Average Error: 37.1 → 0.4
Time: 5.7s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\cos x \cdot \sin \varepsilon - {\sin \varepsilon}^{2} \cdot \frac{\sin x}{1 + \cos \varepsilon}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\cos x \cdot \sin \varepsilon - {\sin \varepsilon}^{2} \cdot \frac{\sin x}{1 + \cos \varepsilon}
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (-
  (* (cos x) (sin eps))
  (* (pow (sin eps) 2.0) (/ (sin x) (+ 1.0 (cos eps))))))
double code(double x, double eps) {
	return sin(x + eps) - sin(x);
}
↓
double code(double x, double eps) {
	return (cos(x) * sin(eps)) - (pow(sin(eps), 2.0) * (sin(x) / (1.0 + cos(eps))));
}

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
Target15.3
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.1

    \[\sin \left(x + \varepsilon\right) - \sin x\]
  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. Simplified21.7

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

  5. Simplified21.7

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

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

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

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

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

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

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

  10. Simplified0.4

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

  12. Applied flip-+_binary640.5

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

  13. Applied associate-*r/_binary640.5

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

  14. Simplified0.4

    \[\leadsto 1 \cdot \left(\cos x \cdot \sin \varepsilon + \frac{\color{blue}{\left(\sin \varepsilon \cdot \sin \varepsilon\right) \cdot \left(-\sin x\right)}}{\cos \varepsilon - -1}\right)\]
  15. Using strategy rm

  16. Applied *-un-lft-identity_binary640.4

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

  17. Applied times-frac_binary640.4

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

  18. Simplified0.4

    \[\leadsto 1 \cdot \left(\cos x \cdot \sin \varepsilon + \color{blue}{{\sin \varepsilon}^{2}} \cdot \frac{-\sin x}{\cos \varepsilon - -1}\right)\]

  19. Simplified0.4

    \[\leadsto 1 \cdot \left(\cos x \cdot \sin \varepsilon + {\sin \varepsilon}^{2} \cdot \color{blue}{\frac{-\sin x}{\cos \varepsilon + 1}}\right)\]

  20. Final simplification0.4

    \[\leadsto \cos x \cdot \sin \varepsilon - {\sin \varepsilon}^{2} \cdot \frac{\sin x}{1 + \cos \varepsilon}\]

Reproduce

herbie shell --seed 2021173 
(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)))