Average Error: 37.2 → 0.2
Time: 6.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \left(\left(-\sin \varepsilon\right) \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \left(\left(-\sin \varepsilon\right) \cdot \tan \left(\frac{\varepsilon}{2}\right)\right) + \cos x \cdot \sin \varepsilon
double code(double x, double eps) {
	return (sin((x + eps)) - sin(x));
}
↓
double code(double x, double eps) {
	return ((sin(x) * (-sin(eps) * tan((eps / 2.0)))) + (cos(x) * sin(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

Original37.2
Target15.1
Herbie0.2
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.2

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

  3. Applied sin-sum22.0

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

  4. Applied associate--l+22.0

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

  5. Taylor expanded around inf 22.0

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

  6. Simplified0.4

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

  8. Applied flip--0.5

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

  9. Simplified0.4

    \[\leadsto \sin x \cdot \frac{\color{blue}{-\sin \varepsilon \cdot \sin \varepsilon}}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]
  10. Using strategy rm

  11. Applied *-un-lft-identity0.4

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

  12. Applied distribute-lft-neg-in0.4

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

  13. Applied times-frac0.4

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

  14. Simplified0.4

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

  15. Simplified0.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2020103 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))