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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.139482297119390755310250707982414236905 \cdot 10^{-45}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;z \le 1.084913602950906459652906504493151559031 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\frac{z}{y}}, \frac{1}{2}, \frac{\frac{y}{x}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;z \le -3.139482297119390755310250707982414236905 \cdot 10^{-45}:\\
\;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\

\mathbf{elif}\;z \le 1.084913602950906459652906504493151559031 \cdot 10^{-11}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{\frac{z}{y}}, \frac{1}{2}, \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 r23381931 = x;
        double r23381932 = cosh(r23381931);
        double r23381933 = y;
        double r23381934 = r23381933 / r23381931;
        double r23381935 = r23381932 * r23381934;
        double r23381936 = z;
        double r23381937 = r23381935 / r23381936;
        return r23381937;
}

↓

double f(double x, double y, double z) {
        double r23381938 = z;
        double r23381939 = -3.139482297119391e-45;
        bool r23381940 = r23381938 <= r23381939;
        double r23381941 = x;
        double r23381942 = cosh(r23381941);
        double r23381943 = y;
        double r23381944 = r23381942 * r23381943;
        double r23381945 = r23381941 * r23381938;
        double r23381946 = r23381944 / r23381945;
        double r23381947 = 1.0849136029509065e-11;
        bool r23381948 = r23381938 <= r23381947;
        double r23381949 = r23381938 / r23381943;
        double r23381950 = r23381941 / r23381949;
        double r23381951 = 0.5;
        double r23381952 = r23381943 / r23381941;
        double r23381953 = r23381952 / r23381938;
        double r23381954 = fma(r23381950, r23381951, r23381953);
        double r23381955 = r23381948 ? r23381954 : r23381946;
        double r23381956 = r23381940 ? r23381946 : r23381955;
        return r23381956;
}

Original	8.0
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 11.4
  \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
2. Using strategy rm
3. Applied associate-*r/11.4
  \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z}\]
4. Applied associate-/l/0.4
  \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}}\]
if -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 22.6
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]
3. Simplified1.3
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\frac{z}{y}}, \frac{1}{2}, \frac{\frac{y}{x}}{z}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.6
\[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.139482297119390755310250707982414236905 \cdot 10^{-45}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;z \le 1.084913602950906459652906504493151559031 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\frac{z}{y}}, \frac{1}{2}, \frac{\frac{y}{x}}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

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