Average Error: 2.9 → 0.2
Time: 7.3s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.579176583523817341408656991031950031801 \cdot 10^{-58} \lor \neg \left(z \le 14047118970.9252910614013671875\right):\\ \;\;\;\;\frac{x \cdot \frac{\frac{1}{y}}{\frac{1}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.579176583523817341408656991031950031801 \cdot 10^{-58} \lor \neg \left(z \le 14047118970.9252910614013671875\right):\\
\;\;\;\;\frac{x \cdot \frac{\frac{1}{y}}{\frac{1}{\sin y}}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r517629 = x;
        double r517630 = y;
        double r517631 = sin(r517630);
        double r517632 = r517631 / r517630;
        double r517633 = r517629 * r517632;
        double r517634 = z;
        double r517635 = r517633 / r517634;
        return r517635;
}
↓
double f(double x, double y, double z) {
        double r517636 = z;
        double r517637 = -1.5791765835238173e-58;
        bool r517638 = r517636 <= r517637;
        double r517639 = 14047118970.925291;
        bool r517640 = r517636 <= r517639;
        double r517641 = !r517640;
        bool r517642 = r517638 || r517641;
        double r517643 = x;
        double r517644 = 1.0;
        double r517645 = y;
        double r517646 = r517644 / r517645;
        double r517647 = sin(r517645);
        double r517648 = r517644 / r517647;
        double r517649 = r517646 / r517648;
        double r517650 = r517643 * r517649;
        double r517651 = r517650 / r517636;
        double r517652 = r517647 / r517645;
        double r517653 = r517636 / r517652;
        double r517654 = r517643 / r517653;
        double r517655 = r517642 ? r517651 : r517654;
        return r517655;
}

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
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;z \lt -4.217372020342714661850238929213415773451 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.5791765835238173e-58 or 14047118970.925291 < z

    1. Initial program 0.2

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

    3. Applied clear-num0.2

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{\frac{y}{\sin y}}}}{z}\]
    4. Using strategy rm

    5. Applied div-inv0.3

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

    6. Applied associate-/r*0.2

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

    if -1.5791765835238173e-58 < z < 14047118970.925291

    1. Initial program 6.6

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

    3. Applied associate-/l*0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.579176583523817341408656991031950031801 \cdot 10^{-58} \lor \neg \left(z \le 14047118970.9252910614013671875\right):\\ \;\;\;\;\frac{x \cdot \frac{\frac{1}{y}}{\frac{1}{\sin y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{\sin y}{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))