Average Error: 7.8 → 1.4
Time: 17.5s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.127846607674739281554471432647215989737 \cdot 10^{141}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{x}}{\frac{z}{y \cdot \left(e^{-x} + e^{x}\right)}}\\ \mathbf{elif}\;y \le 26250898118978766400982614016:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.127846607674739281554471432647215989737 \cdot 10^{141}:\\
\;\;\;\;\frac{\frac{\frac{1}{2}}{x}}{\frac{z}{y \cdot \left(e^{-x} + e^{x}\right)}}\\

\mathbf{elif}\;y \le 26250898118978766400982614016:\\
\;\;\;\;\frac{\frac{\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot y}{x}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\

\end{array}
double f(double x, double y, double z) {
        double r360924 = x;
        double r360925 = cosh(r360924);
        double r360926 = y;
        double r360927 = r360926 / r360924;
        double r360928 = r360925 * r360927;
        double r360929 = z;
        double r360930 = r360928 / r360929;
        return r360930;
}
↓
double f(double x, double y, double z) {
        double r360931 = y;
        double r360932 = -1.1278466076747393e+141;
        bool r360933 = r360931 <= r360932;
        double r360934 = 0.5;
        double r360935 = x;
        double r360936 = r360934 / r360935;
        double r360937 = z;
        double r360938 = -r360935;
        double r360939 = exp(r360938);
        double r360940 = exp(r360935);
        double r360941 = r360939 + r360940;
        double r360942 = r360931 * r360941;
        double r360943 = r360937 / r360942;
        double r360944 = r360936 / r360943;
        double r360945 = 2.6250898118978766e+28;
        bool r360946 = r360931 <= r360945;
        double r360947 = r360934 * r360941;
        double r360948 = r360947 * r360931;
        double r360949 = r360948 / r360935;
        double r360950 = r360949 / r360937;
        double r360951 = r360935 * r360931;
        double r360952 = r360951 / r360937;
        double r360953 = r360934 * r360952;
        double r360954 = r360935 * r360937;
        double r360955 = r360931 / r360954;
        double r360956 = r360953 + r360955;
        double r360957 = r360946 ? r360950 : r360956;
        double r360958 = r360933 ? r360944 : r360957;
        return r360958;
}

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
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.1278466076747393e+141

    1. Initial program 36.1

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

    2. Taylor expanded around inf 36.1

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

    3. Simplified36.1

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

    5. Applied div-inv36.2

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

    6. Applied times-frac36.2

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

    7. Applied associate-/l*0.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2}}{x}}{\frac{z}{\frac{e^{-x} + e^{x}}{\frac{1}{y}}}}}\]

    8. Simplified0.5

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

    if -1.1278466076747393e+141 < y < 2.6250898118978766e+28

    1. Initial program 1.5

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

    2. Taylor expanded around inf 1.5

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

    3. Simplified1.5

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

    5. Applied *-un-lft-identity1.5

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

    6. Applied associate-/r/1.6

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

    7. Applied times-frac9.3

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

    8. Simplified9.3

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

    10. Applied associate-*r/1.6

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

    11. Simplified1.5

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

    if 2.6250898118978766e+28 < y

    1. Initial program 25.0

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

    2. Taylor expanded around inf 25.0

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

    3. Simplified25.0

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

    4. Taylor expanded around 0 1.3

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.127846607674739281554471432647215989737 \cdot 10^{141}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{x}}{\frac{z}{y \cdot \left(e^{-x} + e^{x}\right)}}\\ \mathbf{elif}\;y \le 26250898118978766400982614016:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)\right) \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(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))