Average Error: 13.7 → 0.1
Time: 19.5s
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 r26339593 = x;
        double r26339594 = sin(r26339593);
        double r26339595 = y;
        double r26339596 = sinh(r26339595);
        double r26339597 = r26339594 * r26339596;
        double r26339598 = r26339597 / r26339593;
        return r26339598;
}
↓
double f(double x, double y) {
        double r26339599 = y;
        double r26339600 = sinh(r26339599);
        double r26339601 = x;
        double r26339602 = sin(r26339601);
        double r26339603 = r26339602 / r26339601;
        double r26339604 = r26339600 * r26339603;
        return r26339604;
}

Error

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Bits error versus y

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Results

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Target

Original13.7
Target0.2
Herbie0.1
\[\sin x \cdot \frac{\sinh y}{x}\]

Derivation

  1. Initial program 13.7

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

  3. Applied associate-/l*0.7

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

  4. Taylor expanded around inf 42.8

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"

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

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