Average Error: 7.3 → 0.3
Time: 42.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -0.17773348789390944:\\ \;\;\;\;\frac{y}{\frac{z}{\cosh x} \cdot x}\\ \mathbf{elif}\;z \le 7.165052739410651 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{e^{-x} + e^{x}}{2 \cdot \frac{x}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\cosh x} \cdot x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -0.17773348789390944:\\
\;\;\;\;\frac{y}{\frac{z}{\cosh x} \cdot x}\\

\mathbf{elif}\;z \le 7.165052739410651 \cdot 10^{-33}:\\
\;\;\;\;\frac{\frac{e^{-x} + e^{x}}{2 \cdot \frac{x}{y}}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r17244296 = x;
        double r17244297 = cosh(r17244296);
        double r17244298 = y;
        double r17244299 = r17244298 / r17244296;
        double r17244300 = r17244297 * r17244299;
        double r17244301 = z;
        double r17244302 = r17244300 / r17244301;
        return r17244302;
}
↓
double f(double x, double y, double z) {
        double r17244303 = z;
        double r17244304 = -0.17773348789390944;
        bool r17244305 = r17244303 <= r17244304;
        double r17244306 = y;
        double r17244307 = x;
        double r17244308 = cosh(r17244307);
        double r17244309 = r17244303 / r17244308;
        double r17244310 = r17244309 * r17244307;
        double r17244311 = r17244306 / r17244310;
        double r17244312 = 7.165052739410651e-33;
        bool r17244313 = r17244303 <= r17244312;
        double r17244314 = -r17244307;
        double r17244315 = exp(r17244314);
        double r17244316 = exp(r17244307);
        double r17244317 = r17244315 + r17244316;
        double r17244318 = 2.0;
        double r17244319 = r17244307 / r17244306;
        double r17244320 = r17244318 * r17244319;
        double r17244321 = r17244317 / r17244320;
        double r17244322 = r17244321 / r17244303;
        double r17244323 = r17244313 ? r17244322 : r17244311;
        double r17244324 = r17244305 ? r17244311 : r17244323;
        return r17244324;
}

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
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -0.17773348789390944 or 7.165052739410651e-33 < z

    1. Initial program 10.8

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

    3. Applied *-commutative10.8

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

    4. Applied associate-/l*10.8

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

    6. Applied associate-/l/0.3

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

    if -0.17773348789390944 < z < 7.165052739410651e-33

    1. Initial program 0.3

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

    3. Applied clear-num0.4

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

    4. Applied cosh-def0.4

      \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{1}{\frac{x}{y}}}{z}\]

    5. Applied frac-times0.4

      \[\leadsto \frac{\color{blue}{\frac{\left(e^{x} + e^{-x}\right) \cdot 1}{2 \cdot \frac{x}{y}}}}{z}\]

    6. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -0.17773348789390944:\\ \;\;\;\;\frac{y}{\frac{z}{\cosh x} \cdot x}\\ \mathbf{elif}\;z \le 7.165052739410651 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{e^{-x} + e^{x}}{2 \cdot \frac{x}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\cosh x} \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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))