Average Error: 36.9 → 0.4
Time: 6.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\frac{\sin x \cdot \left({\left(\cos \varepsilon\right)}^{3} - 1\right)}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 + \cos \varepsilon \cdot 1\right)} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\frac{\sin x \cdot \left({\left(\cos \varepsilon\right)}^{3} - 1\right)}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 + \cos \varepsilon \cdot 1\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) * (pow(cos(eps), 3.0) - 1.0)) / ((cos(eps) * cos(eps)) + (1.0 + (cos(eps) * 1.0)))) + (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

Original36.9
Target14.7
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 sin-sum22.1

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

  4. Applied associate--l+22.1

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

  5. Taylor expanded around inf 22.1

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

  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 flip3--0.4

    \[\leadsto \sin x \cdot \color{blue}{\frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}} + \cos x \cdot \sin \varepsilon\]

  9. Applied associate-*r/0.4

    \[\leadsto \color{blue}{\frac{\sin x \cdot \left({\left(\cos \varepsilon\right)}^{3} - {1}^{3}\right)}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}} + \cos x \cdot \sin \varepsilon\]

  10. Simplified0.4

    \[\leadsto \frac{\color{blue}{\sin x \cdot \left({\left(\cos \varepsilon\right)}^{3} - 1\right)}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)} + \cos x \cdot \sin \varepsilon\]

  11. Final simplification0.4

    \[\leadsto \frac{\sin x \cdot \left({\left(\cos \varepsilon\right)}^{3} - 1\right)}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 + \cos \varepsilon \cdot 1\right)} + \cos x \cdot \sin \varepsilon\]

Reproduce

herbie shell --seed 2020053 
(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)))