Average Error: 36.3 → 0.2
Time: 10.3s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\sin \varepsilon \cdot \tan \left(\frac{-\varepsilon}{2}\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\cos x \cdot \sin \varepsilon + \sin x \cdot \left(\sin \varepsilon \cdot \tan \left(\frac{-\varepsilon}{2}\right)\right)
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (+ (* (cos x) (sin eps)) (* (sin x) (* (sin eps) (tan (/ (- eps) 2.0))))))
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) * (sin(eps) * tan(-eps / 2.0)));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.3
Target14.7
Herbie0.2
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.3

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

  3. Applied sin-sum_binary64_191621.4

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

  4. Simplified21.4

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

  5. Simplified21.4

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

  6. Taylor expanded around inf 21.4

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

  7. Simplified0.4

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

  9. Applied flip-+_binary64_17570.5

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

  10. Simplified0.3

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

  11. Simplified0.3

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

  13. Applied *-un-lft-identity_binary64_17830.3

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

  14. Applied times-frac_binary64_17890.3

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

  15. Simplified0.3

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

  16. Simplified0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2021050 
(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)))