Average Error: 8.0 → 0.4
Time: 16.5s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -40510205393232588192078396191825461248:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;y \le 7.039456669498942289348721752920549611426 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -40510205393232588192078396191825461248:\\
\;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\

\mathbf{elif}\;y \le 7.039456669498942289348721752920549611426 \cdot 10^{-17}:\\
\;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r19357081 = x;
        double r19357082 = cosh(r19357081);
        double r19357083 = y;
        double r19357084 = r19357083 / r19357081;
        double r19357085 = r19357082 * r19357084;
        double r19357086 = z;
        double r19357087 = r19357085 / r19357086;
        return r19357087;
}
↓
double f(double x, double y, double z) {
        double r19357088 = y;
        double r19357089 = -4.051020539323259e+37;
        bool r19357090 = r19357088 <= r19357089;
        double r19357091 = x;
        double r19357092 = cosh(r19357091);
        double r19357093 = r19357092 * r19357088;
        double r19357094 = z;
        double r19357095 = r19357091 * r19357094;
        double r19357096 = r19357093 / r19357095;
        double r19357097 = 7.039456669498942e-17;
        bool r19357098 = r19357088 <= r19357097;
        double r19357099 = r19357093 / r19357091;
        double r19357100 = r19357099 / r19357094;
        double r19357101 = r19357098 ? r19357100 : r19357096;
        double r19357102 = r19357090 ? r19357096 : r19357101;
        return r19357102;
}

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

Original8.0
Target0.4
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 y < -4.051020539323259e+37 or 7.039456669498942e-17 < y

    1. Initial program 22.7

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

    3. Applied associate-*r/22.7

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

    4. Applied associate-/l/0.3

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

    if -4.051020539323259e+37 < y < 7.039456669498942e-17

    1. Initial program 0.4

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

    3. Applied div-inv0.5

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

    4. Applied associate-*r*0.5

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

    6. Applied un-div-inv0.4

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -40510205393232588192078396191825461248:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{elif}\;y \le 7.039456669498942289348721752920549611426 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"

  :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))