Average Error: 7.2 → 0.4
Time: 4.0s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -3.58400891098028717 \cdot 10^{108}:\\ \;\;\;\;\frac{1}{\frac{z}{\cosh x \cdot y} \cdot x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.5273947602591855 \cdot 10^{302}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -3.58400891098028717 \cdot 10^{108}:\\
\;\;\;\;\frac{1}{\frac{z}{\cosh x \cdot y} \cdot x}\\

\mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.5273947602591855 \cdot 10^{302}:\\
\;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r569516 = x;
        double r569517 = cosh(r569516);
        double r569518 = y;
        double r569519 = r569518 / r569516;
        double r569520 = r569517 * r569519;
        double r569521 = z;
        double r569522 = r569520 / r569521;
        return r569522;
}
↓
double f(double x, double y, double z) {
        double r569523 = x;
        double r569524 = cosh(r569523);
        double r569525 = y;
        double r569526 = r569525 / r569523;
        double r569527 = r569524 * r569526;
        double r569528 = z;
        double r569529 = r569527 / r569528;
        double r569530 = -3.584008910980287e+108;
        bool r569531 = r569529 <= r569530;
        double r569532 = 1.0;
        double r569533 = r569524 * r569525;
        double r569534 = r569528 / r569533;
        double r569535 = r569534 * r569523;
        double r569536 = r569532 / r569535;
        double r569537 = 1.5273947602591855e+302;
        bool r569538 = r569529 <= r569537;
        double r569539 = r569523 * r569528;
        double r569540 = r569525 / r569539;
        double r569541 = r569524 * r569540;
        double r569542 = r569538 ? r569529 : r569541;
        double r569543 = r569531 ? r569536 : r569542;
        return r569543;
}

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

Original7.2
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 3 regimes

  2. if (/ (* (cosh x) (/ y x)) z) < -3.584008910980287e+108

    1. Initial program 19.0

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

    3. Applied div-inv19.1

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

    4. Applied associate-*r*19.1

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

    6. Applied clear-num19.1

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

    7. Simplified0.6

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

    if -3.584008910980287e+108 < (/ (* (cosh x) (/ y x)) z) < 1.5273947602591855e+302

    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

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

    6. Applied associate-*l*0.3

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

    7. Simplified0.2

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

    if 1.5273947602591855e+302 < (/ (* (cosh x) (/ y x)) z)

    1. Initial program 60.5

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

    3. Applied *-un-lft-identity60.5

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

    4. Applied times-frac60.5

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

    5. Simplified60.5

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

    6. Simplified2.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le -3.58400891098028717 \cdot 10^{108}:\\ \;\;\;\;\frac{1}{\frac{z}{\cosh x \cdot y} \cdot x}\\ \mathbf{elif}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \le 1.5273947602591855 \cdot 10^{302}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))