Average Error: 0.3 → 0.4
Time: 17.5s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{1}{\frac{\cos x}{\tan x \cdot \sin x}}}{1 + \frac{\sqrt[3]{{\left(\sin x\right)}^{6}}}{{\left(\cos x\right)}^{2}}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \frac{1}{\frac{\cos x}{\tan x \cdot \sin x}}}{1 + \frac{\sqrt[3]{{\left(\sin x\right)}^{6}}}{{\left(\cos x\right)}^{2}}}
double f(double x) {
        double r19958 = 1.0;
        double r19959 = x;
        double r19960 = tan(r19959);
        double r19961 = r19960 * r19960;
        double r19962 = r19958 - r19961;
        double r19963 = r19958 + r19961;
        double r19964 = r19962 / r19963;
        return r19964;
}
↓
double f(double x) {
        double r19965 = 1.0;
        double r19966 = 1.0;
        double r19967 = x;
        double r19968 = cos(r19967);
        double r19969 = tan(r19967);
        double r19970 = sin(r19967);
        double r19971 = r19969 * r19970;
        double r19972 = r19968 / r19971;
        double r19973 = r19966 / r19972;
        double r19974 = r19965 - r19973;
        double r19975 = 6.0;
        double r19976 = pow(r19970, r19975);
        double r19977 = cbrt(r19976);
        double r19978 = 2.0;
        double r19979 = pow(r19968, r19978);
        double r19980 = r19977 / r19979;
        double r19981 = r19965 + r19980;
        double r19982 = r19974 / r19981;
        return r19982;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.3

    \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
  2. Using strategy rm

  3. Applied tan-quot0.4

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

  4. Applied associate-*r/0.4

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

  6. Applied clear-num0.4

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

  8. Applied tan-quot0.4

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

  9. Applied tan-quot0.4

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

  10. Applied frac-times0.4

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

  11. Simplified0.4

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

  12. Simplified0.4

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

  14. Applied add-cbrt-cube0.4

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

  15. Simplified0.4

    \[\leadsto \frac{1 - \frac{1}{\frac{\cos x}{\tan x \cdot \sin x}}}{1 + \frac{\sqrt[3]{\color{blue}{{\left(\sin x\right)}^{6}}}}{{\left(\cos x\right)}^{2}}}\]

  16. Final simplification0.4

    \[\leadsto \frac{1 - \frac{1}{\frac{\cos x}{\tan x \cdot \sin x}}}{1 + \frac{\sqrt[3]{{\left(\sin x\right)}^{6}}}{{\left(\cos x\right)}^{2}}}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))