2tan (problem 3.3.2)

Average Error: 37.1 → 0.6

Time: 13.4s

Precision: 64

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

Math

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

↓

\[\left(\mathsf{fma}\left(\frac{\sin x}{{\left(\cos \varepsilon\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x}, \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon}\right) + \log \left(e^{\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x} - \frac{\sin x}{\cos x}}\right)\right) + \tan x \cdot 0\]

C

double f(double x, double eps) {
        double r107928 = x;
        double r107929 = eps;
        double r107930 = r107928 + r107929;
        double r107931 = tan(r107930);
        double r107932 = tan(r107928);
        double r107933 = r107931 - r107932;
        return r107933;
}

↓

double f(double x, double eps) {
        double r107934 = x;
        double r107935 = sin(r107934);
        double r107936 = eps;
        double r107937 = cos(r107936);
        double r107938 = 2.0;
        double r107939 = pow(r107937, r107938);
        double r107940 = r107935 / r107939;
        double r107941 = sin(r107936);
        double r107942 = pow(r107941, r107938);
        double r107943 = 1.0;
        double r107944 = pow(r107935, r107938);
        double r107945 = r107944 * r107942;
        double r107946 = cos(r107934);
        double r107947 = pow(r107946, r107938);
        double r107948 = r107947 * r107939;
        double r107949 = r107945 / r107948;
        double r107950 = r107943 - r107949;
        double r107951 = r107950 * r107946;
        double r107952 = r107942 / r107951;
        double r107953 = r107944 / r107947;
        double r107954 = r107953 + r107943;
        double r107955 = r107950 * r107937;
        double r107956 = r107941 / r107955;
        double r107957 = r107954 * r107956;
        double r107958 = fma(r107940, r107952, r107957);
        double r107959 = r107935 / r107951;
        double r107960 = r107935 / r107946;
        double r107961 = r107959 - r107960;
        double r107962 = exp(r107961);
        double r107963 = log(r107962);
        double r107964 = r107958 + r107963;
        double r107965 = tan(r107934);
        double r107966 = 0.0;
        double r107967 = r107965 * r107966;
        double r107968 = r107964 + r107967;
        return r107968;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.1
Herbie	0.6

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

Derivation

1. Initial program 37.1

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

3. Applied tan-sum 21.8

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

5. Applied add-cube-cbrt 22.3

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

6. Applied flip-- 22.3

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

7. Applied associate-/r/ 22.3

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

8. Applied prod-diff 22.4

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

9. Simplified 22.2

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

10. Simplified 21.8

    \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}, \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right), -\tan x\right) + \color{blue}{\tan x \cdot 0}\]

11. Taylor expanded around inf 22.0

    \[\leadsto \color{blue}{\left(\left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x} + \left(\frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x\right)} + \left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon\right)}\right)\right)\right) - \frac{\sin x}{\cos x}\right)} + \tan x \cdot 0\]

12. Simplified 0.6

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\sin x}{{\left(\cos \varepsilon\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x}, \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon}\right) + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right)\right)} + \tan x \cdot 0\]
13. Using strategy rm

14. Applied add-log-exp 12.7

    \[\leadsto \left(\mathsf{fma}\left(\frac{\sin x}{{\left(\cos \varepsilon\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x}, \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon}\right) + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x} - \color{blue}{\log \left(e^{\frac{\sin x}{\cos x}}\right)}\right)\right) + \tan x \cdot 0\]

15. Applied add-log-exp 0.6

    \[\leadsto \left(\mathsf{fma}\left(\frac{\sin x}{{\left(\cos \varepsilon\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x}, \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon}\right) + \left(\color{blue}{\log \left(e^{\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x}}\right)} - \log \left(e^{\frac{\sin x}{\cos x}}\right)\right)\right) + \tan x \cdot 0\]

16. Applied diff-log 0.6

    \[\leadsto \left(\mathsf{fma}\left(\frac{\sin x}{{\left(\cos \varepsilon\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x}, \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon}\right) + \color{blue}{\log \left(\frac{e^{\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x}}}{e^{\frac{\sin x}{\cos x}}}\right)}\right) + \tan x \cdot 0\]

17. Simplified 0.6

    \[\leadsto \left(\mathsf{fma}\left(\frac{\sin x}{{\left(\cos \varepsilon\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x}, \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon}\right) + \log \color{blue}{\left(e^{\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x} - \frac{\sin x}{\cos x}}\right)}\right) + \tan x \cdot 0\]

18. Final simplification 0.6

    \[\leadsto \left(\mathsf{fma}\left(\frac{\sin x}{{\left(\cos \varepsilon\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x}, \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon}\right) + \log \left(e^{\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x} - \frac{\sin x}{\cos x}}\right)\right) + \tan x \cdot 0\]

Reproduce

herbie shell --seed 2020046 +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)))