Average Error: 7.7 → 0.6
Time: 3.4s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.40953097935408513 \cdot 10^{35} \lor \neg \left(z \le 5.82511916007843312 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.40953097935408513 \cdot 10^{35} \lor \neg \left(z \le 5.82511916007843312 \cdot 10^{-85}\right):\\
\;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r529933 = x;
        double r529934 = cosh(r529933);
        double r529935 = y;
        double r529936 = r529935 / r529933;
        double r529937 = r529934 * r529936;
        double r529938 = z;
        double r529939 = r529937 / r529938;
        return r529939;
}
↓
double f(double x, double y, double z) {
        double r529940 = z;
        double r529941 = -1.4095309793540851e+35;
        bool r529942 = r529940 <= r529941;
        double r529943 = 5.825119160078433e-85;
        bool r529944 = r529940 <= r529943;
        double r529945 = !r529944;
        bool r529946 = r529942 || r529945;
        double r529947 = x;
        double r529948 = cosh(r529947);
        double r529949 = y;
        double r529950 = r529948 * r529949;
        double r529951 = r529947 * r529940;
        double r529952 = r529950 / r529951;
        double r529953 = exp(r529947);
        double r529954 = -r529947;
        double r529955 = exp(r529954);
        double r529956 = r529953 + r529955;
        double r529957 = r529956 * r529949;
        double r529958 = r529957 / r529940;
        double r529959 = 2.0;
        double r529960 = r529959 * r529947;
        double r529961 = r529958 / r529960;
        double r529962 = r529946 ? r529952 : r529961;
        return r529962;
}

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

Original7.7
Target0.4
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.0385305359351529 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.4095309793540851e+35 or 5.825119160078433e-85 < z

    1. Initial program 11.2

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

    3. Applied div-inv11.2

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

    5. Applied associate-*r/11.2

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

    6. Applied frac-times0.7

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

    7. Simplified0.7

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

    if -1.4095309793540851e+35 < z < 5.825119160078433e-85

    1. Initial program 0.7

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

    3. Applied div-inv0.8

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

    5. Applied cosh-def0.8

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

    6. Applied frac-times0.8

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

    7. Applied associate-*l/0.5

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

    8. Simplified0.4

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.40953097935408513 \cdot 10^{35} \lor \neg \left(z \le 5.82511916007843312 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(e^{x} + e^{-x}\right) \cdot y}{z}}{2 \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))