Average Error: 8.0 → 0.4
Time: 6.4s
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 r458201 = x;
        double r458202 = cosh(r458201);
        double r458203 = y;
        double r458204 = r458203 / r458201;
        double r458205 = r458202 * r458204;
        double r458206 = z;
        double r458207 = r458205 / r458206;
        return r458207;
}
↓
double f(double x, double y, double z) {
        double r458208 = z;
        double r458209 = -3.7750383628585267e+18;
        bool r458210 = r458208 <= r458209;
        double r458211 = 7.25860780760428e-28;
        bool r458212 = r458208 <= r458211;
        double r458213 = !r458212;
        bool r458214 = r458210 || r458213;
        double r458215 = x;
        double r458216 = cosh(r458215);
        double r458217 = y;
        double r458218 = r458215 * r458208;
        double r458219 = r458217 / r458218;
        double r458220 = r458216 * r458219;
        double r458221 = r458217 / r458215;
        double r458222 = r458216 * r458221;
        double r458223 = 1.0;
        double r458224 = r458223 / r458208;
        double r458225 = r458222 * r458224;
        double r458226 = r458214 ? r458220 : r458225;
        return r458226;
}

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))