Average Error: 0.3 → 0.4
Time: 4.4s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{\frac{{\left(\sin x\right)}^{2}}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos x\right)}^{2}\right)\right)} + 1}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{\frac{{\left(\sin x\right)}^{2}}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos x\right)}^{2}\right)\right)} + 1}
double f(double x) {
        double r11306 = 1.0;
        double r11307 = x;
        double r11308 = tan(r11307);
        double r11309 = r11308 * r11308;
        double r11310 = r11306 - r11309;
        double r11311 = r11306 + r11309;
        double r11312 = r11310 / r11311;
        return r11312;
}
↓
double f(double x) {
        double r11313 = 1.0;
        double r11314 = x;
        double r11315 = sin(r11314);
        double r11316 = 2.0;
        double r11317 = pow(r11315, r11316);
        double r11318 = cos(r11314);
        double r11319 = pow(r11318, r11316);
        double r11320 = r11317 / r11319;
        double r11321 = r11313 - r11320;
        double r11322 = expm1(r11319);
        double r11323 = log1p(r11322);
        double r11324 = r11317 / r11323;
        double r11325 = r11324 + r11313;
        double r11326 = r11321 / r11325;
        return r11326;
}

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. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1}}\]
  3. Using strategy rm

  4. Applied log1p-expm1-u0.4

    \[\leadsto \frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{\frac{{\left(\sin x\right)}^{2}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos x\right)}^{2}\right)\right)}} + 1}\]

  5. Final simplification0.4

    \[\leadsto \frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{\frac{{\left(\sin x\right)}^{2}}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\cos x\right)}^{2}\right)\right)} + 1}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))