Average Error: 37.1 → 0.4
Time: 6.1s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)
double code(double x, double eps) {
	return ((double) (((double) sin(((double) (x + eps)))) - ((double) sin(x))));
}
↓
double code(double x, double eps) {
	return ((double) (((double) (((double) cos(x)) * ((double) sin(eps)))) + ((double) (((double) sin(x)) * ((double) (((double) cos(eps)) + -1.0))))));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target14.7
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program Error: 37.1 bits

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  2. Using strategy rm

  3. Applied sin-sumError: 22.2 bits

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

  4. Taylor expanded around inf Error: 22.2 bits

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

  5. SimplifiedError: 0.4 bits

    \[\leadsto \color{blue}{\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)}\]

  6. Final simplificationError: 0.4 bits

    \[\leadsto \cos x \cdot \sin \varepsilon + \sin x \cdot \left(\cos \varepsilon + -1\right)\]

Reproduce

herbie shell --seed 2020200 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))