Average Error: 36.9 → 0.4
Time: 5.7s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \frac{\sqrt[3]{{\left({\left(\cos \varepsilon\right)}^{3} - 1\right)}^{3}}}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \frac{\sqrt[3]{{\left({\left(\cos \varepsilon\right)}^{3} - 1\right)}^{3}}}{\cos \varepsilon \cdot \left(\cos \varepsilon + 1\right) + 1} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r127037 = x;
        double r127038 = eps;
        double r127039 = r127037 + r127038;
        double r127040 = sin(r127039);
        double r127041 = sin(r127037);
        double r127042 = r127040 - r127041;
        return r127042;
}
↓
double f(double x, double eps) {
        double r127043 = x;
        double r127044 = sin(r127043);
        double r127045 = eps;
        double r127046 = cos(r127045);
        double r127047 = 3.0;
        double r127048 = pow(r127046, r127047);
        double r127049 = 1.0;
        double r127050 = r127048 - r127049;
        double r127051 = pow(r127050, r127047);
        double r127052 = cbrt(r127051);
        double r127053 = r127046 + r127049;
        double r127054 = r127046 * r127053;
        double r127055 = r127054 + r127049;
        double r127056 = r127052 / r127055;
        double r127057 = r127044 * r127056;
        double r127058 = cos(r127043);
        double r127059 = sin(r127045);
        double r127060 = r127058 * r127059;
        double r127061 = r127057 + r127060;
        return r127061;
}

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
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 36.9

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

  3. Applied sin-sum21.4

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

  4. Applied associate--l+21.4

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

  5. Taylor expanded around inf 21.4

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

  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. Simplified0.4

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

  10. Simplified0.4

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

  12. Applied add-cbrt-cube0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2020065 
(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)))