Average Error: 8.0 → 0.4
Time: 15.9s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.108045198460169534729664699359786352891 \cdot 10^{-61} \lor \neg \left(y \le 4.48169197924863956927649820888772435422 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.108045198460169534729664699359786352891 \cdot 10^{-61} \lor \neg \left(y \le 4.48169197924863956927649820888772435422 \cdot 10^{-26}\right):\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r329325 = x;
        double r329326 = cosh(r329325);
        double r329327 = y;
        double r329328 = r329327 / r329325;
        double r329329 = r329326 * r329328;
        double r329330 = z;
        double r329331 = r329329 / r329330;
        return r329331;
}
↓
double f(double x, double y, double z) {
        double r329332 = y;
        double r329333 = -2.1080451984601695e-61;
        bool r329334 = r329332 <= r329333;
        double r329335 = 4.4816919792486396e-26;
        bool r329336 = r329332 <= r329335;
        double r329337 = !r329336;
        bool r329338 = r329334 || r329337;
        double r329339 = x;
        double r329340 = cosh(r329339);
        double r329341 = r329340 * r329332;
        double r329342 = z;
        double r329343 = r329341 / r329342;
        double r329344 = r329343 / r329339;
        double r329345 = r329341 / r329339;
        double r329346 = r329345 / r329342;
        double r329347 = r329338 ? r329344 : r329346;
        return r329347;
}

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

Original8.0
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 y < -2.1080451984601695e-61 or 4.4816919792486396e-26 < y

    1. Initial program 18.1

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

    3. Applied div-inv18.1

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

    4. Applied associate-*r*18.1

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

    6. Applied clear-num18.2

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

    7. Simplified0.9

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

    9. Applied associate-/r*0.9

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

    10. Simplified0.6

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

    if -2.1080451984601695e-61 < y < 4.4816919792486396e-26

    1. Initial program 0.3

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

    3. Applied div-inv0.4

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

    4. Applied associate-*r*0.4

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

    6. Applied associate-*r/0.3

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

    7. Simplified0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.108045198460169534729664699359786352891 \cdot 10^{-61} \lor \neg \left(y \le 4.48169197924863956927649820888772435422 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(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))