Average Error: 39.5 → 0.4
Time: 16.0s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot -2\right)
double f(double x, double eps) {
        double r44370 = x;
        double r44371 = eps;
        double r44372 = r44370 + r44371;
        double r44373 = cos(r44372);
        double r44374 = cos(r44370);
        double r44375 = r44373 - r44374;
        return r44375;
}
↓
double f(double x, double eps) {
        double r44376 = 0.5;
        double r44377 = eps;
        double r44378 = r44376 * r44377;
        double r44379 = sin(r44378);
        double r44380 = x;
        double r44381 = cos(r44380);
        double r44382 = r44379 * r44381;
        double r44383 = expm1(r44382);
        double r44384 = log1p(r44383);
        double r44385 = cos(r44378);
        double r44386 = sin(r44380);
        double r44387 = r44385 * r44386;
        double r44388 = r44384 + r44387;
        double r44389 = -2.0;
        double r44390 = r44388 * r44389;
        double r44391 = r44379 * r44390;
        return r44391;
}

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 39.5

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

  3. Applied diff-cos34.0

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

  4. Simplified14.5

    \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

  5. Taylor expanded around inf 14.5

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

  6. Simplified14.5

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

  8. Applied fma-udef14.5

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

  9. Applied sin-sum0.4

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

  11. Applied log1p-expm1-u0.4

    \[\leadsto \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\right)} + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot -2\right)\]

  12. Final simplification0.4

    \[\leadsto \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right)\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot -2\right)\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))