Average Error: 8.1 → 0.4
Time: 6.0s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -9.323363828121578113609366669881517175807 \cdot 10^{189}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{1}{x} \cdot \frac{y}{z}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 3.443490109674912910497508208705102086219 \cdot 10^{276}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{z \cdot x} \cdot y\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -9.323363828121578113609366669881517175807 \cdot 10^{189}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{1}{x} \cdot \frac{y}{z}\\

\mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 3.443490109674912910497508208705102086219 \cdot 10^{276}:\\
\;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r332021 = x;
        double r332022 = cosh(r332021);
        double r332023 = y;
        double r332024 = r332023 / r332021;
        double r332025 = r332022 * r332024;
        double r332026 = z;
        double r332027 = r332025 / r332026;
        return r332027;
}
↓
double f(double x, double y, double z) {
        double r332028 = x;
        double r332029 = cosh(r332028);
        double r332030 = y;
        double r332031 = r332030 / r332028;
        double r332032 = r332029 * r332031;
        double r332033 = -9.323363828121578e+189;
        bool r332034 = r332032 <= r332033;
        double r332035 = 0.5;
        double r332036 = r332028 * r332030;
        double r332037 = z;
        double r332038 = r332036 / r332037;
        double r332039 = r332035 * r332038;
        double r332040 = 1.0;
        double r332041 = r332040 / r332028;
        double r332042 = r332030 / r332037;
        double r332043 = r332041 * r332042;
        double r332044 = r332039 + r332043;
        double r332045 = 3.443490109674913e+276;
        bool r332046 = r332032 <= r332045;
        double r332047 = r332032 / r332037;
        double r332048 = -1.0;
        double r332049 = r332048 * r332028;
        double r332050 = exp(r332049);
        double r332051 = exp(r332028);
        double r332052 = r332050 + r332051;
        double r332053 = r332035 * r332052;
        double r332054 = r332037 * r332028;
        double r332055 = r332053 / r332054;
        double r332056 = r332055 * r332030;
        double r332057 = r332046 ? r332047 : r332056;
        double r332058 = r332034 ? r332044 : r332057;
        return r332058;
}

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

Original8.1
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 3 regimes

  2. if (* (cosh x) (/ y x)) < -9.323363828121578e+189

    1. Initial program 28.2

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

    2. Taylor expanded around 0 1.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity1.4

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

    5. Applied times-frac1.2

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

    if -9.323363828121578e+189 < (* (cosh x) (/ y x)) < 3.443490109674913e+276

    1. Initial program 0.3

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

    if 3.443490109674913e+276 < (* (cosh x) (/ y x))

    1. Initial program 49.4

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.4

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

    5. Applied associate-/r/0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \le -9.323363828121578113609366669881517175807 \cdot 10^{189}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{x \cdot y}{z} + \frac{1}{x} \cdot \frac{y}{z}\\ \mathbf{elif}\;\cosh x \cdot \frac{y}{x} \le 3.443490109674912910497508208705102086219 \cdot 10^{276}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(e^{-1 \cdot x} + e^{x}\right)}{z \cdot x} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(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))