Average Error: 36.9 → 0.4
Time: 5.8s
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 r99159 = x;
        double r99160 = eps;
        double r99161 = r99159 + r99160;
        double r99162 = sin(r99161);
        double r99163 = sin(r99159);
        double r99164 = r99162 - r99163;
        return r99164;
}
↓
double f(double x, double eps) {
        double r99165 = x;
        double r99166 = sin(r99165);
        double r99167 = eps;
        double r99168 = cos(r99167);
        double r99169 = 3.0;
        double r99170 = pow(r99168, r99169);
        double r99171 = 1.0;
        double r99172 = r99170 - r99171;
        double r99173 = r99166 * r99172;
        double r99174 = r99168 * r99168;
        double r99175 = r99168 * r99171;
        double r99176 = r99171 + r99175;
        double r99177 = r99174 + r99176;
        double r99178 = r99173 / r99177;
        double r99179 = cos(r99165);
        double r99180 = sin(r99167);
        double r99181 = r99179 * r99180;
        double r99182 = r99178 + r99181;
        return r99182;
}

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
Target15.2
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-sum21.6

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

  5. Applied *-un-lft-identity21.6

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

  6. Applied *-un-lft-identity21.6

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

  7. Applied distribute-lft-out--21.6

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

  8. Simplified0.4

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

  10. Applied flip3--0.4

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

  11. Applied associate-*r/0.4

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

  12. 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 2020020 
(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)))