Average Error: 7.8 → 0.4
Time: 15.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -5.260389744678724087641010398547091919543 \cdot 10^{-15}:\\ \;\;\;\;\frac{y \cdot \frac{\frac{1}{2} \cdot e^{x} + \frac{\frac{1}{2}}{e^{x}}}{z}}{x}\\ \mathbf{elif}\;y \le 1.743933658318041395875089693172887195128 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(x \cdot 2\right)}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -5.260389744678724087641010398547091919543 \cdot 10^{-15}:\\
\;\;\;\;\frac{y \cdot \frac{\frac{1}{2} \cdot e^{x} + \frac{\frac{1}{2}}{e^{x}}}{z}}{x}\\

\mathbf{elif}\;y \le 1.743933658318041395875089693172887195128 \cdot 10^{-58}:\\
\;\;\;\;\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{x}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(x \cdot 2\right)}\\

\end{array}
double f(double x, double y, double z) {
        double r26089881 = x;
        double r26089882 = cosh(r26089881);
        double r26089883 = y;
        double r26089884 = r26089883 / r26089881;
        double r26089885 = r26089882 * r26089884;
        double r26089886 = z;
        double r26089887 = r26089885 / r26089886;
        return r26089887;
}
↓
double f(double x, double y, double z) {
        double r26089888 = y;
        double r26089889 = -5.260389744678724e-15;
        bool r26089890 = r26089888 <= r26089889;
        double r26089891 = 0.5;
        double r26089892 = x;
        double r26089893 = exp(r26089892);
        double r26089894 = r26089891 * r26089893;
        double r26089895 = r26089891 / r26089893;
        double r26089896 = r26089894 + r26089895;
        double r26089897 = z;
        double r26089898 = r26089896 / r26089897;
        double r26089899 = r26089888 * r26089898;
        double r26089900 = r26089899 / r26089892;
        double r26089901 = 1.7439336583180414e-58;
        bool r26089902 = r26089888 <= r26089901;
        double r26089903 = cosh(r26089892);
        double r26089904 = r26089903 * r26089888;
        double r26089905 = 1.0;
        double r26089906 = r26089905 / r26089892;
        double r26089907 = r26089904 * r26089906;
        double r26089908 = r26089907 / r26089897;
        double r26089909 = -r26089892;
        double r26089910 = exp(r26089909);
        double r26089911 = r26089893 + r26089910;
        double r26089912 = r26089911 * r26089888;
        double r26089913 = 2.0;
        double r26089914 = r26089892 * r26089913;
        double r26089915 = r26089897 * r26089914;
        double r26089916 = r26089912 / r26089915;
        double r26089917 = r26089902 ? r26089908 : r26089916;
        double r26089918 = r26089890 ? r26089900 : r26089917;
        return r26089918;
}

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

Derivation

  1. Split input into 3 regimes

  2. if y < -5.260389744678724e-15

    1. Initial program 20.2

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

      \[\leadsto \color{blue}{\frac{y \cdot \frac{\frac{\frac{1}{2}}{e^{x}} + \frac{1}{2} \cdot e^{x}}{z}}{x}}\]

    if -5.260389744678724e-15 < y < 1.7439336583180414e-58

    1. Initial program 0.2

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

    3. Applied div-inv0.3

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

    4. Applied associate-*r*0.3

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

    if 1.7439336583180414e-58 < y

    1. Initial program 16.7

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

    3. Applied cosh-def16.7

      \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}} \cdot \frac{y}{x}}{z}\]

    4. Applied frac-times16.7

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

    5. Applied associate-/l/0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.260389744678724087641010398547091919543 \cdot 10^{-15}:\\ \;\;\;\;\frac{y \cdot \frac{\frac{1}{2} \cdot e^{x} + \frac{\frac{1}{2}}{e^{x}}}{z}}{x}\\ \mathbf{elif}\;y \le 1.743933658318041395875089693172887195128 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z \cdot \left(x \cdot 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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))