Average Error: 7.8 → 0.4
Time: 18.0s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.706524267514688817080607810109361259464 \cdot 10^{-14} \lor \neg \left(z \le 1.345654323883122095521379973183563704723 \cdot 10^{50}\right):\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)}{x} \cdot \frac{y}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.706524267514688817080607810109361259464 \cdot 10^{-14} \lor \neg \left(z \le 1.345654323883122095521379973183563704723 \cdot 10^{50}\right):\\
\;\;\;\;\frac{\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot y}{x \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)}{x} \cdot \frac{y}{z}\\

\end{array}
double f(double x, double y, double z) {
        double r758912 = x;
        double r758913 = cosh(r758912);
        double r758914 = y;
        double r758915 = r758914 / r758912;
        double r758916 = r758913 * r758915;
        double r758917 = z;
        double r758918 = r758916 / r758917;
        return r758918;
}
↓
double f(double x, double y, double z) {
        double r758919 = z;
        double r758920 = -1.7065242675146888e-14;
        bool r758921 = r758919 <= r758920;
        double r758922 = 1.345654323883122e+50;
        bool r758923 = r758919 <= r758922;
        double r758924 = !r758923;
        bool r758925 = r758921 || r758924;
        double r758926 = 0.5;
        double r758927 = x;
        double r758928 = -r758927;
        double r758929 = exp(r758928);
        double r758930 = exp(r758927);
        double r758931 = r758929 + r758930;
        double r758932 = r758926 * r758931;
        double r758933 = y;
        double r758934 = r758932 * r758933;
        double r758935 = r758927 * r758919;
        double r758936 = r758934 / r758935;
        double r758937 = r758932 / r758927;
        double r758938 = r758933 / r758919;
        double r758939 = r758937 * r758938;
        double r758940 = r758925 ? r758936 : r758939;
        return r758940;
}

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
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.7065242675146888e-14 or 1.345654323883122e+50 < z

    1. Initial program 12.1

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]

    2. Taylor expanded around inf 0.3

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

    3. Simplified12.2

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

    5. Applied *-un-lft-identity12.2

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

    6. Applied associate-/r/12.2

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

    7. Applied times-frac11.4

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

    8. Simplified11.4

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

    10. Applied frac-times0.3

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

    if -1.7065242675146888e-14 < z < 1.345654323883122e+50

    1. Initial program 0.6

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]

    2. Taylor expanded around inf 17.6

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

    3. Simplified0.7

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

    5. Applied *-un-lft-identity0.7

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

    6. Applied associate-/r/0.7

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

    7. Applied times-frac0.5

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

    8. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.706524267514688817080607810109361259464 \cdot 10^{-14} \lor \neg \left(z \le 1.345654323883122095521379973183563704723 \cdot 10^{50}\right):\\ \;\;\;\;\frac{\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)}{x} \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +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 1.03853053593515302e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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