Average Error: 7.6 → 0.4
Time: 12.8s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.90737577331802458 \cdot 10^{-57} \lor \neg \left(z \le 2.54423027784147743 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.90737577331802458 \cdot 10^{-57} \lor \neg \left(z \le 2.54423027784147743 \cdot 10^{-6}\right):\\
\;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r507111 = x;
        double r507112 = cosh(r507111);
        double r507113 = y;
        double r507114 = r507113 / r507111;
        double r507115 = r507112 * r507114;
        double r507116 = z;
        double r507117 = r507115 / r507116;
        return r507117;
}
↓
double f(double x, double y, double z) {
        double r507118 = z;
        double r507119 = -5.907375773318025e-57;
        bool r507120 = r507118 <= r507119;
        double r507121 = 2.5442302778414774e-06;
        bool r507122 = r507118 <= r507121;
        double r507123 = !r507122;
        bool r507124 = r507120 || r507123;
        double r507125 = x;
        double r507126 = cosh(r507125);
        double r507127 = y;
        double r507128 = r507126 * r507127;
        double r507129 = r507125 * r507118;
        double r507130 = r507128 / r507129;
        double r507131 = r507118 / r507127;
        double r507132 = r507125 * r507131;
        double r507133 = r507126 / r507132;
        double r507134 = r507124 ? r507130 : r507133;
        return r507134;
}

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

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

Derivation

  1. Split input into 2 regimes

  2. if z < -5.907375773318025e-57 or 2.5442302778414774e-06 < z

    1. Initial program 10.7

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

    3. Applied associate-*r/10.7

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

    4. Applied associate-/l/0.4

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

    5. Simplified0.4

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

    if -5.907375773318025e-57 < z < 2.5442302778414774e-06

    1. Initial program 0.3

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

    3. Applied associate-/l*0.4

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

    4. Simplified0.4

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

    5. Taylor expanded around 0 21.9

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

    6. Simplified0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.90737577331802458 \cdot 10^{-57} \lor \neg \left(z \le 2.54423027784147743 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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 1.0385305359351529e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

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