Average Error: 14.4 → 0.1
Time: 13.1s
Precision: 64
\[\frac{\sin x \cdot \sinh y}{x}\]
↓
\[\frac{\frac{2 \cdot \sinh y}{\frac{x}{\sin x}}}{2}\]
\frac{\sin x \cdot \sinh y}{x}
↓
\frac{\frac{2 \cdot \sinh y}{\frac{x}{\sin x}}}{2}
double f(double x, double y) {
        double r524562 = x;
        double r524563 = sin(r524562);
        double r524564 = y;
        double r524565 = sinh(r524564);
        double r524566 = r524563 * r524565;
        double r524567 = r524566 / r524562;
        return r524567;
}
↓
double f(double x, double y) {
        double r524568 = 2.0;
        double r524569 = y;
        double r524570 = sinh(r524569);
        double r524571 = r524568 * r524570;
        double r524572 = x;
        double r524573 = sin(r524572);
        double r524574 = r524572 / r524573;
        double r524575 = r524571 / r524574;
        double r524576 = r524575 / r524568;
        return r524576;
}

Error

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Results

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Target

Original14.4
Target0.2
Herbie0.1
\[\sin x \cdot \frac{\sinh y}{x}\]

Derivation

  1. Initial program 14.4

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

  3. Applied *-un-lft-identity14.4

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

  4. Applied times-frac0.2

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

  5. Simplified0.2

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

  7. Applied div-inv0.3

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

  9. Applied sinh-def43.8

    \[\leadsto \sin x \cdot \left(\color{blue}{\frac{e^{y} - e^{-y}}{2}} \cdot \frac{1}{x}\right)\]

  10. Applied associate-*l/43.8

    \[\leadsto \sin x \cdot \color{blue}{\frac{\left(e^{y} - e^{-y}\right) \cdot \frac{1}{x}}{2}}\]

  11. Applied associate-*r/43.8

    \[\leadsto \color{blue}{\frac{\sin x \cdot \left(\left(e^{y} - e^{-y}\right) \cdot \frac{1}{x}\right)}{2}}\]

  12. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{2 \cdot \sinh y}{\frac{x}{\sin x}}}}{2}\]

  13. Final simplification0.1

    \[\leadsto \frac{\frac{2 \cdot \sinh y}{\frac{x}{\sin x}}}{2}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (* (sin x) (/ (sinh y) x))

  (/ (* (sin x) (sinh y)) x))