Average Error: 2.9 → 2.3
Time: 4.6s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -50225608.0783285126:\\ \;\;\;\;\frac{\left(x \cdot \sin y\right) \cdot \frac{1}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{1}{\frac{\sin y}{y}}}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -50225608.0783285126:\\
\;\;\;\;\frac{\left(x \cdot \sin y\right) \cdot \frac{1}{y}}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{z \cdot \frac{1}{\frac{\sin y}{y}}}\\

\end{array}
double f(double x, double y, double z) {
        double r526875 = x;
        double r526876 = y;
        double r526877 = sin(r526876);
        double r526878 = r526877 / r526876;
        double r526879 = r526875 * r526878;
        double r526880 = z;
        double r526881 = r526879 / r526880;
        return r526881;
}
↓
double f(double x, double y, double z) {
        double r526882 = z;
        double r526883 = -50225608.07832851;
        bool r526884 = r526882 <= r526883;
        double r526885 = x;
        double r526886 = y;
        double r526887 = sin(r526886);
        double r526888 = r526885 * r526887;
        double r526889 = 1.0;
        double r526890 = r526889 / r526886;
        double r526891 = r526888 * r526890;
        double r526892 = r526891 / r526882;
        double r526893 = r526887 / r526886;
        double r526894 = r526889 / r526893;
        double r526895 = r526882 * r526894;
        double r526896 = r526885 / r526895;
        double r526897 = r526884 ? r526892 : r526896;
        return r526897;
}

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

Original2.9
Target0.4
Herbie2.3
\[\begin{array}{l} \mathbf{if}\;z \lt -4.21737202034271466 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -50225608.07832851

    1. Initial program 0.1

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

    3. Applied div-inv0.2

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

    4. Applied associate-*r*2.5

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

    if -50225608.07832851 < z

    1. Initial program 3.8

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

    3. Applied associate-/l*2.2

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{\frac{\sin y}{y}}}}\]
    4. Using strategy rm

    5. Applied div-inv2.3

      \[\leadsto \frac{x}{\color{blue}{z \cdot \frac{1}{\frac{\sin y}{y}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -50225608.0783285126:\\ \;\;\;\;\frac{\left(x \cdot \sin y\right) \cdot \frac{1}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{1}{\frac{\sin y}{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))