Average Error: 14.3 → 0.2
Time: 34.3s
Precision: 64
\[\frac{\sin x \cdot \sinh y}{x}\]
↓
\[\frac{\sinh y}{x} \cdot \sin x\]
\frac{\sin x \cdot \sinh y}{x}
↓
\frac{\sinh y}{x} \cdot \sin x
double f(double x, double y) {
        double r359620 = x;
        double r359621 = sin(r359620);
        double r359622 = y;
        double r359623 = sinh(r359622);
        double r359624 = r359621 * r359623;
        double r359625 = r359624 / r359620;
        return r359625;
}
↓
double f(double x, double y) {
        double r359626 = y;
        double r359627 = sinh(r359626);
        double r359628 = x;
        double r359629 = r359627 / r359628;
        double r359630 = sin(r359628);
        double r359631 = r359629 * r359630;
        return r359631;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original14.3
Target0.2
Herbie0.2
\[\sin x \cdot \frac{\sinh y}{x}\]

Derivation

  1. Initial program 14.3

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

  2. Taylor expanded around inf 43.7

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019326 
(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))