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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.139482297119391 \cdot 10^{-45}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;z \le 1.0849136029509065 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{x}{z} \cdot y, \frac{\frac{y}{x}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;z \le -3.139482297119391 \cdot 10^{-45}:\\
\;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\

\mathbf{elif}\;z \le 1.0849136029509065 \cdot 10^{-11}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{x}{z} \cdot y, \frac{\frac{y}{x}}{z}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\

\end{array}

double f(double x, double y, double z) {
        double r20374088 = x;
        double r20374089 = cosh(r20374088);
        double r20374090 = y;
        double r20374091 = r20374090 / r20374088;
        double r20374092 = r20374089 * r20374091;
        double r20374093 = z;
        double r20374094 = r20374092 / r20374093;
        return r20374094;
}

↓

double f(double x, double y, double z) {
        double r20374095 = z;
        double r20374096 = -3.139482297119391e-45;
        bool r20374097 = r20374095 <= r20374096;
        double r20374098 = x;
        double r20374099 = cosh(r20374098);
        double r20374100 = y;
        double r20374101 = r20374099 * r20374100;
        double r20374102 = r20374098 * r20374095;
        double r20374103 = r20374101 / r20374102;
        double r20374104 = 1.0849136029509065e-11;
        bool r20374105 = r20374095 <= r20374104;
        double r20374106 = 0.5;
        double r20374107 = r20374098 / r20374095;
        double r20374108 = r20374107 * r20374100;
        double r20374109 = r20374100 / r20374098;
        double r20374110 = r20374109 / r20374095;
        double r20374111 = fma(r20374106, r20374108, r20374110);
        double r20374112 = r20374105 ? r20374111 : r20374103;
        double r20374113 = r20374097 ? r20374103 : r20374112;
        return r20374113;
}

Original	7.5
Target	0.4
Herbie	0.6

Derivation

Split input into 2 regimes
if z < -3.139482297119391e-45 or 1.0849136029509065e-11 < z
1. Initial program 10.7
  \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
2. Using strategy rm
3. Applied associate-*r/10.7
  \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z}\]
4. Applied associate-/l/0.4
  \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}}\]
if -3.139482297119391e-45 < z < 1.0849136029509065e-11
1. Initial program 0.3
  \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
2. Taylor expanded around 0 21.3
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]
3. Simplified1.3
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot y}{z}, \frac{1}{2}, \frac{\frac{y}{z}}{x}\right)}\]
4. Taylor expanded around 0 21.3
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]
5. Simplified1.3
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{x}{z} \cdot y, \frac{\frac{y}{x}}{z}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.6
\[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.139482297119391 \cdot 10^{-45}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;z \le 1.0849136029509065 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{x}{z} \cdot y, \frac{\frac{y}{x}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))

Error

Target

Derivation

`if z < -3.139482297119391e-45 or 1.0849136029509065e-11 < z`

`if -3.139482297119391e-45 < z < 1.0849136029509065e-11`

Reproduce