Average Error: 37.8 → 0.4
Time: 11.6s
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)
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (+ (* (cos x) (sin eps)) (* (sin x) (+ (cos eps) -1.0))))
double code(double x, double eps) {
	return sin(x + eps) - sin(x);
}
↓
double code(double x, double eps) {
	return (cos(x) * sin(eps)) + (sin(x) * (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

Original37.8
Target15.8
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.8

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

  3. Applied sin-sum_binary64_225721.9

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

  4. Applied associate--l+_binary64_206121.9

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

  5. Taylor expanded around inf 21.9

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2021007 
(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)))