2tan (problem 3.3.2)

Average Error: 36.9 → 0.6

Time: 33.6s

Precision: 64

Math

\[\tan \left(x + \varepsilon\right) - \tan x\]

↓

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

C

double f(double x, double eps) {
        double r4179710 = x;
        double r4179711 = eps;
        double r4179712 = r4179710 + r4179711;
        double r4179713 = tan(r4179712);
        double r4179714 = tan(r4179710);
        double r4179715 = r4179713 - r4179714;
        return r4179715;
}

↓

double f(double x, double eps) {
        double r4179716 = x;
        double r4179717 = sin(r4179716);
        double r4179718 = cos(r4179716);
        double r4179719 = r4179717 / r4179718;
        double r4179720 = 1.0;
        double r4179721 = r4179717 * r4179717;
        double r4179722 = eps;
        double r4179723 = cos(r4179722);
        double r4179724 = r4179723 * r4179723;
        double r4179725 = r4179721 / r4179724;
        double r4179726 = sin(r4179722);
        double r4179727 = r4179726 * r4179726;
        double r4179728 = r4179718 * r4179718;
        double r4179729 = r4179727 / r4179728;
        double r4179730 = r4179725 * r4179729;
        double r4179731 = r4179720 - r4179730;
        double r4179732 = r4179719 / r4179731;
        double r4179733 = r4179732 - r4179719;
        double r4179734 = r4179726 / r4179723;
        double r4179735 = r4179734 / r4179731;
        double r4179736 = r4179721 / r4179728;
        double r4179737 = r4179736 + r4179720;
        double r4179738 = r4179735 * r4179737;
        double r4179739 = r4179717 * r4179727;
        double r4179740 = r4179739 / r4179718;
        double r4179741 = r4179740 / r4179724;
        double r4179742 = r4179741 / r4179731;
        double r4179743 = r4179738 + r4179742;
        double r4179744 = r4179733 + r4179743;
        return r4179744;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	15.5
Herbie	0.6

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

Derivation

1. Initial program 36.9

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

3. Applied tan-sum 21.4

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm

5. Applied flip-- 21.4

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

6. Applied associate-/r/ 21.4

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

8. Applied distribute-lft-in 21.4

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

9. Applied associate--l+ 21.4

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

10. Taylor expanded around inf 21.5

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

11. Simplified 0.6

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

12. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))