Average Error: 8.1 → 0.7
Time: 34.0s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -8.986365873619643 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{elif}\;y \le 1.0982237141506651 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -8.986365873619643 \cdot 10^{-102}:\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\

\mathbf{elif}\;y \le 1.0982237141506651 \cdot 10^{-11}:\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r22184247 = x;
        double r22184248 = cosh(r22184247);
        double r22184249 = y;
        double r22184250 = r22184249 / r22184247;
        double r22184251 = r22184248 * r22184250;
        double r22184252 = z;
        double r22184253 = r22184251 / r22184252;
        return r22184253;
}
↓
double f(double x, double y, double z) {
        double r22184254 = y;
        double r22184255 = -8.986365873619643e-102;
        bool r22184256 = r22184254 <= r22184255;
        double r22184257 = x;
        double r22184258 = cosh(r22184257);
        double r22184259 = r22184258 * r22184254;
        double r22184260 = z;
        double r22184261 = r22184259 / r22184260;
        double r22184262 = r22184261 / r22184257;
        double r22184263 = 1.0982237141506651e-11;
        bool r22184264 = r22184254 <= r22184263;
        double r22184265 = r22184259 / r22184257;
        double r22184266 = r22184265 / r22184260;
        double r22184267 = r22184257 * r22184260;
        double r22184268 = r22184259 / r22184267;
        double r22184269 = r22184264 ? r22184266 : r22184268;
        double r22184270 = r22184256 ? r22184262 : r22184269;
        return r22184270;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original8.1
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -8.986365873619643e-102

    1. Initial program 14.6

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

    3. Applied associate-*r/14.7

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

    4. Applied associate-/l/1.4

      \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}}\]
    5. Using strategy rm

    6. Applied associate-/r*1.6

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

    if -8.986365873619643e-102 < y < 1.0982237141506651e-11

    1. Initial program 0.3

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

    3. Applied associate-*r/0.3

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

    4. Applied associate-/l/12.2

      \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}}\]
    5. Using strategy rm

    6. Applied associate-/r*11.2

      \[\leadsto \color{blue}{\frac{\frac{\cosh x \cdot y}{z}}{x}}\]
    7. Using strategy rm

    8. Applied div-inv11.3

      \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z} \cdot \frac{1}{x}}\]
    9. Using strategy rm

    10. Applied associate-*l/0.4

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

    11. Simplified0.3

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

    if 1.0982237141506651e-11 < y

    1. Initial program 19.9

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

    3. Applied associate-*r/19.9

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

    4. Applied associate-/l/0.4

      \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -8.986365873619643 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{elif}\;y \le 1.0982237141506651 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +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))