Average Error: 2.8 → 0.7
Time: 18.3s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -140933690204890492166891680825344 \lor \neg \left(z \le 1.663038927321811361009908848030056352037 \cdot 10^{-155}\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{1}{\frac{\sin y}{y}}}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -140933690204890492166891680825344 \lor \neg \left(z \le 1.663038927321811361009908848030056352037 \cdot 10^{-155}\right):\\
\;\;\;\;\frac{x \cdot \frac{\sin y}{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 r257703 = x;
        double r257704 = y;
        double r257705 = sin(r257704);
        double r257706 = r257705 / r257704;
        double r257707 = r257703 * r257706;
        double r257708 = z;
        double r257709 = r257707 / r257708;
        return r257709;
}
↓
double f(double x, double y, double z) {
        double r257710 = z;
        double r257711 = -1.409336902048905e+32;
        bool r257712 = r257710 <= r257711;
        double r257713 = 1.6630389273218114e-155;
        bool r257714 = r257710 <= r257713;
        double r257715 = !r257714;
        bool r257716 = r257712 || r257715;
        double r257717 = x;
        double r257718 = y;
        double r257719 = sin(r257718);
        double r257720 = r257719 / r257718;
        double r257721 = r257717 * r257720;
        double r257722 = r257721 / r257710;
        double r257723 = 1.0;
        double r257724 = r257723 / r257720;
        double r257725 = r257710 * r257724;
        double r257726 = r257717 / r257725;
        double r257727 = r257716 ? r257722 : r257726;
        return r257727;
}

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

Original2.8
Target0.4
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;z \lt -4.217372020342714661850238929213415773451 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.409336902048905e+32 or 1.6630389273218114e-155 < z

    1. Initial program 0.8

      \[\frac{x \cdot \frac{\sin y}{y}}{z}\]

    if -1.409336902048905e+32 < z < 1.6630389273218114e-155

    1. Initial program 6.6

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

    3. Applied associate-/l*0.4

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

    4. Simplified0.4

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

    6. Applied clear-num0.4

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -140933690204890492166891680825344 \lor \neg \left(z \le 1.663038927321811361009908848030056352037 \cdot 10^{-155}\right):\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{1}{\frac{\sin y}{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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))