2tan (problem 3.3.2)

Average Error: 37.1 → 15.9

Time: 11.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.85969210690957729 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \tan x \cdot \tan \varepsilon, 1\right), \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \mathbf{elif}\;\varepsilon \le 2.69186416397234008 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r103998 = x;
        double r103999 = eps;
        double r104000 = r103998 + r103999;
        double r104001 = tan(r104000);
        double r104002 = tan(r103998);
        double r104003 = r104001 - r104002;
        return r104003;
}

↓

double f(double x, double eps) {
        double r104004 = eps;
        double r104005 = -2.859692106909577e-20;
        bool r104006 = r104004 <= r104005;
        double r104007 = x;
        double r104008 = tan(r104007);
        double r104009 = tan(r104004);
        double r104010 = 1.0;
        double r104011 = fma(r104008, r104009, r104010);
        double r104012 = r104008 * r104009;
        double r104013 = fma(r104011, r104012, r104010);
        double r104014 = r104008 + r104009;
        double r104015 = 3.0;
        double r104016 = pow(r104012, r104015);
        double r104017 = r104010 - r104016;
        double r104018 = r104014 / r104017;
        double r104019 = -r104008;
        double r104020 = fma(r104013, r104018, r104019);
        double r104021 = fma(r104019, r104010, r104008);
        double r104022 = r104020 + r104021;
        double r104023 = 2.69186416397234e-43;
        bool r104024 = r104004 <= r104023;
        double r104025 = 2.0;
        double r104026 = pow(r104004, r104025);
        double r104027 = pow(r104007, r104025);
        double r104028 = fma(r104004, r104027, r104004);
        double r104029 = fma(r104026, r104007, r104028);
        double r104030 = sin(r104007);
        double r104031 = r104030 * r104009;
        double r104032 = cos(r104007);
        double r104033 = r104031 / r104032;
        double r104034 = r104012 * r104033;
        double r104035 = r104010 - r104034;
        double r104036 = r104014 / r104035;
        double r104037 = r104010 + r104012;
        double r104038 = fma(r104036, r104037, r104019);
        double r104039 = r104024 ? r104029 : r104038;
        double r104040 = r104006 ? r104022 : r104039;
        return r104040;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	14.5
Herbie	15.9

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

Derivation

1. Split input into 3 regimes

2. if eps < -2.859692106909577e-20

   1. Initial program 29.0

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

   3. Applied tan-sum 1.2

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

   5. Applied add-cube-cbrt 1.5

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

   6. Applied flip3-- 1.5

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

   7. Applied associate-/r/ 1.5

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

   8. Applied prod-diff 1.5

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

   9. Simplified 1.2

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

   10. Simplified 1.2

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

   if -2.859692106909577e-20 < eps < 2.69186416397234e-43

   1. Initial program 45.9

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

   2. Taylor expanded around 0 31.7

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

   3. Simplified 31.7

      \[\leadsto \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)}\]

   if 2.69186416397234e-43 < eps

   1. Initial program 30.0

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

   3. Applied tan-sum 3.7

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

   5. Applied flip-- 3.7

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

   6. Applied associate-/r/ 3.7

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

   7. Applied fma-neg 3.7

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

   9. Applied tan-quot 3.7

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

   10. Applied associate-*l/ 3.7

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

4. Final simplification 15.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.85969210690957729 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), \tan x \cdot \tan \varepsilon, 1\right), \frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, -\tan x\right) + \mathsf{fma}\left(-\tan x, 1, \tan x\right)\\ \mathbf{elif}\;\varepsilon \le 2.69186416397234008 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left({\varepsilon}^{2}, x, \mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \frac{\sin x \cdot \tan \varepsilon}{\cos x}}, 1 + \tan x \cdot \tan \varepsilon, -\tan x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(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)))