2tan (problem 3.3.2)

Average Error: 36.9 → 15.6

Time: 31.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.035403109286184130586353726641947804415 \cdot 10^{-76}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \tan \varepsilon\right)}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \cos x} \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\ \mathbf{elif}\;\varepsilon \le 3.290261811358081045825245977755062406771 \cdot 10^{-26}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \left(x \cdot \left(\varepsilon + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \tan \varepsilon\right)}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \cos x} \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\ \end{array}\]

C

double f(double x, double eps) {
        double r8750522 = x;
        double r8750523 = eps;
        double r8750524 = r8750522 + r8750523;
        double r8750525 = tan(r8750524);
        double r8750526 = tan(r8750522);
        double r8750527 = r8750525 - r8750526;
        return r8750527;
}

↓

double f(double x, double eps) {
        double r8750528 = eps;
        double r8750529 = -1.0354031092861841e-76;
        bool r8750530 = r8750528 <= r8750529;
        double r8750531 = x;
        double r8750532 = tan(r8750531);
        double r8750533 = tan(r8750528);
        double r8750534 = r8750532 + r8750533;
        double r8750535 = 1.0;
        double r8750536 = sin(r8750531);
        double r8750537 = sin(r8750528);
        double r8750538 = r8750536 * r8750537;
        double r8750539 = r8750536 * r8750533;
        double r8750540 = r8750538 * r8750539;
        double r8750541 = cos(r8750531);
        double r8750542 = cos(r8750528);
        double r8750543 = r8750541 * r8750542;
        double r8750544 = r8750543 * r8750541;
        double r8750545 = r8750540 / r8750544;
        double r8750546 = r8750532 * r8750533;
        double r8750547 = r8750545 * r8750546;
        double r8750548 = r8750535 - r8750547;
        double r8750549 = r8750534 / r8750548;
        double r8750550 = r8750546 * r8750546;
        double r8750551 = r8750546 + r8750550;
        double r8750552 = r8750535 + r8750551;
        double r8750553 = r8750549 * r8750552;
        double r8750554 = r8750553 - r8750532;
        double r8750555 = 3.290261811358081e-26;
        bool r8750556 = r8750528 <= r8750555;
        double r8750557 = r8750528 + r8750531;
        double r8750558 = r8750531 * r8750557;
        double r8750559 = r8750528 * r8750558;
        double r8750560 = r8750528 + r8750559;
        double r8750561 = r8750556 ? r8750560 : r8750554;
        double r8750562 = r8750530 ? r8750554 : r8750561;
        return r8750562;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	15.3
Herbie	15.6

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

Derivation

1. Split input into 2 regimes

2. if eps < -1.0354031092861841e-76 or 3.290261811358081e-26 < eps

   1. Initial program 30.2

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

   3. Applied tan-sum 4.2

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

   5. Applied flip3-- 4.3

      \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x\]

   6. Applied associate-/r/ 4.3

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)} - \tan x\]

   7. Simplified 4.3

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

   9. Applied tan-quot 4.3

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

   10. Applied associate-*l/ 4.3

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

   11. Applied tan-quot 4.3

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

   12. Applied tan-quot 4.3

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

   13. Applied frac-times 4.3

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

   14. Applied frac-times 4.3

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

   if -1.0354031092861841e-76 < eps < 3.290261811358081e-26

   1. Initial program 46.5

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

   2. Taylor expanded around 0 32.0

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

   3. Simplified 32.0

      \[\leadsto \color{blue}{\varepsilon + \varepsilon \cdot \left(x \cdot \left(\varepsilon + x\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 15.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.035403109286184130586353726641947804415 \cdot 10^{-76}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \tan \varepsilon\right)}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \cos x} \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\ \mathbf{elif}\;\varepsilon \le 3.290261811358081045825245977755062406771 \cdot 10^{-26}:\\ \;\;\;\;\varepsilon + \varepsilon \cdot \left(x \cdot \left(\varepsilon + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \tan \varepsilon\right)}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \cos x} \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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