Average Error: 0.3 → 0.4
Time: 16.9s
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 r20072 = 1.0;
        double r20073 = x;
        double r20074 = tan(r20073);
        double r20075 = r20074 * r20074;
        double r20076 = r20072 - r20075;
        double r20077 = r20072 + r20075;
        double r20078 = r20076 / r20077;
        return r20078;
}
↓
double f(double x) {
        double r20079 = 1.0;
        double r20080 = 1.0;
        double r20081 = x;
        double r20082 = cos(r20081);
        double r20083 = tan(r20081);
        double r20084 = sin(r20081);
        double r20085 = r20083 * r20084;
        double r20086 = r20082 / r20085;
        double r20087 = r20080 / r20086;
        double r20088 = r20079 - r20087;
        double r20089 = 6.0;
        double r20090 = pow(r20084, r20089);
        double r20091 = cbrt(r20090);
        double r20092 = 2.0;
        double r20093 = pow(r20082, r20092);
        double r20094 = r20091 / r20093;
        double r20095 = r20079 + r20094;
        double r20096 = r20088 / r20095;
        return r20096;
}

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)))))