Average Error: 40.0 → 0.4
Time: 37.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)
double f(double x, double eps) {
        double r2481801 = x;
        double r2481802 = eps;
        double r2481803 = r2481801 + r2481802;
        double r2481804 = cos(r2481803);
        double r2481805 = cos(r2481801);
        double r2481806 = r2481804 - r2481805;
        return r2481806;
}
↓
double f(double x, double eps) {
        double r2481807 = 0.5;
        double r2481808 = eps;
        double r2481809 = r2481807 * r2481808;
        double r2481810 = sin(r2481809);
        double r2481811 = x;
        double r2481812 = cos(r2481811);
        double r2481813 = r2481810 * r2481812;
        double r2481814 = cos(r2481809);
        double r2481815 = sin(r2481811);
        double r2481816 = r2481814 * r2481815;
        double r2481817 = r2481813 + r2481816;
        double r2481818 = expm1(r2481817);
        double r2481819 = log1p(r2481818);
        double r2481820 = -2.0;
        double r2481821 = r2481810 * r2481820;
        double r2481822 = r2481819 * r2481821;
        return r2481822;
}

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 40.0

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

  3. Applied diff-cos34.3

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

  4. Simplified15.1

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

  6. Applied associate-*r*15.1

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

  8. Applied log1p-expm1-u15.1

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

  9. Simplified15.1

    \[\leadsto \left(-2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\sin \left(\mathsf{fma}\left(\varepsilon, \frac{1}{2}, x\right)\right)\right)}\right)\]
  10. Using strategy rm

  11. Applied fma-udef15.1

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

  12. Applied sin-sum0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))