Average Error: 37.3 → 0.5
Time: 6.0s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\sin x \cdot \frac{\cos \varepsilon \cdot \cos \varepsilon - 1}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\sin x \cdot \frac{\cos \varepsilon \cdot \cos \varepsilon - 1}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon
double f(double x, double eps) {
        double r121734 = x;
        double r121735 = eps;
        double r121736 = r121734 + r121735;
        double r121737 = sin(r121736);
        double r121738 = sin(r121734);
        double r121739 = r121737 - r121738;
        return r121739;
}
↓
double f(double x, double eps) {
        double r121740 = x;
        double r121741 = sin(r121740);
        double r121742 = eps;
        double r121743 = cos(r121742);
        double r121744 = r121743 * r121743;
        double r121745 = 1.0;
        double r121746 = r121744 - r121745;
        double r121747 = r121743 + r121745;
        double r121748 = r121746 / r121747;
        double r121749 = r121741 * r121748;
        double r121750 = cos(r121740);
        double r121751 = sin(r121742);
        double r121752 = r121750 * r121751;
        double r121753 = r121749 + r121752;
        return r121753;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target15.3
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.3

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

  3. Applied sin-sum21.9

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

  4. Applied associate--l+21.9

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

  5. Taylor expanded around inf 21.9

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

  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 flip--0.5

    \[\leadsto \sin x \cdot \color{blue}{\frac{\cos \varepsilon \cdot \cos \varepsilon - 1 \cdot 1}{\cos \varepsilon + 1}} + \cos x \cdot \sin \varepsilon\]

  9. Simplified0.5

    \[\leadsto \sin x \cdot \frac{\color{blue}{\cos \varepsilon \cdot \cos \varepsilon - 1}}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]

  10. Final simplification0.5

    \[\leadsto \sin x \cdot \frac{\cos \varepsilon \cdot \cos \varepsilon - 1}{\cos \varepsilon + 1} + \cos x \cdot \sin \varepsilon\]

Reproduce

herbie shell --seed 2020049 
(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)))