Average Error: 36.9 → 0.4
Time: 5.5s
Precision: binary64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \frac{\left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) - \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}{\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right) + \sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)}\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \frac{\left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) - \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}{\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right) + \sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)}\right)
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
↓
(FPCore (x eps)
 :precision binary64
 (*
  2.0
  (*
   (sin (/ eps 2.0))
   (/
    (-
     (* (* (cos x) (cos (* eps 0.5))) (* (cos x) (cos (* eps 0.5))))
     (* (* (sin x) (sin (* eps 0.5))) (* (sin x) (sin (* eps 0.5)))))
    (+ (* (cos x) (cos (* eps 0.5))) (* (sin x) (sin (* eps 0.5))))))))
double code(double x, double eps) {
	return sin(x + eps) - sin(x);
}
↓
double code(double x, double eps) {
	return 2.0 * (sin(eps / 2.0) * ((((cos(x) * cos(eps * 0.5)) * (cos(x) * cos(eps * 0.5))) - ((sin(x) * sin(eps * 0.5)) * (sin(x) * sin(eps * 0.5)))) / ((cos(x) * cos(eps * 0.5)) + (sin(x) * sin(eps * 0.5)))));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
Target14.8
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

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

  3. Applied diff-sin_binary6437.2

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

  4. Simplified14.8

    \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube_binary6415.0

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)}}\right)\]

  7. Simplified15.0

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\color{blue}{{\cos \left(x + \varepsilon \cdot 0.5\right)}^{3}}}\right)\]
  8. Using strategy rm

  9. Applied cos-sum_binary640.6

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{{\color{blue}{\left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right) - \sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}}^{3}}\right)\]
  10. Using strategy rm

  11. Applied flip--_binary640.6

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{{\color{blue}{\left(\frac{\left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) - \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}{\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right) + \sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)}\right)}}^{3}}\right)\]

  12. Applied cube-div_binary640.6

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sqrt[3]{\color{blue}{\frac{{\left(\left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) - \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right)}^{3}}{{\left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right) + \sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}^{3}}}}\right)\]

  13. Applied cbrt-div_binary640.7

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\frac{\sqrt[3]{{\left(\left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) - \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\right)}^{3}}}{\sqrt[3]{{\left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right) + \sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}^{3}}}}\right)\]

  14. Simplified0.6

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \frac{\color{blue}{\left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) - \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}}{\sqrt[3]{{\left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right) + \sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}^{3}}}\right)\]

  15. Simplified0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \frac{\left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) - \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}{\color{blue}{\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right) + \sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)}}\right)\]

  16. Final simplification0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \frac{\left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right)\right) - \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \cdot \left(\sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)}{\cos x \cdot \cos \left(\varepsilon \cdot 0.5\right) + \sin x \cdot \sin \left(\varepsilon \cdot 0.5\right)}\right)\]

Reproduce

herbie shell --seed 2020232 
(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)))