Average Error: 29.9 → 29.8
Time: 6.6s
Precision: binary64
\[\frac{\left(\frac{\sin \left(\left(x + \varepsilon\right) - \varepsilon\right)}{\cos x \cdot \cos \varepsilon} \cdot \cos x\right) \cdot \cos \varepsilon}{\sin \varepsilon}\]
↓
\[\frac{\sin \left(\left(x + \varepsilon\right) - \varepsilon\right)}{\sin \varepsilon}\]
\frac{\left(\frac{\sin \left(\left(x + \varepsilon\right) - \varepsilon\right)}{\cos x \cdot \cos \varepsilon} \cdot \cos x\right) \cdot \cos \varepsilon}{\sin \varepsilon}
↓
\frac{\sin \left(\left(x + \varepsilon\right) - \varepsilon\right)}{\sin \varepsilon}
double code(double x, double eps) {
	return ((double) (((double) (((double) (((double) (((double) sin(((double) (((double) (x + eps)) - eps)))) / ((double) (((double) cos(x)) * ((double) cos(eps)))))) * ((double) cos(x)))) * ((double) cos(eps)))) / ((double) sin(eps))));
}
↓
double code(double x, double eps) {
	return ((double) (((double) sin(((double) (((double) (x + eps)) - eps)))) / ((double) sin(eps))));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.9

    \[\frac{\left(\frac{\sin \left(\left(x + \varepsilon\right) - \varepsilon\right)}{\cos x \cdot \cos \varepsilon} \cdot \cos x\right) \cdot \cos \varepsilon}{\sin \varepsilon}\]

  2. Simplified29.8

    \[\leadsto \color{blue}{\frac{\sin \left(\left(x + \varepsilon\right) - \varepsilon\right)}{\sin \varepsilon}}\]

  3. Final simplification29.8

    \[\leadsto \frac{\sin \left(\left(x + \varepsilon\right) - \varepsilon\right)}{\sin \varepsilon}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (x eps)
  :name "(/ (* (* (/ (sin (- (+ x eps) eps)) (* (cos x) (cos eps))) (cos x)) (cos eps)) (sin eps))"
  :precision binary64
  (/ (* (* (/ (sin (- (+ x eps) eps)) (* (cos x) (cos eps))) (cos x)) (cos eps)) (sin eps)))