Average Error: 2.8 → 0.3
Time: 4.5s
Precision: 64
\[\frac{x \cdot \frac{\sin y}{y}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -87678001982141.2969 \lor \neg \left(z \le 4.87378218103819904 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\sin y} \cdot z}\\ \end{array}\]
\frac{x \cdot \frac{\sin y}{y}}{z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -87678001982141.2969 \lor \neg \left(z \le 4.87378218103819904 \cdot 10^{-93}\right):\\
\;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r464391 = x;
        double r464392 = y;
        double r464393 = sin(r464392);
        double r464394 = r464393 / r464392;
        double r464395 = r464391 * r464394;
        double r464396 = z;
        double r464397 = r464395 / r464396;
        return r464397;
}
↓
double f(double x, double y, double z) {
        double r464398 = z;
        double r464399 = -87678001982141.3;
        bool r464400 = r464398 <= r464399;
        double r464401 = 4.873782181038199e-93;
        bool r464402 = r464398 <= r464401;
        double r464403 = !r464402;
        bool r464404 = r464400 || r464403;
        double r464405 = x;
        double r464406 = r464405 / r464398;
        double r464407 = y;
        double r464408 = sin(r464407);
        double r464409 = r464407 / r464408;
        double r464410 = r464406 / r464409;
        double r464411 = r464409 * r464398;
        double r464412 = r464405 / r464411;
        double r464413 = r464404 ? r464410 : r464412;
        return r464413;
}

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 < -87678001982141.3 or 4.873782181038199e-93 < z

    1. Initial program 0.4

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

    3. Applied clear-num0.4

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{\frac{y}{\sin y}}}}{z}\]
    4. Using strategy rm

    5. Applied div-inv0.5

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\frac{y}{\sin y}}\right) \cdot \frac{1}{z}}\]
    6. Using strategy rm

    7. Applied un-div-inv0.5

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

    8. Applied associate-*l/0.4

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

    9. Simplified0.3

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

    if -87678001982141.3 < z < 4.873782181038199e-93

    1. Initial program 6.6

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

    3. Applied clear-num6.6

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{\frac{y}{\sin y}}}}{z}\]
    4. Using strategy rm

    5. Applied div-inv6.8

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\frac{y}{\sin y}}\right) \cdot \frac{1}{z}}\]
    6. Using strategy rm

    7. Applied un-div-inv6.7

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

    8. Applied frac-times0.3

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

    9. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -87678001982141.2969 \lor \neg \left(z \le 4.87378218103819904 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\sin y} \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +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))