Average Error: 2.9 → 1.5
Time: 7.6s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \le -5.40252227994517841 \cdot 10^{-206}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{1}{\frac{y}{\sin y}}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \le -5.40252227994517841 \cdot 10^{-206}:\\
\;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r474978 = x;
        double r474979 = y;
        double r474980 = sin(r474979);
        double r474981 = r474980 / r474979;
        double r474982 = r474978 * r474981;
        double r474983 = z;
        double r474984 = r474982 / r474983;
        return r474984;
}
↓
double f(double x, double y, double z) {
        double r474985 = x;
        double r474986 = y;
        double r474987 = sin(r474986);
        double r474988 = r474987 / r474986;
        double r474989 = r474985 * r474988;
        double r474990 = z;
        double r474991 = r474989 / r474990;
        double r474992 = -5.402522279945178e-206;
        bool r474993 = r474991 <= r474992;
        double r474994 = r474985 / r474990;
        double r474995 = 1.0;
        double r474996 = r474986 / r474987;
        double r474997 = r474995 / r474996;
        double r474998 = r474994 * r474997;
        double r474999 = r474993 ? r474991 : r474998;
        return r474999;
}

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
Target0.3
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;z \lt -4.21737202034271466 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* x (/ (sin y) y)) z) < -5.402522279945178e-206

    1. Initial program 0.3

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

    if -5.402522279945178e-206 < (/ (* x (/ (sin y) y)) z)

    1. Initial program 4.1

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

    3. Applied associate-/l*3.0

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

    4. Simplified3.0

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

    6. Applied associate-/r*2.0

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

    8. Applied div-inv2.0

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \le -5.40252227994517841 \cdot 10^{-206}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{1}{\frac{y}{\sin y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))