Average Error: 8.0 → 0.5
Time: 5.2s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.74391345329689551 \cdot 10^{-32}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;z \le 1.76815495392722571 \cdot 10^{26}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{z \cdot x}{y}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.74391345329689551 \cdot 10^{-32}:\\
\;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\

\mathbf{elif}\;z \le 1.76815495392722571 \cdot 10^{26}:\\
\;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r537157 = x;
        double r537158 = cosh(r537157);
        double r537159 = y;
        double r537160 = r537159 / r537157;
        double r537161 = r537158 * r537160;
        double r537162 = z;
        double r537163 = r537161 / r537162;
        return r537163;
}
↓
double f(double x, double y, double z) {
        double r537164 = z;
        double r537165 = -2.7439134532968955e-32;
        bool r537166 = r537164 <= r537165;
        double r537167 = x;
        double r537168 = cosh(r537167);
        double r537169 = y;
        double r537170 = r537167 * r537164;
        double r537171 = r537169 / r537170;
        double r537172 = r537168 * r537171;
        double r537173 = 1.7681549539272257e+26;
        bool r537174 = r537164 <= r537173;
        double r537175 = exp(r537167);
        double r537176 = -r537167;
        double r537177 = exp(r537176);
        double r537178 = r537175 + r537177;
        double r537179 = r537178 * r537169;
        double r537180 = r537179 / r537164;
        double r537181 = 2.0;
        double r537182 = r537181 * r537167;
        double r537183 = r537180 / r537182;
        double r537184 = 0.5;
        double r537185 = -1.0;
        double r537186 = r537185 * r537167;
        double r537187 = exp(r537186);
        double r537188 = r537187 + r537175;
        double r537189 = r537184 * r537188;
        double r537190 = r537164 * r537167;
        double r537191 = r537190 / r537169;
        double r537192 = r537189 / r537191;
        double r537193 = r537174 ? r537183 : r537192;
        double r537194 = r537166 ? r537172 : r537193;
        return r537194;
}

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
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.7439134532968955e-32

    1. Initial program 11.3

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

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

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

    4. Applied times-frac11.2

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

    5. Simplified11.2

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

    6. Simplified0.3

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

    if -2.7439134532968955e-32 < z < 1.7681549539272257e+26

    1. Initial program 0.4

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

    3. Applied div-inv0.5

      \[\leadsto \color{blue}{\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}}\]
    4. Using strategy rm

    5. Applied cosh-def0.5

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

    6. Applied frac-times0.5

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

    7. Applied associate-*l/0.4

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

    8. Simplified0.3

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

    if 1.7681549539272257e+26 < z

    1. Initial program 12.8

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.74391345329689551 \cdot 10^{-32}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{elif}\;z \le 1.76815495392722571 \cdot 10^{26}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{\frac{z \cdot x}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))