2tan (problem 3.3.2)

Average Error: 37.6 → 0.6

Time: 2.0m

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r21808681 = x;
        double r21808682 = eps;
        double r21808683 = r21808681 + r21808682;
        double r21808684 = tan(r21808683);
        double r21808685 = tan(r21808681);
        double r21808686 = r21808684 - r21808685;
        return r21808686;
}

↓

double f(double x, double eps) {
        double r21808687 = eps;
        double r21808688 = sin(r21808687);
        double r21808689 = cos(r21808687);
        double r21808690 = r21808688 / r21808689;
        double r21808691 = r21808690 * r21808690;
        double r21808692 = x;
        double r21808693 = sin(r21808692);
        double r21808694 = cos(r21808692);
        double r21808695 = r21808693 / r21808694;
        double r21808696 = 1.0;
        double r21808697 = r21808690 * r21808695;
        double r21808698 = r21808697 * r21808697;
        double r21808699 = r21808696 - r21808698;
        double r21808700 = r21808695 / r21808699;
        double r21808701 = r21808691 * r21808700;
        double r21808702 = r21808693 * r21808693;
        double r21808703 = r21808694 * r21808694;
        double r21808704 = r21808702 / r21808703;
        double r21808705 = r21808704 + r21808696;
        double r21808706 = r21808705 * r21808690;
        double r21808707 = r21808706 / r21808699;
        double r21808708 = r21808700 - r21808695;
        double r21808709 = r21808707 + r21808708;
        double r21808710 = r21808701 + r21808709;
        return r21808710;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.6
Target	15.3
Herbie	0.6

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

Derivation

1. Initial program 37.6

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

3. Applied tan-sum 22.2

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

5. Applied flip-- 22.2

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

6. Applied associate-/r/ 22.2

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

7. Taylor expanded around -inf 22.3

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

8. Simplified 0.6

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

10. Applied frac-times 0.6

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

11. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019128 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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