Average Error: 0.3 → 0.5
Time: 21.2s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\left(1 + \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) - \tan x \cdot \tan x\right)\right) \cdot \frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(\left(\tan x \cdot \tan x\right) \cdot \tan x, \left(\tan x \cdot \tan x\right) \cdot \tan x, 1\right)}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\left(1 + \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) - \tan x \cdot \tan x\right)\right) \cdot \frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(\left(\tan x \cdot \tan x\right) \cdot \tan x, \left(\tan x \cdot \tan x\right) \cdot \tan x, 1\right)}
double f(double x) {
        double r871245 = 1.0;
        double r871246 = x;
        double r871247 = tan(r871246);
        double r871248 = r871247 * r871247;
        double r871249 = r871245 - r871248;
        double r871250 = r871245 + r871248;
        double r871251 = r871249 / r871250;
        return r871251;
}
↓
double f(double x) {
        double r871252 = 1.0;
        double r871253 = x;
        double r871254 = tan(r871253);
        double r871255 = r871254 * r871254;
        double r871256 = r871255 * r871255;
        double r871257 = r871256 - r871255;
        double r871258 = r871252 + r871257;
        double r871259 = r871252 - r871255;
        double r871260 = r871255 * r871254;
        double r871261 = fma(r871260, r871260, r871252);
        double r871262 = r871259 / r871261;
        double r871263 = r871258 * r871262;
        return r871263;
}

Error

Bits error versus x

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 flip3-+0.4

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

  4. Applied associate-/r/0.4

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

  5. Simplified0.5

    \[\leadsto \color{blue}{\frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(\left(\tan x \cdot \tan x\right) \cdot \tan x, \left(\tan x \cdot \tan x\right) \cdot \tan x, 1\right)}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) - 1 \cdot \left(\tan x \cdot \tan x\right)\right)\right)\]

  6. Final simplification0.5

    \[\leadsto \left(1 + \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) - \tan x \cdot \tan x\right)\right) \cdot \frac{1 - \tan x \cdot \tan x}{\mathsf{fma}\left(\left(\tan x \cdot \tan x\right) \cdot \tan x, \left(\tan x \cdot \tan x\right) \cdot \tan x, 1\right)}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))