Average Error: 37.5 → 0.5
Time: 6.2s
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 f(double x, double eps) {
        double r110061 = x;
        double r110062 = eps;
        double r110063 = r110061 + r110062;
        double r110064 = sin(r110063);
        double r110065 = sin(r110061);
        double r110066 = r110064 - r110065;
        return r110066;
}
↓
double f(double x, double eps) {
        double r110067 = x;
        double r110068 = sin(r110067);
        double r110069 = eps;
        double r110070 = cos(r110069);
        double r110071 = 3.0;
        double r110072 = pow(r110070, r110071);
        double r110073 = 1.0;
        double r110074 = r110072 - r110073;
        double r110075 = r110068 * r110074;
        double r110076 = r110070 * r110070;
        double r110077 = r110070 * r110073;
        double r110078 = r110073 + r110077;
        double r110079 = r110076 + r110078;
        double r110080 = r110075 / r110079;
        double r110081 = cos(r110067);
        double r110082 = sin(r110069);
        double r110083 = r110081 * r110082;
        double r110084 = r110080 + r110083;
        return r110084;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.5
Target14.9
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.5

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

  3. Applied sin-sum22.5

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

  5. Applied *-un-lft-identity22.5

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

  6. Applied *-un-lft-identity22.5

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

  7. Applied distribute-lft-out--22.5

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

  8. Simplified0.4

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \cos x \cdot \sin \varepsilon\right)}\]
  9. Using strategy rm

  10. Applied fma-udef0.4

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

  12. Applied flip3--0.5

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

  13. Applied associate-*r/0.5

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

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(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)))