Average Error: 7.9 → 0.5
Time: 25.2s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -15778803037148576273221211758983643136:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;z \le 2.108633642705728546500571940335929006581 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{\frac{x}{y} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -15778803037148576273221211758983643136:\\
\;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\

\mathbf{elif}\;z \le 2.108633642705728546500571940335929006581 \cdot 10^{-48}:\\
\;\;\;\;\frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{\frac{x}{y} \cdot z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r16607245 = x;
        double r16607246 = cosh(r16607245);
        double r16607247 = y;
        double r16607248 = r16607247 / r16607245;
        double r16607249 = r16607246 * r16607248;
        double r16607250 = z;
        double r16607251 = r16607249 / r16607250;
        return r16607251;
}
↓
double f(double x, double y, double z) {
        double r16607252 = z;
        double r16607253 = -1.5778803037148576e+37;
        bool r16607254 = r16607252 <= r16607253;
        double r16607255 = x;
        double r16607256 = cosh(r16607255);
        double r16607257 = y;
        double r16607258 = r16607256 * r16607257;
        double r16607259 = r16607255 * r16607252;
        double r16607260 = r16607258 / r16607259;
        double r16607261 = 2.1086336427057285e-48;
        bool r16607262 = r16607252 <= r16607261;
        double r16607263 = exp(r16607255);
        double r16607264 = 0.5;
        double r16607265 = r16607264 / r16607263;
        double r16607266 = fma(r16607263, r16607264, r16607265);
        double r16607267 = r16607255 / r16607257;
        double r16607268 = r16607267 * r16607252;
        double r16607269 = r16607266 / r16607268;
        double r16607270 = r16607262 ? r16607269 : r16607260;
        double r16607271 = r16607254 ? r16607260 : r16607270;
        return r16607271;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.9
Target0.4
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 z < -1.5778803037148576e+37 or 2.1086336427057285e-48 < z

    1. Initial program 12.0

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

    3. Applied associate-*r/12.0

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

    4. Applied associate-/l/0.5

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

    if -1.5778803037148576e+37 < z < 2.1086336427057285e-48

    1. Initial program 0.4

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.5

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{\frac{x}{y}}}}{z}\]
    4. Using strategy rm

    5. Applied associate-/l/0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -15778803037148576273221211758983643136:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;z \le 2.108633642705728546500571940335929006581 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{x}, \frac{1}{2}, \frac{\frac{1}{2}}{e^{x}}\right)}{\frac{x}{y} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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))