Average Error: 8.0 → 0.5
Time: 14.7s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -280311605560231821312:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \mathbf{elif}\;y \le 3.959780483839766565551118859398086167973 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(y \cdot \cosh x\right) \cdot \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -280311605560231821312:\\
\;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\

\mathbf{elif}\;y \le 3.959780483839766565551118859398086167973 \cdot 10^{-53}:\\
\;\;\;\;\frac{\left(y \cdot \cosh x\right) \cdot \frac{1}{x}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r17594219 = x;
        double r17594220 = cosh(r17594219);
        double r17594221 = y;
        double r17594222 = r17594221 / r17594219;
        double r17594223 = r17594220 * r17594222;
        double r17594224 = z;
        double r17594225 = r17594223 / r17594224;
        return r17594225;
}
↓
double f(double x, double y, double z) {
        double r17594226 = y;
        double r17594227 = -2.8031160556023182e+20;
        bool r17594228 = r17594226 <= r17594227;
        double r17594229 = z;
        double r17594230 = r17594226 / r17594229;
        double r17594231 = x;
        double r17594232 = exp(r17594231);
        double r17594233 = 0.5;
        double r17594234 = r17594233 / r17594232;
        double r17594235 = fma(r17594232, r17594233, r17594234);
        double r17594236 = r17594230 * r17594235;
        double r17594237 = r17594236 / r17594231;
        double r17594238 = 3.9597804838397666e-53;
        bool r17594239 = r17594226 <= r17594238;
        double r17594240 = cosh(r17594231);
        double r17594241 = r17594226 * r17594240;
        double r17594242 = 1.0;
        double r17594243 = r17594242 / r17594231;
        double r17594244 = r17594241 * r17594243;
        double r17594245 = r17594244 / r17594229;
        double r17594246 = r17594239 ? r17594245 : r17594237;
        double r17594247 = r17594228 ? r17594237 : r17594246;
        return r17594247;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original8.0
Target0.5
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -2.8031160556023182e+20 or 3.9597804838397666e-53 < y

    1. Initial program 19.6

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

    3. Applied associate-*r/19.6

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

    4. Applied associate-/l/0.7

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

    5. Taylor expanded around inf 0.7

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

    6. Simplified0.6

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

    if -2.8031160556023182e+20 < y < 3.9597804838397666e-53

    1. Initial program 0.4

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

    3. Applied div-inv0.5

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

    4. Applied associate-*r*0.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -280311605560231821312:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \mathbf{elif}\;y \le 3.959780483839766565551118859398086167973 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(y \cdot \cosh x\right) \cdot \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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))