Average Error: 39.9 → 0.3
Time: 27.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right) \cdot \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right) \cdot \left(\cos x \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)
double f(double x, double eps) {
        double r6359128 = x;
        double r6359129 = eps;
        double r6359130 = r6359128 + r6359129;
        double r6359131 = cos(r6359130);
        double r6359132 = cos(r6359128);
        double r6359133 = r6359131 - r6359132;
        return r6359133;
}
↓
double f(double x, double eps) {
        double r6359134 = 0.5;
        double r6359135 = eps;
        double r6359136 = r6359134 * r6359135;
        double r6359137 = sin(r6359136);
        double r6359138 = -2.0;
        double r6359139 = r6359137 * r6359138;
        double r6359140 = x;
        double r6359141 = cos(r6359140);
        double r6359142 = r6359141 * r6359137;
        double r6359143 = cos(r6359136);
        double r6359144 = sin(r6359140);
        double r6359145 = r6359143 * r6359144;
        double r6359146 = r6359142 + r6359145;
        double r6359147 = r6359139 * r6359146;
        return r6359147;
}

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 39.9

    \[\cos \left(x + \varepsilon\right) - \cos x\]
  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. Simplified15.1

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

  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

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

  8. Applied sin-sum0.3

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

  10. Applied *-commutative0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019173 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))