Average Error: 39.7 → 0.4
Time: 26.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\frac{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x + \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)}{\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x} \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\frac{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x + \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)}{\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x} \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)
double f(double x, double eps) {
        double r48406 = x;
        double r48407 = eps;
        double r48408 = r48406 + r48407;
        double r48409 = cos(r48408);
        double r48410 = cos(r48406);
        double r48411 = r48409 - r48410;
        return r48411;
}
↓
double f(double x, double eps) {
        double r48412 = 0.5;
        double r48413 = eps;
        double r48414 = r48412 * r48413;
        double r48415 = sin(r48414);
        double r48416 = x;
        double r48417 = cos(r48416);
        double r48418 = r48415 * r48417;
        double r48419 = cos(r48414);
        double r48420 = sin(r48416);
        double r48421 = r48419 * r48420;
        double r48422 = r48418 - r48421;
        double r48423 = r48421 + r48418;
        double r48424 = r48422 * r48423;
        double r48425 = r48424 / r48422;
        double r48426 = -2.0;
        double r48427 = r48415 * r48426;
        double r48428 = r48425 * r48427;
        return r48428;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 39.7

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

  3. Applied diff-cos34.1

    \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

  4. Simplified15.0

    \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

  5. Taylor expanded around inf 15.0

    \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)}\]

  6. Simplified15.0

    \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \varepsilon + x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)}\]
  7. Using strategy rm

  8. Applied sin-sum0.4

    \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)} \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]
  9. Using strategy rm

  10. Applied flip-+0.4

    \[\leadsto \color{blue}{\frac{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) - \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)}{\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}} \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]

  11. Simplified0.4

    \[\leadsto \frac{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x + \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)}}{\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x} \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]

  12. Final simplification0.4

    \[\leadsto \frac{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x + \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)}{\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x} \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]

Reproduce

herbie shell --seed 2019347 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))