Average Error: 10.2 → 10.2
Time: 2.8s
Precision: binary64
\[x - \frac{\sin x - \frac{x}{2}}{\cos x - \frac{1}{2}}\]
↓
\[x - \frac{\sin x - \frac{x}{2}}{\cos x - \frac{1}{2}}\]
x - \frac{\sin x - \frac{x}{2}}{\cos x - \frac{1}{2}}
↓
x - \frac{\sin x - \frac{x}{2}}{\cos x - \frac{1}{2}}
double code(double x) {
	return ((double) (x - ((double) (((double) (((double) sin(x)) - ((double) (x / 2.0)))) / ((double) (((double) cos(x)) - ((double) (1.0 / 2.0))))))));
}
↓
double code(double x) {
	return ((double) (x - ((double) (((double) (((double) sin(x)) - ((double) (x / 2.0)))) / ((double) (((double) cos(x)) - ((double) (1.0 / 2.0))))))));
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 10.2

    \[x - \frac{\sin x - \frac{x}{2}}{\cos x - \frac{1}{2}}\]

  2. Final simplification10.2

    \[\leadsto x - \frac{\sin x - \frac{x}{2}}{\cos x - \frac{1}{2}}\]

Reproduce

herbie shell --seed 2020153 
(FPCore (x)
  :name "(- x (/ (- (sin x) (/ x 2)) (- (cos x) (/ 1 2))))"
  :precision binary64
  (- x (/ (- (sin x) (/ x 2.0)) (- (cos x) (/ 1.0 2.0)))))