Average Error: 0.3 → 0.4
Time: 17.6s
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 r19947 = 1.0;
        double r19948 = x;
        double r19949 = tan(r19948);
        double r19950 = r19949 * r19949;
        double r19951 = r19947 - r19950;
        double r19952 = r19947 + r19950;
        double r19953 = r19951 / r19952;
        return r19953;
}
↓
double f(double x) {
        double r19954 = 1.0;
        double r19955 = 1.0;
        double r19956 = x;
        double r19957 = cos(r19956);
        double r19958 = tan(r19956);
        double r19959 = sin(r19956);
        double r19960 = r19958 * r19959;
        double r19961 = r19957 / r19960;
        double r19962 = r19955 / r19961;
        double r19963 = r19954 - r19962;
        double r19964 = 6.0;
        double r19965 = pow(r19959, r19964);
        double r19966 = cbrt(r19965);
        double r19967 = 2.0;
        double r19968 = pow(r19957, r19967);
        double r19969 = r19966 / r19968;
        double r19970 = r19954 + r19969;
        double r19971 = r19963 / r19970;
        return r19971;
}

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)))))