Average Error: 37.5 → 0.4
Time: 10.0s
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

Original37.5
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 37.5

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

  3. Applied sin-sum_binary6421.7

    \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
  4. Using strategy rm

  5. Applied *-un-lft-identity_binary6421.7

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

  6. Applied *-un-lft-identity_binary6421.7

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

  7. Applied distribute-lft-out--_binary6421.7

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020288 
(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)))