Average Error: 8.0 → 0.4
Time: 7.2s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3775038362858526700 \lor \neg \left(z \le 7.25860780760428 \cdot 10^{-28}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3775038362858526700 \lor \neg \left(z \le 7.25860780760428 \cdot 10^{-28}\right):\\
\;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}\\

\end{array}
double f(double x, double y, double z) {
        double r468216 = x;
        double r468217 = cosh(r468216);
        double r468218 = y;
        double r468219 = r468218 / r468216;
        double r468220 = r468217 * r468219;
        double r468221 = z;
        double r468222 = r468220 / r468221;
        return r468222;
}
↓
double f(double x, double y, double z) {
        double r468223 = z;
        double r468224 = -3.7750383628585267e+18;
        bool r468225 = r468223 <= r468224;
        double r468226 = 7.25860780760428e-28;
        bool r468227 = r468223 <= r468226;
        double r468228 = !r468227;
        bool r468229 = r468225 || r468228;
        double r468230 = x;
        double r468231 = cosh(r468230);
        double r468232 = y;
        double r468233 = r468230 * r468223;
        double r468234 = r468232 / r468233;
        double r468235 = r468231 * r468234;
        double r468236 = r468232 / r468230;
        double r468237 = r468231 * r468236;
        double r468238 = 1.0;
        double r468239 = r468238 / r468223;
        double r468240 = r468237 * r468239;
        double r468241 = r468229 ? r468235 : r468240;
        return r468241;
}

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

Original8.0
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 2 regimes

  2. if z < -3.7750383628585267e+18 or 7.25860780760428e-28 < z

    1. Initial program 11.9

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

    3. Applied *-un-lft-identity11.9

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

    4. Applied times-frac11.9

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

    5. Simplified11.9

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

    6. Simplified0.3

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

    if -3.7750383628585267e+18 < z < 7.25860780760428e-28

    1. Initial program 0.4

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

    3. Applied div-inv0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3775038362858526700 \lor \neg \left(z \le 7.25860780760428 \cdot 10^{-28}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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))