Average Error: 14.6 → 0.1
Time: 13.1s
Precision: 64
\[\frac{\sin x \cdot \sinh y}{x}\]
↓
\[\sinh y \cdot \frac{\sin x}{x}\]
\frac{\sin x \cdot \sinh y}{x}
↓
\sinh y \cdot \frac{\sin x}{x}
double f(double x, double y) {
        double r588837 = x;
        double r588838 = sin(r588837);
        double r588839 = y;
        double r588840 = sinh(r588839);
        double r588841 = r588838 * r588840;
        double r588842 = r588841 / r588837;
        return r588842;
}
↓
double f(double x, double y) {
        double r588843 = y;
        double r588844 = sinh(r588843);
        double r588845 = x;
        double r588846 = sin(r588845);
        double r588847 = r588846 / r588845;
        double r588848 = r588844 * r588847;
        return r588848;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original14.6
Target0.2
Herbie0.1
\[\sin x \cdot \frac{\sinh y}{x}\]

Derivation

  1. Initial program 14.6

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

  2. Taylor expanded around inf 43.6

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

  3. Simplified0.2

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

  5. Applied div-inv0.3

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

  6. Applied associate-*l*0.2

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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))