Average Error: 39.2 → 0.5
Time: 20.2s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\left(\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right)\right) \cdot \sqrt[3]{\cos x} + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\left(\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right)\right) \cdot \sqrt[3]{\cos x} + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)
double f(double x, double eps) {
        double r52128 = x;
        double r52129 = eps;
        double r52130 = r52128 + r52129;
        double r52131 = cos(r52130);
        double r52132 = cos(r52128);
        double r52133 = r52131 - r52132;
        return r52133;
}
↓
double f(double x, double eps) {
        double r52134 = eps;
        double r52135 = 0.5;
        double r52136 = r52134 * r52135;
        double r52137 = sin(r52136);
        double r52138 = x;
        double r52139 = cos(r52138);
        double r52140 = cbrt(r52139);
        double r52141 = r52140 * r52140;
        double r52142 = r52137 * r52141;
        double r52143 = r52142 * r52140;
        double r52144 = cos(r52136);
        double r52145 = sin(r52138);
        double r52146 = r52144 * r52145;
        double r52147 = r52143 + r52146;
        double r52148 = -2.0;
        double r52149 = r52135 * r52134;
        double r52150 = sin(r52149);
        double r52151 = r52148 * r52150;
        double r52152 = r52147 * r52151;
        return r52152;
}

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 39.2

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

  3. Applied diff-cos33.5

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

  4. Simplified14.6

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

  5. Taylor expanded around inf 14.6

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

  6. Simplified14.5

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

  8. Applied fma-udef14.5

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

  9. Applied sin-sum0.4

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

  11. Applied add-cube-cbrt0.5

    \[\leadsto \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}\right)} + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

  12. Applied associate-*r*0.5

    \[\leadsto \left(\color{blue}{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right)\right) \cdot \sqrt[3]{\cos x}} + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

  13. Final simplification0.5

    \[\leadsto \left(\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right)\right) \cdot \sqrt[3]{\cos x} + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) \cdot \left(-2 \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))