cosh x}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -6.394362921332223351644270586304115137557 \cdot 10^{-33}:\\
\;\;\;\;\frac{y}{z \cdot \frac{x}{\cosh x}}\\

\mathbf{elif}\;z \le 2.035540255879025646360089979447494101805 \cdot 10^{-6}:\\
\;\;\;\;\frac{y}{\frac{x}{\cosh x}} \cdot \frac{1}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r26042325 = x;
        double r26042326 = cosh(r26042325);
        double r26042327 = y;
        double r26042328 = r26042327 / r26042325;
        double r26042329 = r26042326 * r26042328;
        double r26042330 = z;
        double r26042331 = r26042329 / r26042330;
        return r26042331;
}
↓
double f(double x, double y, double z) {
        double r26042332 = z;
        double r26042333 = -6.394362921332223e-33;
        bool r26042334 = r26042332 <= r26042333;
        double r26042335 = y;
        double r26042336 = x;
        double r26042337 = cosh(r26042336);
        double r26042338 = r26042336 / r26042337;
        double r26042339 = r26042332 * r26042338;
        double r26042340 = r26042335 / r26042339;
        double r26042341 = 2.0355402558790256e-06;
        bool r26042342 = r26042332 <= r26042341;
        double r26042343 = r26042335 / r26042338;
        double r26042344 = 1.0;
        double r26042345 = r26042344 / r26042332;
        double r26042346 = r26042343 * r26042345;
        double r26042347 = r26042342 ? r26042346 : r26042340;
        double r26042348 = r26042334 ? r26042340 : r26042347;
        return r26042348;
}

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

Original7.9
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -6.394362921332223e-33 or 2.0355402558790256e-06 < z

    1. Initial program 11.5

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

    3. Applied *-un-lft-identity11.5

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

    4. Applied associate-/r*11.5

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

    5. Simplified11.5

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

    7. Applied associate-/l/0.3

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

    if -6.394362921332223e-33 < z < 2.0355402558790256e-06

    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 associate-/r*0.3

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

    5. Simplified0.3

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

    7. Applied div-inv0.4

      \[\leadsto \color{blue}{\frac{y}{\frac{x}{\cosh x}} \cdot \frac{1}{z}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.394362921332223351644270586304115137557 \cdot 10^{-33}:\\ \;\;\;\;\frac{y}{z \cdot \frac{x}{\cosh x}}\\ \mathbf{elif}\;z \le 2.035540255879025646360089979447494101805 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{\frac{x}{\cosh x}} \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{x}{\cosh x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))