Average Error: 7.8 → 0.4
Time: 3.6s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -5.71438945780871019 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{z}}{x} \cdot y\\ \mathbf{elif}\;y \le 2.3191721167167056 \cdot 10^{-29}:\\ \;\;\;\;\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -5.71438945780871019 \cdot 10^{-5}:\\
\;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{z}}{x} \cdot y\\

\mathbf{elif}\;y \le 2.3191721167167056 \cdot 10^{-29}:\\
\;\;\;\;\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r573724 = x;
        double r573725 = cosh(r573724);
        double r573726 = y;
        double r573727 = r573726 / r573724;
        double r573728 = r573725 * r573727;
        double r573729 = z;
        double r573730 = r573728 / r573729;
        return r573730;
}
↓
double f(double x, double y, double z) {
        double r573731 = y;
        double r573732 = -5.71438945780871e-05;
        bool r573733 = r573731 <= r573732;
        double r573734 = 0.5;
        double r573735 = -1.0;
        double r573736 = x;
        double r573737 = r573735 * r573736;
        double r573738 = exp(r573737);
        double r573739 = exp(r573736);
        double r573740 = r573738 + r573739;
        double r573741 = r573734 * r573740;
        double r573742 = z;
        double r573743 = r573741 / r573742;
        double r573744 = r573743 / r573736;
        double r573745 = r573744 * r573731;
        double r573746 = 2.3191721167167056e-29;
        bool r573747 = r573731 <= r573746;
        double r573748 = cosh(r573736);
        double r573749 = r573731 / r573736;
        double r573750 = r573748 * r573749;
        double r573751 = 1.0;
        double r573752 = r573751 / r573742;
        double r573753 = r573750 * r573752;
        double r573754 = r573748 / r573736;
        double r573755 = r573731 / r573742;
        double r573756 = r573754 * r573755;
        double r573757 = r573747 ? r573753 : r573756;
        double r573758 = r573733 ? r573745 : r573757;
        return r573758;
}

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

Original7.8
Target0.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -5.71438945780871e-05

    1. Initial program 21.5

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

    5. Applied associate-/r/0.4

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

    7. Applied associate-/r*0.4

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

    if -5.71438945780871e-05 < y < 2.3191721167167056e-29

    1. Initial program 0.2

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

    3. Applied div-inv0.3

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

    if 2.3191721167167056e-29 < y

    1. Initial program 18.6

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

    3. Applied *-un-lft-identity18.6

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

    4. Applied times-frac18.6

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

    5. Simplified18.6

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

    6. Simplified0.4

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

    8. Applied *-un-lft-identity0.4

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

    9. Applied times-frac0.6

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

    10. Applied associate-*r*0.6

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

    11. Simplified0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.71438945780871019 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{z}}{x} \cdot y\\ \mathbf{elif}\;y \le 2.3191721167167056 \cdot 10^{-29}:\\ \;\;\;\;\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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))