Average Error: 7.3 → 0.3
Time: 3.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -12403175825184520:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{z}{y}}}{x}\\ \mathbf{elif}\;y \le 4.70989780554481168 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{\cosh x \cdot y}{z}}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -12403175825184520:\\
\;\;\;\;\frac{\frac{\cosh x}{\frac{z}{y}}}{x}\\

\mathbf{elif}\;y \le 4.70989780554481168 \cdot 10^{-7}:\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{x}{\frac{\cosh x \cdot y}{z}}}\\

\end{array}
double f(double x, double y, double z) {
        double r503787 = x;
        double r503788 = cosh(r503787);
        double r503789 = y;
        double r503790 = r503789 / r503787;
        double r503791 = r503788 * r503790;
        double r503792 = z;
        double r503793 = r503791 / r503792;
        return r503793;
}
↓
double f(double x, double y, double z) {
        double r503794 = y;
        double r503795 = -1.240317582518452e+16;
        bool r503796 = r503794 <= r503795;
        double r503797 = x;
        double r503798 = cosh(r503797);
        double r503799 = z;
        double r503800 = r503799 / r503794;
        double r503801 = r503798 / r503800;
        double r503802 = r503801 / r503797;
        double r503803 = 4.7098978055448117e-07;
        bool r503804 = r503794 <= r503803;
        double r503805 = r503798 * r503794;
        double r503806 = r503805 / r503797;
        double r503807 = r503806 / r503799;
        double r503808 = 1.0;
        double r503809 = r503805 / r503799;
        double r503810 = r503797 / r503809;
        double r503811 = r503808 / r503810;
        double r503812 = r503804 ? r503807 : r503811;
        double r503813 = r503796 ? r503802 : r503812;
        return r503813;
}

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

Original7.3
Target0.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.240317582518452e+16

    1. Initial program 22.3

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

    3. Applied associate-*r/22.3

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

    4. Applied associate-/l/0.3

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

    6. Applied associate-/r*0.3

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

    8. Applied associate-/l*0.4

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

    if -1.240317582518452e+16 < y < 4.7098978055448117e-07

    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/11.1

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

    6. Applied associate-/r*10.3

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

    8. Applied clear-num10.4

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

    10. Applied associate-/r/0.7

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\cosh x \cdot y} \cdot z}}\]

    11. Applied associate-/r*0.3

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{x}{\cosh x \cdot y}}}{z}}\]

    12. Simplified0.3

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

    if 4.7098978055448117e-07 < y

    1. Initial program 20.3

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

    3. Applied associate-*r/20.4

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

    4. Applied associate-/l/0.5

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

    6. Applied associate-/r*0.5

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

    8. Applied clear-num0.6

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -12403175825184520:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{z}{y}}}{x}\\ \mathbf{elif}\;y \le 4.70989780554481168 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{\cosh x \cdot y}{z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.0385305359351529e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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