Average Error: 7.6 → 1.0
Time: 21.4s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.570530855841579352725070352970716566831 \cdot 10^{-31} \lor \neg \left(y \le 1.168432178035094477207947711711724571865 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{x}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.570530855841579352725070352970716566831 \cdot 10^{-31} \lor \neg \left(y \le 1.168432178035094477207947711711724571865 \cdot 10^{-120}\right):\\
\;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r355113 = x;
        double r355114 = cosh(r355113);
        double r355115 = y;
        double r355116 = r355115 / r355113;
        double r355117 = r355114 * r355116;
        double r355118 = z;
        double r355119 = r355117 / r355118;
        return r355119;
}
↓
double f(double x, double y, double z) {
        double r355120 = y;
        double r355121 = -1.5705308558415794e-31;
        bool r355122 = r355120 <= r355121;
        double r355123 = 1.1684321780350945e-120;
        bool r355124 = r355120 <= r355123;
        double r355125 = !r355124;
        bool r355126 = r355122 || r355125;
        double r355127 = x;
        double r355128 = cosh(r355127);
        double r355129 = r355128 * r355120;
        double r355130 = z;
        double r355131 = r355130 * r355127;
        double r355132 = r355129 / r355131;
        double r355133 = r355120 / r355127;
        double r355134 = r355133 / r355130;
        double r355135 = r355128 * r355134;
        double r355136 = r355126 ? r355132 : r355135;
        return r355136;
}

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
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935152855236908684227749499669 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.5705308558415794e-31 or 1.1684321780350945e-120 < y

    1. Initial program 15.1

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

    3. Applied associate-*r/15.1

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

    4. Applied associate-/l/1.6

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

    if -1.5705308558415794e-31 < y < 1.1684321780350945e-120

    1. Initial program 0.3

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

    3. Applied *-un-lft-identity0.3

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

    4. Applied times-frac0.3

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

    5. Simplified0.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.570530855841579352725070352970716566831 \cdot 10^{-31} \lor \neg \left(y \le 1.168432178035094477207947711711724571865 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{x}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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))