Average Error: 7.6 → 2.1
Time: 23.5s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.72160496452561263986704681669141916127 \cdot 10^{202} \lor \neg \left(z \le 1.342227135368572564573491428767395770614 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{y}{x \cdot z} \cdot \cosh x\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.72160496452561263986704681669141916127 \cdot 10^{202} \lor \neg \left(z \le 1.342227135368572564573491428767395770614 \cdot 10^{-58}\right):\\
\;\;\;\;\frac{y}{x \cdot z} \cdot \cosh x\\

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

\end{array}
double f(double x, double y, double z) {
        double r406299 = x;
        double r406300 = cosh(r406299);
        double r406301 = y;
        double r406302 = r406301 / r406299;
        double r406303 = r406300 * r406302;
        double r406304 = z;
        double r406305 = r406303 / r406304;
        return r406305;
}
↓
double f(double x, double y, double z) {
        double r406306 = z;
        double r406307 = -1.7216049645256126e+202;
        bool r406308 = r406306 <= r406307;
        double r406309 = 1.3422271353685726e-58;
        bool r406310 = r406306 <= r406309;
        double r406311 = !r406310;
        bool r406312 = r406308 || r406311;
        double r406313 = y;
        double r406314 = x;
        double r406315 = r406314 * r406306;
        double r406316 = r406313 / r406315;
        double r406317 = cosh(r406314);
        double r406318 = r406316 * r406317;
        double r406319 = r406313 / r406306;
        double r406320 = r406319 / r406314;
        double r406321 = r406317 * r406320;
        double r406322 = r406312 ? r406318 : r406321;
        return r406322;
}

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

Original7.6
Target0.4
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.7216049645256126e+202 or 1.3422271353685726e-58 < z

    1. Initial program 11.7

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

    3. Applied associate-/l*12.2

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

    4. Simplified10.8

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

    6. Applied div-inv10.8

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

    7. Simplified10.7

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

    9. Applied *-un-lft-identity10.7

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

    10. Applied *-un-lft-identity10.7

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

    11. Applied *-un-lft-identity10.7

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

    12. Applied times-frac10.7

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

    13. Applied times-frac10.7

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

    14. Simplified10.7

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

    15. Simplified0.5

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

    if -1.7216049645256126e+202 < z < 1.3422271353685726e-58

    1. Initial program 3.8

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

    3. Applied associate-/l*3.9

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

    4. Simplified3.6

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

    6. Applied div-inv3.6

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

    7. Simplified3.5

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.72160496452561263986704681669141916127 \cdot 10^{202} \lor \neg \left(z \le 1.342227135368572564573491428767395770614 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{y}{x \cdot z} \cdot \cosh x\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(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))