2tan (problem 3.3.2)

Average Error: 36.9 → 5.6

Time: 25.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.41386816340647429 \cdot 10^{-17} \lor \neg \left(\varepsilon \le 8.14994350532359335 \cdot 10^{-19}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, 1 + \left(\sqrt[3]{{\left(\tan \varepsilon \cdot \tan x\right)}^{6}} + \tan x \cdot \tan \varepsilon\right), -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, \mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left(\frac{1}{3}, {\varepsilon}^{3}, \varepsilon\right)\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r141016 = x;
        double r141017 = eps;
        double r141018 = r141016 + r141017;
        double r141019 = tan(r141018);
        double r141020 = tan(r141016);
        double r141021 = r141019 - r141020;
        return r141021;
}

↓

double f(double x, double eps) {
        double r141022 = eps;
        double r141023 = -2.4138681634064743e-17;
        bool r141024 = r141022 <= r141023;
        double r141025 = 8.149943505323593e-19;
        bool r141026 = r141022 <= r141025;
        double r141027 = !r141026;
        bool r141028 = r141024 || r141027;
        double r141029 = x;
        double r141030 = tan(r141029);
        double r141031 = tan(r141022);
        double r141032 = r141030 + r141031;
        double r141033 = 1.0;
        double r141034 = 3.0;
        double r141035 = pow(r141033, r141034);
        double r141036 = r141030 * r141031;
        double r141037 = pow(r141036, r141034);
        double r141038 = r141035 - r141037;
        double r141039 = r141032 / r141038;
        double r141040 = r141031 * r141030;
        double r141041 = 6.0;
        double r141042 = pow(r141040, r141041);
        double r141043 = cbrt(r141042);
        double r141044 = r141043 + r141036;
        double r141045 = r141033 + r141044;
        double r141046 = -r141030;
        double r141047 = fma(r141039, r141045, r141046);
        double r141048 = sin(r141029);
        double r141049 = 2.0;
        double r141050 = pow(r141048, r141049);
        double r141051 = cos(r141022);
        double r141052 = r141050 / r141051;
        double r141053 = sin(r141022);
        double r141054 = cos(r141029);
        double r141055 = pow(r141054, r141049);
        double r141056 = r141048 * r141053;
        double r141057 = pow(r141056, r141034);
        double r141058 = r141054 * r141051;
        double r141059 = pow(r141058, r141034);
        double r141060 = r141057 / r141059;
        double r141061 = r141033 - r141060;
        double r141062 = r141055 * r141061;
        double r141063 = r141053 / r141062;
        double r141064 = pow(r141022, r141049);
        double r141065 = 0.3333333333333333;
        double r141066 = pow(r141022, r141034);
        double r141067 = fma(r141065, r141066, r141022);
        double r141068 = fma(r141029, r141064, r141067);
        double r141069 = fma(r141052, r141063, r141068);
        double r141070 = r141028 ? r141047 : r141069;
        return r141070;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	15.2
Herbie	5.6

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

Derivation

1. Split input into 2 regimes

2. if eps < -2.4138681634064743e-17 or 8.149943505323593e-19 < eps

   1. Initial program 29.5

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

   3. Applied tan-sum 0.9

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

   5. Applied flip3-- 1.0

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

   6. Applied associate-/r/ 1.0

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

   7. Applied fma-neg 1.0

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

   9. Applied add-cbrt-cube 1.0

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

   10. Applied add-cbrt-cube 1.0

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

   11. Applied cbrt-unprod 1.0

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

   12. Applied add-cbrt-cube 1.0

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

   13. Applied add-cbrt-cube 1.0

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

   14. Applied cbrt-unprod 1.0

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

   15. Applied cbrt-unprod 1.0

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

   16. Simplified 1.0

       \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, 1 \cdot 1 + \left(\sqrt[3]{\color{blue}{{\left(\tan \varepsilon \cdot \tan x\right)}^{6}}} + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right), -\tan x\right)\]

   if -2.4138681634064743e-17 < eps < 8.149943505323593e-19

   1. Initial program 45.4

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

   3. Applied tan-sum 45.4

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

   5. Applied flip3-- 45.4

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

   6. Applied associate-/r/ 45.4

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

   7. Applied fma-neg 45.4

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

   8. Taylor expanded around -inf 45.4

      \[\leadsto \color{blue}{\left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}\right)} + \left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \left(\frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \left({\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{3}\right)} + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{3}\right)} + \frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \left(\cos x \cdot {\left(\cos \varepsilon\right)}^{2}\right)}\right)\right)\right)\right)\right) - \frac{\sin x}{\cos x}}\]

   9. Simplified 40.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, \left(\left(\frac{\frac{\sin \varepsilon}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}}{\cos \varepsilon} + \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{3}}, \frac{\frac{\sin x}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}}{\cos x}\right)\right) + \frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)} \cdot \left(\frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \frac{\sin x}{\cos x}\right)\right) - \frac{\sin x}{\cos x}\right)}\]

   10. Taylor expanded around 0 10.8

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, \color{blue}{x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)}\right)\]

   11. Simplified 10.8

       \[\leadsto \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, \color{blue}{\mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left(\frac{1}{3}, {\varepsilon}^{3}, \varepsilon\right)\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.41386816340647429 \cdot 10^{-17} \lor \neg \left(\varepsilon \le 8.14994350532359335 \cdot 10^{-19}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, 1 + \left(\sqrt[3]{{\left(\tan \varepsilon \cdot \tan x\right)}^{6}} + \tan x \cdot \tan \varepsilon\right), -\tan x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, \mathsf{fma}\left(x, {\varepsilon}^{2}, \mathsf{fma}\left(\frac{1}{3}, {\varepsilon}^{3}, \varepsilon\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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)))