2tan (problem 3.3.2)

Average Error: 37.4 → 0.6

Time: 13.4s

Precision: 64

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

Math

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

↓

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

C

double f(double x, double eps) {
        double r170788 = x;
        double r170789 = eps;
        double r170790 = r170788 + r170789;
        double r170791 = tan(r170790);
        double r170792 = tan(r170788);
        double r170793 = r170791 - r170792;
        return r170793;
}

↓

double f(double x, double eps) {
        double r170794 = x;
        double r170795 = sin(r170794);
        double r170796 = eps;
        double r170797 = cos(r170796);
        double r170798 = 2.0;
        double r170799 = pow(r170797, r170798);
        double r170800 = r170795 / r170799;
        double r170801 = sin(r170796);
        double r170802 = pow(r170801, r170798);
        double r170803 = 1.0;
        double r170804 = pow(r170795, r170798);
        double r170805 = r170804 * r170802;
        double r170806 = cos(r170794);
        double r170807 = pow(r170806, r170798);
        double r170808 = r170807 * r170799;
        double r170809 = r170805 / r170808;
        double r170810 = r170803 - r170809;
        double r170811 = r170810 * r170806;
        double r170812 = r170802 / r170811;
        double r170813 = r170804 / r170807;
        double r170814 = r170813 + r170803;
        double r170815 = r170810 * r170797;
        double r170816 = r170801 / r170815;
        double r170817 = r170814 * r170816;
        double r170818 = fma(r170800, r170812, r170817);
        double r170819 = r170803 / r170810;
        double r170820 = r170795 / r170806;
        double r170821 = -r170820;
        double r170822 = fma(r170819, r170820, r170821);
        double r170823 = r170818 + r170822;
        double r170824 = tan(r170794);
        double r170825 = 0.0;
        double r170826 = r170824 * r170825;
        double r170827 = r170823 + r170826;
        return r170827;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.4
Target	15.2
Herbie	0.6

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

Derivation

1. Initial program 37.4

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

3. Applied tan-sum 22.1

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

5. Applied add-cube-cbrt 22.7

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

6. Applied flip-- 22.7

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

7. Applied associate-/r/ 22.7

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

8. Applied prod-diff 22.7

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

9. Simplified 22.5

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

10. Simplified 22.2

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

11. Taylor expanded around inf 22.3

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

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 *-un-lft-identity 0.6

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

15. Applied times-frac 0.6

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

16. Applied fma-neg 0.6

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

17. Final simplification 0.6

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

Reproduce

herbie shell --seed 2020047 +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)))