Average Error: 2.5 → 2.6
Time: 7.6s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \le -4.8358356144772814 \cdot 10^{-280}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \mathbf{elif}\;\frac{\sin y}{y} \le 4.8956053390658603 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{1}{\frac{y}{\sin y}}}}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\sin y}{y} \le -4.8358356144772814 \cdot 10^{-280}:\\
\;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\

\mathbf{elif}\;\frac{\sin y}{y} \le 4.8956053390658603 \cdot 10^{-95}:\\
\;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r484964 = x;
        double r484965 = y;
        double r484966 = sin(r484965);
        double r484967 = r484966 / r484965;
        double r484968 = r484964 * r484967;
        double r484969 = z;
        double r484970 = r484968 / r484969;
        return r484970;
}
↓
double f(double x, double y, double z) {
        double r484971 = y;
        double r484972 = sin(r484971);
        double r484973 = r484972 / r484971;
        double r484974 = -4.835835614477281e-280;
        bool r484975 = r484973 <= r484974;
        double r484976 = x;
        double r484977 = z;
        double r484978 = r484972 / r484977;
        double r484979 = r484978 / r484971;
        double r484980 = r484976 * r484979;
        double r484981 = 4.89560533906586e-95;
        bool r484982 = r484973 <= r484981;
        double r484983 = r484976 * r484972;
        double r484984 = r484983 / r484971;
        double r484985 = r484984 / r484977;
        double r484986 = 1.0;
        double r484987 = r484971 / r484972;
        double r484988 = r484986 / r484987;
        double r484989 = r484977 / r484988;
        double r484990 = r484976 / r484989;
        double r484991 = r484982 ? r484985 : r484990;
        double r484992 = r484975 ? r484980 : r484991;
        return r484992;
}

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

Original2.5
Target0.3
Herbie2.6
\[\begin{array}{l} \mathbf{if}\;z \lt -4.21737202034271466 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (sin y) y) < -4.835835614477281e-280

    1. Initial program 4.3

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

    3. Applied associate-/l*4.9

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

    5. Applied div-inv5.0

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

    6. Simplified4.4

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

    if -4.835835614477281e-280 < (/ (sin y) y) < 4.89560533906586e-95

    1. Initial program 6.1

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

    3. Applied associate-*r/6.2

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

    if 4.89560533906586e-95 < (/ (sin y) y)

    1. Initial program 0.3

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

    3. Applied associate-/l*0.4

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

    5. Applied clear-num0.4

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

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \le -4.8358356144772814 \cdot 10^{-280}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \mathbf{elif}\;\frac{\sin y}{y} \le 4.8956053390658603 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{x \cdot \sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{1}{\frac{y}{\sin y}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))