Average Error: 2.7 → 0.2
Time: 4.9s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \le -4.5545988867587666 \cdot 10^{-295}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;x \cdot \frac{\sin y}{y} \le 7.56936427636939772 \cdot 10^{-202}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1}{y}}{\frac{1}{\sin y}}}{z}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot \frac{\sin y}{y} \le -4.5545988867587666 \cdot 10^{-295}:\\
\;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\

\mathbf{elif}\;x \cdot \frac{\sin y}{y} \le 7.56936427636939772 \cdot 10^{-202}:\\
\;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r557659 = x;
        double r557660 = y;
        double r557661 = sin(r557660);
        double r557662 = r557661 / r557660;
        double r557663 = r557659 * r557662;
        double r557664 = z;
        double r557665 = r557663 / r557664;
        return r557665;
}
↓
double f(double x, double y, double z) {
        double r557666 = x;
        double r557667 = y;
        double r557668 = sin(r557667);
        double r557669 = r557668 / r557667;
        double r557670 = r557666 * r557669;
        double r557671 = -4.554598886758767e-295;
        bool r557672 = r557670 <= r557671;
        double r557673 = 1.0;
        double r557674 = r557667 / r557668;
        double r557675 = r557673 / r557674;
        double r557676 = r557666 * r557675;
        double r557677 = z;
        double r557678 = r557676 / r557677;
        double r557679 = 7.569364276369398e-202;
        bool r557680 = r557670 <= r557679;
        double r557681 = r557669 / r557677;
        double r557682 = r557666 * r557681;
        double r557683 = r557673 / r557667;
        double r557684 = r557673 / r557668;
        double r557685 = r557683 / r557684;
        double r557686 = r557666 * r557685;
        double r557687 = r557686 / r557677;
        double r557688 = r557680 ? r557682 : r557687;
        double r557689 = r557672 ? r557678 : r557688;
        return r557689;
}

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

Original2.7
Target0.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;z \lt -4.21737202034271466 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x (/ (sin y) y)) < -4.554598886758767e-295

    1. Initial program 0.2

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

    3. Applied clear-num0.2

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

    if -4.554598886758767e-295 < (* x (/ (sin y) y)) < 7.569364276369398e-202

    1. Initial program 10.7

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

    3. Applied *-un-lft-identity10.7

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

    4. Applied times-frac0.3

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

    5. Simplified0.3

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

    if 7.569364276369398e-202 < (* x (/ (sin y) y))

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \le -4.5545988867587666 \cdot 10^{-295}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;x \cdot \frac{\sin y}{y} \le 7.56936427636939772 \cdot 10^{-202}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1}{y}}{\frac{1}{\sin y}}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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))