Average Error: 39.8 → 0.4
Time: 15.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\left(\sin x \cdot -2\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\left(\sin x \cdot -2\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r19477 = x;
        double r19478 = eps;
        double r19479 = r19477 + r19478;
        double r19480 = cos(r19479);
        double r19481 = cos(r19477);
        double r19482 = r19480 - r19481;
        return r19482;
}
↓
double f(double x, double eps) {
        double r19483 = 0.5;
        double r19484 = eps;
        double r19485 = r19483 * r19484;
        double r19486 = sin(r19485);
        double r19487 = x;
        double r19488 = cos(r19487);
        double r19489 = r19486 * r19488;
        double r19490 = -2.0;
        double r19491 = r19490 * r19486;
        double r19492 = r19489 * r19491;
        double r19493 = cos(r19485);
        double r19494 = sin(r19487);
        double r19495 = r19494 * r19490;
        double r19496 = r19495 * r19486;
        double r19497 = r19493 * r19496;
        double r19498 = r19492 + r19497;
        return r19498;
}

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 39.8

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

  3. Applied diff-cos34.2

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

  4. Simplified15.5

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

  5. Taylor expanded around inf 15.5

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

  6. Simplified15.5

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

  8. Applied sin-sum0.4

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

  9. Applied distribute-lft-in0.4

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

  10. Simplified0.4

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

  11. Simplified0.4

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

  13. Applied associate-*l*0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019209 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))