Average Error: 7.8 → 0.3
Time: 21.8s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.749876674636675285330697870290708089216 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{elif}\;y \le 521563463103.41900634765625:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{y}{x}}{2}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\frac{z \cdot x}{y}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.749876674636675285330697870290708089216 \cdot 10^{-11}:\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\

\mathbf{elif}\;y \le 521563463103.41900634765625:\\
\;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{y}{x}}{2}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r363428 = x;
        double r363429 = cosh(r363428);
        double r363430 = y;
        double r363431 = r363430 / r363428;
        double r363432 = r363429 * r363431;
        double r363433 = z;
        double r363434 = r363432 / r363433;
        return r363434;
}
↓
double f(double x, double y, double z) {
        double r363435 = y;
        double r363436 = -1.7498766746366753e-11;
        bool r363437 = r363435 <= r363436;
        double r363438 = x;
        double r363439 = cosh(r363438);
        double r363440 = r363439 * r363435;
        double r363441 = z;
        double r363442 = r363440 / r363441;
        double r363443 = r363442 / r363438;
        double r363444 = 521563463103.419;
        bool r363445 = r363435 <= r363444;
        double r363446 = exp(r363438);
        double r363447 = -r363438;
        double r363448 = exp(r363447);
        double r363449 = r363446 + r363448;
        double r363450 = r363435 / r363438;
        double r363451 = r363449 * r363450;
        double r363452 = 2.0;
        double r363453 = r363451 / r363452;
        double r363454 = r363453 / r363441;
        double r363455 = r363441 * r363438;
        double r363456 = r363455 / r363435;
        double r363457 = r363439 / r363456;
        double r363458 = r363445 ? r363454 : r363457;
        double r363459 = r363437 ? r363443 : r363458;
        return r363459;
}

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
Target0.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.7498766746366753e-11

    1. Initial program 20.4

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

    3. Applied associate-*r/20.4

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

    4. Applied associate-/l/0.3

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

    6. Applied associate-/r*0.3

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

    if -1.7498766746366753e-11 < y < 521563463103.419

    1. Initial program 0.3

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

    3. Applied cosh-def0.3

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

    4. Applied associate-*l/0.3

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

    if 521563463103.419 < y

    1. Initial program 23.6

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

    3. Applied associate-*r/23.6

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

    4. Applied associate-/l/0.3

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

    6. Applied associate-/l*0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.749876674636675285330697870290708089216 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{elif}\;y \le 521563463103.41900634765625:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot \frac{y}{x}}{2}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{\frac{z \cdot x}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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 1.03853053593515302e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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