Average Error: 37.8 → 0.4
Time: 10.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\cos x \cdot \sin \varepsilon + \frac{\sin x}{\frac{\left(\cos \varepsilon \cdot \cos \varepsilon + 1\right) + \cos \varepsilon}{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\cos x \cdot \sin \varepsilon + \frac{\sin x}{\frac{\left(\cos \varepsilon \cdot \cos \varepsilon + 1\right) + \cos \varepsilon}{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}}
double f(double x, double eps) {
        double r104565 = x;
        double r104566 = eps;
        double r104567 = r104565 + r104566;
        double r104568 = sin(r104567);
        double r104569 = sin(r104565);
        double r104570 = r104568 - r104569;
        return r104570;
}
↓
double f(double x, double eps) {
        double r104571 = x;
        double r104572 = cos(r104571);
        double r104573 = eps;
        double r104574 = sin(r104573);
        double r104575 = r104572 * r104574;
        double r104576 = sin(r104571);
        double r104577 = cos(r104573);
        double r104578 = r104577 * r104577;
        double r104579 = 1.0;
        double r104580 = r104578 + r104579;
        double r104581 = r104580 + r104577;
        double r104582 = 3.0;
        double r104583 = pow(r104577, r104582);
        double r104584 = pow(r104579, r104582);
        double r104585 = r104583 - r104584;
        double r104586 = r104581 / r104585;
        double r104587 = r104576 / r104586;
        double r104588 = r104575 + r104587;
        return r104588;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.8
Target15.4
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.8

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

  3. Applied sin-sum22.3

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

  4. Applied associate--l+22.3

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

  5. Taylor expanded around inf 22.3

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

  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. Final simplification0.4

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

Reproduce

herbie shell --seed 197574269 
(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)))