Average Error: 14.1 → 0.2
Time: 38.6s
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 r443516 = x;
        double r443517 = sin(r443516);
        double r443518 = y;
        double r443519 = sinh(r443518);
        double r443520 = r443517 * r443519;
        double r443521 = r443520 / r443516;
        return r443521;
}
↓
double f(double x, double y) {
        double r443522 = y;
        double r443523 = sinh(r443522);
        double r443524 = x;
        double r443525 = r443523 / r443524;
        double r443526 = sin(r443524);
        double r443527 = r443525 * r443526;
        return r443527;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

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

Derivation

  1. Initial program 14.1

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

  2. Taylor expanded around inf 43.9

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

  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 2019306 
(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))