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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.87988751982939395 \cdot 10^{-85} \lor \neg \left(y \le 2.9332543178834714 \cdot 10^{-108}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{1}{2}, e^{-1 \cdot x}, \frac{1}{2} \cdot e^{x}\right)}{\frac{x}{y}}}{z}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;y \le -1.87988751982939395 \cdot 10^{-85} \lor \neg \left(y \le 2.9332543178834714 \cdot 10^{-108}\right):\\
\;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\

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

\end{array}

double f(double x, double y, double z) {
        double r615166 = x;
        double r615167 = cosh(r615166);
        double r615168 = y;
        double r615169 = r615168 / r615166;
        double r615170 = r615167 * r615169;
        double r615171 = z;
        double r615172 = r615170 / r615171;
        return r615172;
}

↓

double f(double x, double y, double z) {
        double r615173 = y;
        double r615174 = -1.879887519829394e-85;
        bool r615175 = r615173 <= r615174;
        double r615176 = 2.9332543178834714e-108;
        bool r615177 = r615173 <= r615176;
        double r615178 = !r615177;
        bool r615179 = r615175 || r615178;
        double r615180 = x;
        double r615181 = cosh(r615180);
        double r615182 = z;
        double r615183 = r615180 * r615182;
        double r615184 = r615173 / r615183;
        double r615185 = r615181 * r615184;
        double r615186 = 0.5;
        double r615187 = -1.0;
        double r615188 = r615187 * r615180;
        double r615189 = exp(r615188);
        double r615190 = exp(r615180);
        double r615191 = r615186 * r615190;
        double r615192 = fma(r615186, r615189, r615191);
        double r615193 = r615180 / r615173;
        double r615194 = r615192 / r615193;
        double r615195 = r615194 / r615182;
        double r615196 = r615179 ? r615185 : r615195;
        return r615196;
}

Original	7.8
Target	0.4
Herbie	1.1

Derivation

Split input into 2 regimes
if y < -1.879887519829394e-85 or 2.9332543178834714e-108 < y
1. Initial program 14.4
  \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
2. Using strategy rm
3. Applied *-un-lft-identity14.4
  \[\leadsto \frac{\cosh x \cdot \frac{y}{x}}{\color{blue}{1 \cdot z}}\]
4. Applied times-frac14.3
  \[\leadsto \color{blue}{\frac{\cosh x}{1} \cdot \frac{\frac{y}{x}}{z}}\]
5. Simplified14.3
  \[\leadsto \color{blue}{\cosh x} \cdot \frac{\frac{y}{x}}{z}\]
6. Simplified1.7
  \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}}\]
if -1.879887519829394e-85 < y < 2.9332543178834714e-108
1. Initial program 0.3
  \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
2. Taylor expanded around inf 0.3
  \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot e^{x} + \frac{1}{2} \cdot e^{-x}\right)}{x}}}{z}\]
3. Simplified0.4
  \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(\frac{1}{2}, e^{-1 \cdot x}, \frac{1}{2} \cdot e^{x}\right)}{\frac{x}{y}}}}{z}\]
Recombined 2 regimes into one program.
Final simplification1.1
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.87988751982939395 \cdot 10^{-85} \lor \neg \left(y \le 2.9332543178834714 \cdot 10^{-108}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{1}{2}, e^{-1 \cdot x}, \frac{1}{2} \cdot e^{x}\right)}{\frac{x}{y}}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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))

Error

Target

Derivation

`if y < -1.879887519829394e-85 or 2.9332543178834714e-108 < y`

`if -1.879887519829394e-85 < y < 2.9332543178834714e-108`

Reproduce