Average Error: 7.6 → 0.7
Time: 5.6s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.3159694493115124 \cdot 10^{88}:\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{elif}\;z \le 3.7180943296009127 \cdot 10^{45}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{x}}{z}}{\frac{1}{y}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.3159694493115124 \cdot 10^{88}:\\
\;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\

\mathbf{elif}\;z \le 3.7180943296009127 \cdot 10^{45}:\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{x}}{z}}{\frac{1}{y}}\\

\end{array}
double f(double x, double y, double z) {
        double r2237 = x;
        double r2238 = cosh(r2237);
        double r2239 = y;
        double r2240 = r2239 / r2237;
        double r2241 = r2238 * r2240;
        double r2242 = z;
        double r2243 = r2241 / r2242;
        return r2243;
}
↓
double f(double x, double y, double z) {
        double r2244 = z;
        double r2245 = -1.3159694493115124e+88;
        bool r2246 = r2244 <= r2245;
        double r2247 = x;
        double r2248 = cosh(r2247);
        double r2249 = y;
        double r2250 = r2248 * r2249;
        double r2251 = r2244 * r2247;
        double r2252 = r2250 / r2251;
        double r2253 = 3.7180943296009127e+45;
        bool r2254 = r2244 <= r2253;
        double r2255 = r2250 / r2244;
        double r2256 = r2255 / r2247;
        double r2257 = 0.5;
        double r2258 = exp(r2247);
        double r2259 = -r2247;
        double r2260 = exp(r2259);
        double r2261 = r2258 + r2260;
        double r2262 = r2257 * r2261;
        double r2263 = r2262 / r2247;
        double r2264 = r2263 / r2244;
        double r2265 = 1.0;
        double r2266 = r2265 / r2249;
        double r2267 = r2264 / r2266;
        double r2268 = r2254 ? r2256 : r2267;
        double r2269 = r2246 ? r2252 : r2268;
        return r2269;
}

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

Original7.6
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 3 regimes

  2. if z < -1.3159694493115124e+88

    1. Initial program 13.4

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

    3. Applied associate-*r/13.4

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

    4. Applied associate-/l/0.4

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

    if -1.3159694493115124e+88 < z < 3.7180943296009127e+45

    1. Initial program 1.6

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

    3. Applied associate-*r/1.6

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

    4. Applied associate-/l/14.0

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

    6. Applied associate-/r*1.1

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

    if 3.7180943296009127e+45 < z

    1. Initial program 12.9

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

    3. Applied associate-*r/13.0

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

    4. Applied associate-/l/0.3

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

    6. Applied associate-/l*0.8

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

    8. Applied div-inv0.9

      \[\leadsto \frac{\cosh x}{\color{blue}{\left(z \cdot x\right) \cdot \frac{1}{y}}}\]

    9. Applied associate-/r*0.5

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

    10. Taylor expanded around inf 0.5

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

    11. Simplified0.4

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.3159694493115124 \cdot 10^{88}:\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{elif}\;z \le 3.7180943296009127 \cdot 10^{45}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{2} \cdot \left(e^{x} + e^{-x}\right)}{x}}{z}}{\frac{1}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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))