Average Error: 2.8 → 0.3
Time: 4.3s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.3412211236133659 \cdot 10^{-51} \lor \neg \left(z \le 9.6931285533073065 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{x \cdot \left(\sin y \cdot \frac{1}{y}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -2.3412211236133659 \cdot 10^{-51} \lor \neg \left(z \le 9.6931285533073065 \cdot 10^{-43}\right):\\
\;\;\;\;\frac{x \cdot \left(\sin y \cdot \frac{1}{y}\right)}{z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r509027 = x;
        double r509028 = y;
        double r509029 = sin(r509028);
        double r509030 = r509029 / r509028;
        double r509031 = r509027 * r509030;
        double r509032 = z;
        double r509033 = r509031 / r509032;
        return r509033;
}
↓
double f(double x, double y, double z) {
        double r509034 = z;
        double r509035 = -2.341221123613366e-51;
        bool r509036 = r509034 <= r509035;
        double r509037 = 9.693128553307307e-43;
        bool r509038 = r509034 <= r509037;
        double r509039 = !r509038;
        bool r509040 = r509036 || r509039;
        double r509041 = x;
        double r509042 = y;
        double r509043 = sin(r509042);
        double r509044 = 1.0;
        double r509045 = r509044 / r509042;
        double r509046 = r509043 * r509045;
        double r509047 = r509041 * r509046;
        double r509048 = r509047 / r509034;
        double r509049 = r509043 / r509042;
        double r509050 = r509049 / r509034;
        double r509051 = r509041 * r509050;
        double r509052 = r509040 ? r509048 : r509051;
        return r509052;
}

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

Derivation

  1. Split input into 2 regimes

  2. if z < -2.341221123613366e-51 or 9.693128553307307e-43 < z

    1. Initial program 0.3

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

    3. Applied div-inv0.4

      \[\leadsto \frac{x \cdot \color{blue}{\left(\sin y \cdot \frac{1}{y}\right)}}{z}\]

    if -2.341221123613366e-51 < z < 9.693128553307307e-43

    1. Initial program 6.9

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

    3. Applied *-un-lft-identity6.9

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

    4. Applied times-frac0.3

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

    5. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.3412211236133659 \cdot 10^{-51} \lor \neg \left(z \le 9.6931285533073065 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{x \cdot \left(\sin y \cdot \frac{1}{y}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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))