Average Error: 37.0 → 0.3
Time: 5.9s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right) - \sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right) - \sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right)
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (*
  2.0
  (*
   (sin (/ eps 2.0))
   (- (* (cos x) (cos (* eps 0.5))) (* (sin x) (sin (* eps 0.5)))))))
double code(double x, double eps) {
	return sin(x + eps) - sin(x);
}
↓
double code(double x, double eps) {
	return 2.0 * (sin(eps / 2.0) * ((cos(x) * cos(eps * 0.5)) - (sin(x) * sin(eps * 0.5))));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target15.3
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.0

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

  3. Applied diff-sin_binary64_194937.4

    \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

  4. Simplified15.3

    \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]

  5. Taylor expanded around inf 15.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\cos \left(0.5 \cdot \left(2 \cdot x + \varepsilon\right)\right)}\right)\]

  6. Simplified15.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\cos \left(x + \varepsilon \cdot 0.5\right)}\right)\]
  7. Using strategy rm

  8. Applied cos-sum_binary64_19330.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right) - \sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}\right)\]

  9. Final simplification0.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right) - \sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right)\]

Reproduce

herbie shell --seed 2020280 
(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)))