2tan (problem 3.3.2)

Average Error: 36.9 → 0.5

Time: 8.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r135046 = x;
        double r135047 = eps;
        double r135048 = r135046 + r135047;
        double r135049 = tan(r135048);
        double r135050 = tan(r135046);
        double r135051 = r135049 - r135050;
        return r135051;
}

↓

double f(double x, double eps) {
        double r135052 = x;
        double r135053 = sin(r135052);
        double r135054 = 2.0;
        double r135055 = pow(r135053, r135054);
        double r135056 = eps;
        double r135057 = cos(r135056);
        double r135058 = r135055 / r135057;
        double r135059 = sin(r135056);
        double r135060 = 1.0;
        double r135061 = 3.0;
        double r135062 = pow(r135053, r135061);
        double r135063 = pow(r135059, r135061);
        double r135064 = r135062 * r135063;
        double r135065 = cos(r135052);
        double r135066 = pow(r135065, r135061);
        double r135067 = pow(r135057, r135061);
        double r135068 = r135066 * r135067;
        double r135069 = r135064 / r135068;
        double r135070 = r135060 - r135069;
        double r135071 = pow(r135065, r135054);
        double r135072 = r135070 * r135071;
        double r135073 = r135059 / r135072;
        double r135074 = r135070 * r135057;
        double r135075 = r135059 / r135074;
        double r135076 = fma(r135058, r135073, r135075);
        double r135077 = r135071 * r135070;
        double r135078 = r135055 / r135077;
        double r135079 = r135059 / r135057;
        double r135080 = pow(r135079, r135061);
        double r135081 = r135078 * r135080;
        double r135082 = pow(r135059, r135054);
        double r135083 = pow(r135057, r135054);
        double r135084 = r135070 * r135083;
        double r135085 = r135082 / r135084;
        double r135086 = r135053 / r135065;
        double r135087 = r135062 / r135066;
        double r135088 = r135086 + r135087;
        double r135089 = r135085 * r135088;
        double r135090 = r135081 + r135089;
        double r135091 = r135070 * r135065;
        double r135092 = r135053 / r135091;
        double r135093 = r135090 + r135092;
        double r135094 = r135093 - r135086;
        double r135095 = r135076 + r135094;
        return r135095;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	15.5
Herbie	0.5

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

Derivation

1. Initial program 36.9

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

3. Applied tan-sum 21.4

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

5. Applied tan-quot 21.4

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

6. Applied associate-*r/ 21.4

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

8. Applied flip3-- 21.4

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

9. Applied associate-/r/ 21.4

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

10. Applied fma-neg 21.4

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

11. Taylor expanded around -inf 21.5

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

12. Simplified 0.5

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

13. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020060 +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)))