2tan (problem 3.3.2)

Average Error: 36.9 → 15.8

Time: 11.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.2497976182633953 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \log \left(e^{\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}\right)\right) - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.1114736186435238 \cdot 10^{-119}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\sin x\right) \cdot \left(1 + \left(-\tan x \cdot \tan \varepsilon\right)\right) + \left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \cos x\right) \cdot \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}{\cos x \cdot \left(1 + \left(-\tan x \cdot \tan \varepsilon\right)\right)}\\ \end{array}\]

C

double f(double x, double eps) {
        double r133717 = x;
        double r133718 = eps;
        double r133719 = r133717 + r133718;
        double r133720 = tan(r133719);
        double r133721 = tan(r133717);
        double r133722 = r133720 - r133721;
        return r133722;
}

↓

double f(double x, double eps) {
        double r133723 = eps;
        double r133724 = -4.2497976182633953e-14;
        bool r133725 = r133723 <= r133724;
        double r133726 = x;
        double r133727 = tan(r133726);
        double r133728 = tan(r133723);
        double r133729 = r133727 + r133728;
        double r133730 = 1.0;
        double r133731 = r133727 * r133728;
        double r133732 = r133731 * r133731;
        double r133733 = r133730 - r133732;
        double r133734 = r133729 / r133733;
        double r133735 = sin(r133726);
        double r133736 = sin(r133723);
        double r133737 = r133735 * r133736;
        double r133738 = cos(r133726);
        double r133739 = cos(r133723);
        double r133740 = r133738 * r133739;
        double r133741 = r133737 / r133740;
        double r133742 = exp(r133741);
        double r133743 = log(r133742);
        double r133744 = r133730 + r133743;
        double r133745 = r133734 * r133744;
        double r133746 = r133745 - r133727;
        double r133747 = 2.1114736186435238e-119;
        bool r133748 = r133723 <= r133747;
        double r133749 = r133726 * r133723;
        double r133750 = r133723 + r133726;
        double r133751 = r133749 * r133750;
        double r133752 = r133751 + r133723;
        double r133753 = -r133735;
        double r133754 = -r133731;
        double r133755 = r133730 + r133754;
        double r133756 = r133753 * r133755;
        double r133757 = r133734 * r133738;
        double r133758 = r133757 * r133733;
        double r133759 = r133756 + r133758;
        double r133760 = r133738 * r133755;
        double r133761 = r133759 / r133760;
        double r133762 = r133748 ? r133752 : r133761;
        double r133763 = r133725 ? r133746 : r133762;
        return r133763;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	15.1
Herbie	15.8

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

Derivation

1. Split input into 3 regimes

2. if eps < -4.2497976182633953e-14

   1. Initial program 29.9

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

   3. Applied tan-sum 0.8

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

   5. Applied flip-- 0.9

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

   6. Applied associate-/r/ 0.8

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

   7. Simplified 0.8

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

   9. Applied add-log-exp 1.0

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

   11. Applied tan-quot 1.0

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

   12. Applied tan-quot 1.0

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

   13. Applied frac-times 1.0

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

   if -4.2497976182633953e-14 < eps < 2.1114736186435238e-119

   1. Initial program 46.4

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

   3. Applied tan-sum 46.3

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

   4. Taylor expanded around 0 31.1

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

   5. Simplified 30.9

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

   if 2.1114736186435238e-119 < eps

   1. Initial program 31.5

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

   3. Applied tan-sum 10.0

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

   5. Applied flip-- 10.0

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

   6. Applied associate-/r/ 10.0

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

   7. Simplified 10.0

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

   9. Applied add-log-exp 10.0

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

   11. Applied tan-quot 10.1

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

   12. Applied flip-+ 10.1

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

   13. Applied associate-*r/ 10.0

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

   14. Applied frac-sub 10.1

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

   15. Simplified 10.1

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

   16. Simplified 10.1

       \[\leadsto \frac{\left(-\sin x\right) \cdot \left(1 + \left(-\tan x \cdot \tan \varepsilon\right)\right) + \left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \cos x\right) \cdot \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}{\color{blue}{\cos x \cdot \left(1 + \left(-\tan x \cdot \tan \varepsilon\right)\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 15.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.2497976182633953 \cdot 10^{-14}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \log \left(e^{\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}\right)\right) - \tan x\\ \mathbf{elif}\;\varepsilon \le 2.1114736186435238 \cdot 10^{-119}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\sin x\right) \cdot \left(1 + \left(-\tan x \cdot \tan \varepsilon\right)\right) + \left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \cos x\right) \cdot \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}{\cos x \cdot \left(1 + \left(-\tan x \cdot \tan \varepsilon\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

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

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