Average Error: 7.7 → 0.4
Time: 4.3s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -739634522879.62769 \lor \neg \left(z \le 1.25575211935324887 \cdot 10^{23}\right):\\ \;\;\;\;\frac{\left(\left(e^{x} + e^{-x}\right) \cdot y\right) \cdot 1}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{x}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -739634522879.62769 \lor \neg \left(z \le 1.25575211935324887 \cdot 10^{23}\right):\\
\;\;\;\;\frac{\left(\left(e^{x} + e^{-x}\right) \cdot y\right) \cdot 1}{z \cdot \left(2 \cdot x\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{x}}{z}\\

\end{array}
double f(double x, double y, double z) {
        double r477042 = x;
        double r477043 = cosh(r477042);
        double r477044 = y;
        double r477045 = r477044 / r477042;
        double r477046 = r477043 * r477045;
        double r477047 = z;
        double r477048 = r477046 / r477047;
        return r477048;
}
↓
double f(double x, double y, double z) {
        double r477049 = z;
        double r477050 = -739634522879.6277;
        bool r477051 = r477049 <= r477050;
        double r477052 = 1.2557521193532489e+23;
        bool r477053 = r477049 <= r477052;
        double r477054 = !r477053;
        bool r477055 = r477051 || r477054;
        double r477056 = x;
        double r477057 = exp(r477056);
        double r477058 = -r477056;
        double r477059 = exp(r477058);
        double r477060 = r477057 + r477059;
        double r477061 = y;
        double r477062 = r477060 * r477061;
        double r477063 = 1.0;
        double r477064 = r477062 * r477063;
        double r477065 = 2.0;
        double r477066 = r477065 * r477056;
        double r477067 = r477049 * r477066;
        double r477068 = r477064 / r477067;
        double r477069 = cosh(r477056);
        double r477070 = r477069 * r477061;
        double r477071 = r477063 / r477056;
        double r477072 = r477070 * r477071;
        double r477073 = r477072 / r477049;
        double r477074 = r477055 ? r477068 : r477073;
        return r477074;
}

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

Original7.7
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 2 regimes

  2. if z < -739634522879.6277 or 1.2557521193532489e+23 < z

    1. Initial program 12.1

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

    3. Applied div-inv12.2

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

    4. Applied associate-*r*12.2

      \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot y\right) \cdot \frac{1}{x}}}{z}\]
    5. Using strategy rm

    6. Applied cosh-def12.2

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

    7. Applied associate-*l/12.2

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

    8. Applied frac-times12.1

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

    9. Applied associate-/l/0.3

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

    if -739634522879.6277 < z < 1.2557521193532489e+23

    1. Initial program 0.4

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

    3. Applied div-inv0.5

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

    4. Applied associate-*r*0.5

      \[\leadsto \frac{\color{blue}{\left(\cosh x \cdot y\right) \cdot \frac{1}{x}}}{z}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -739634522879.62769 \lor \neg \left(z \le 1.25575211935324887 \cdot 10^{23}\right):\\ \;\;\;\;\frac{\left(\left(e^{x} + e^{-x}\right) \cdot y\right) \cdot 1}{z \cdot \left(2 \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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))