Average Error: 7.5 → 0.4
Time: 25.0s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.499151431129345690915331400810285626293 \cdot 10^{-22} \lor \neg \left(z \le 3.123371243102039260157787505142381731931 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{y}{\frac{z \cdot x}{\left(e^{x} + e^{-x}\right) \cdot \frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{e^{x} + e^{-x}}}}{\frac{x}{\frac{1}{2}}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.499151431129345690915331400810285626293 \cdot 10^{-22} \lor \neg \left(z \le 3.123371243102039260157787505142381731931 \cdot 10^{-43}\right):\\
\;\;\;\;\frac{y}{\frac{z \cdot x}{\left(e^{x} + e^{-x}\right) \cdot \frac{1}{2}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{y}{\frac{z}{e^{x} + e^{-x}}}}{\frac{x}{\frac{1}{2}}}\\

\end{array}
double f(double x, double y, double z) {
        double r384061 = x;
        double r384062 = cosh(r384061);
        double r384063 = y;
        double r384064 = r384063 / r384061;
        double r384065 = r384062 * r384064;
        double r384066 = z;
        double r384067 = r384065 / r384066;
        return r384067;
}
↓
double f(double x, double y, double z) {
        double r384068 = z;
        double r384069 = -7.499151431129346e-22;
        bool r384070 = r384068 <= r384069;
        double r384071 = 3.1233712431020393e-43;
        bool r384072 = r384068 <= r384071;
        double r384073 = !r384072;
        bool r384074 = r384070 || r384073;
        double r384075 = y;
        double r384076 = x;
        double r384077 = r384068 * r384076;
        double r384078 = exp(r384076);
        double r384079 = -r384076;
        double r384080 = exp(r384079);
        double r384081 = r384078 + r384080;
        double r384082 = 0.5;
        double r384083 = r384081 * r384082;
        double r384084 = r384077 / r384083;
        double r384085 = r384075 / r384084;
        double r384086 = r384068 / r384081;
        double r384087 = r384075 / r384086;
        double r384088 = r384076 / r384082;
        double r384089 = r384087 / r384088;
        double r384090 = r384074 ? r384085 : r384089;
        return r384090;
}

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

Original7.5
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 2 regimes

  2. if z < -7.499151431129346e-22 or 3.1233712431020393e-43 < z

    1. Initial program 10.7

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

    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

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

    if -7.499151431129346e-22 < z < 3.1233712431020393e-43

    1. Initial program 0.3

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

    2. Taylor expanded around inf 22.1

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

    3. Simplified22.1

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

    5. Applied times-frac22.1

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

    6. Applied associate-/r*0.3

      \[\leadsto \color{blue}{\frac{\frac{y}{\frac{z}{e^{x} + e^{-x}}}}{\frac{x}{\frac{1}{2}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.499151431129345690915331400810285626293 \cdot 10^{-22} \lor \neg \left(z \le 3.123371243102039260157787505142381731931 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{y}{\frac{z \cdot x}{\left(e^{x} + e^{-x}\right) \cdot \frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{e^{x} + e^{-x}}}}{\frac{x}{\frac{1}{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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