Average Error: 36.8 → 0.4
Time: 10.9s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x \]
↓
\[\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right) \]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (+ (* (sin eps) (cos x)) (* (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 (sin(eps) * cos(x)) + (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

Original36.8
Target15.0
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \]

Derivation

  1. Initial program 36.8

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

  3. Applied sin-sum_binary6421.6

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

  4. Simplified21.6

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

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

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

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

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

  8. Applied distribute-lft-out--_binary6421.6

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2021206 
(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)))