Average Error: 14.3 → 0.1
Time: 3.8s
Precision: 64
\[\frac{\sin x \cdot \sinh y}{x}\]
↓
\[\frac{\sin x}{\frac{x}{1}} \cdot \sinh y\]
\frac{\sin x \cdot \sinh y}{x}
↓
\frac{\sin x}{\frac{x}{1}} \cdot \sinh y
double code(double x, double y) {
	return ((sin(x) * sinh(y)) / x);
}
↓
double code(double x, double y) {
	return ((sin(x) / (x / 1.0)) * sinh(y));
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original14.3
Target0.2
Herbie0.1
\[\sin x \cdot \frac{\sinh y}{x}\]

Derivation

  1. Initial program 14.3

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

  2. Taylor expanded around inf 43.9

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

  3. Simplified0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020078 +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))