Average Error: 40.0 → 0.4
Time: 1.7m
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\frac{\left(\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) \cdot \left(\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) - \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos x\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos x\right)}{\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x - \sin \left(\frac{\varepsilon}{2}\right) \cdot \cos x} \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\frac{\left(\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) \cdot \left(\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) - \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos x\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos x\right)}{\cos \left(\frac{\varepsilon}{2}\right) \cdot \sin x - \sin \left(\frac{\varepsilon}{2}\right) \cdot \cos x} \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)
double f(double x, double eps) {
        double r1839561 = x;
        double r1839562 = eps;
        double r1839563 = r1839561 + r1839562;
        double r1839564 = cos(r1839563);
        double r1839565 = cos(r1839561);
        double r1839566 = r1839564 - r1839565;
        return r1839566;
}
↓
double f(double x, double eps) {
        double r1839567 = eps;
        double r1839568 = 2.0;
        double r1839569 = r1839567 / r1839568;
        double r1839570 = cos(r1839569);
        double r1839571 = x;
        double r1839572 = sin(r1839571);
        double r1839573 = r1839570 * r1839572;
        double r1839574 = r1839573 * r1839573;
        double r1839575 = sin(r1839569);
        double r1839576 = cos(r1839571);
        double r1839577 = r1839575 * r1839576;
        double r1839578 = r1839577 * r1839577;
        double r1839579 = r1839574 - r1839578;
        double r1839580 = r1839573 - r1839577;
        double r1839581 = r1839579 / r1839580;
        double r1839582 = -2.0;
        double r1839583 = r1839582 * r1839575;
        double r1839584 = r1839581 * r1839583;
        return r1839584;
}

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

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

  5. Taylor expanded around inf 15.1

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

  6. Simplified15.1

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

  8. Applied sin-sum0.4

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

  10. Applied flip-+0.4

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

  11. Final simplification0.4

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

Reproduce

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