Average Error: 0.3 → 0.3
Time: 5.6s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{\tan x \cdot \sin x}{\cos x}}{1 + \frac{1}{\frac{\cos x}{\tan x \cdot \sin x}}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \frac{\tan x \cdot \sin x}{\cos x}}{1 + \frac{1}{\frac{\cos x}{\tan x \cdot \sin x}}}
double f(double x) {
        double r12818 = 1.0;
        double r12819 = x;
        double r12820 = tan(r12819);
        double r12821 = r12820 * r12820;
        double r12822 = r12818 - r12821;
        double r12823 = r12818 + r12821;
        double r12824 = r12822 / r12823;
        return r12824;
}
↓
double f(double x) {
        double r12825 = 1.0;
        double r12826 = x;
        double r12827 = tan(r12826);
        double r12828 = sin(r12826);
        double r12829 = r12827 * r12828;
        double r12830 = cos(r12826);
        double r12831 = r12829 / r12830;
        double r12832 = r12825 - r12831;
        double r12833 = 1.0;
        double r12834 = r12830 / r12829;
        double r12835 = r12833 / r12834;
        double r12836 = r12825 + r12835;
        double r12837 = r12832 / r12836;
        return r12837;
}

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 tan-quot0.4

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

  7. Applied associate-*r/0.3

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

  9. Applied clear-num0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))