Average Error: 0.3 → 0.4
Time: 4.9s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\left(1 - \frac{\tan x \cdot \sin x}{\cos x}\right) \cdot \frac{1}{1 + \tan x \cdot \tan x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\left(1 - \frac{\tan x \cdot \sin x}{\cos x}\right) \cdot \frac{1}{1 + \tan x \cdot \tan x}
double f(double x) {
        double r12260 = 1.0;
        double r12261 = x;
        double r12262 = tan(r12261);
        double r12263 = r12262 * r12262;
        double r12264 = r12260 - r12263;
        double r12265 = r12260 + r12263;
        double r12266 = r12264 / r12265;
        return r12266;
}
↓
double f(double x) {
        double r12267 = 1.0;
        double r12268 = x;
        double r12269 = tan(r12268);
        double r12270 = sin(r12268);
        double r12271 = r12269 * r12270;
        double r12272 = cos(r12268);
        double r12273 = r12271 / r12272;
        double r12274 = r12267 - r12273;
        double r12275 = 1.0;
        double r12276 = r12269 * r12269;
        double r12277 = r12267 + r12276;
        double r12278 = r12275 / r12277;
        double r12279 = r12274 * r12278;
        return r12279;
}

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 div-inv0.4

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

  5. Applied tan-quot0.4

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

  6. Applied associate-*r/0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))