Average Error: 7.7 → 0.7
Time: 14.7s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.93577210121831744 \cdot 10^{80} \lor \neg \left(z \le 2.60686388496247717 \cdot 10^{-42}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\frac{z}{y} \cdot x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.93577210121831744 \cdot 10^{80} \lor \neg \left(z \le 2.60686388496247717 \cdot 10^{-42}\right):\\
\;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r456215 = x;
        double r456216 = cosh(r456215);
        double r456217 = y;
        double r456218 = r456217 / r456215;
        double r456219 = r456216 * r456218;
        double r456220 = z;
        double r456221 = r456219 / r456220;
        return r456221;
}
↓
double f(double x, double y, double z) {
        double r456222 = z;
        double r456223 = -1.9357721012183174e+80;
        bool r456224 = r456222 <= r456223;
        double r456225 = 2.6068638849624772e-42;
        bool r456226 = r456222 <= r456225;
        double r456227 = !r456226;
        bool r456228 = r456224 || r456227;
        double r456229 = x;
        double r456230 = cosh(r456229);
        double r456231 = y;
        double r456232 = r456229 * r456222;
        double r456233 = r456231 / r456232;
        double r456234 = r456230 * r456233;
        double r456235 = r456222 / r456231;
        double r456236 = r456235 * r456229;
        double r456237 = r456230 / r456236;
        double r456238 = r456228 ? r456234 : r456237;
        return r456238;
}

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

Original7.7
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.9357721012183174e+80 or 2.6068638849624772e-42 < z

    1. Initial program 12.0

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

    3. Applied associate-/l*12.4

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

    4. Simplified11.3

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

    6. Applied div-inv11.3

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

    7. Simplified0.4

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

    if -1.9357721012183174e+80 < z < 2.6068638849624772e-42

    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. Simplified1.2

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.93577210121831744 \cdot 10^{80} \lor \neg \left(z \le 2.60686388496247717 \cdot 10^{-42}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\frac{z}{y} \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))