Average Error: 2.7 → 1.6
Time: 6.0s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.4086238538771711 \cdot 10^{84}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.4086238538771711 \cdot 10^{84}:\\
\;\;\;\;\frac{1}{\frac{z}{x \cdot \frac{\sin y}{y}}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r580640 = x;
        double r580641 = y;
        double r580642 = sin(r580641);
        double r580643 = r580642 / r580641;
        double r580644 = r580640 * r580643;
        double r580645 = z;
        double r580646 = r580644 / r580645;
        return r580646;
}
↓
double f(double x, double y, double z) {
        double r580647 = x;
        double r580648 = -1.408623853877171e+84;
        bool r580649 = r580647 <= r580648;
        double r580650 = 1.0;
        double r580651 = z;
        double r580652 = y;
        double r580653 = sin(r580652);
        double r580654 = r580653 / r580652;
        double r580655 = r580647 * r580654;
        double r580656 = r580651 / r580655;
        double r580657 = r580650 / r580656;
        double r580658 = r580654 / r580651;
        double r580659 = r580647 * r580658;
        double r580660 = r580649 ? r580657 : r580659;
        return r580660;
}

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
Target0.2
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;z \lt -4.21737202034271466 \cdot 10^{-29}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{\frac{y}{\sin y}}}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.408623853877171e+84

    1. Initial program 0.2

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

    3. Applied clear-num0.4

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

    if -1.408623853877171e+84 < x

    1. Initial program 3.1

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

    3. Applied *-un-lft-identity3.1

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

    4. Applied times-frac1.9

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

    5. Simplified1.9

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.4086238538771711 \cdot 10^{84}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \frac{\sin y}{y}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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))