Average Error: 0.3 → 0.5
Time: 5.1s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[1 \cdot \frac{1}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1} - \frac{{\left(\sin x\right)}^{2}}{\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot {\left(\cos x\right)}^{2}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
1 \cdot \frac{1}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1} - \frac{{\left(\sin x\right)}^{2}}{\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot {\left(\cos x\right)}^{2}}
double f(double x) {
        double r11392 = 1.0;
        double r11393 = x;
        double r11394 = tan(r11393);
        double r11395 = r11394 * r11394;
        double r11396 = r11392 - r11395;
        double r11397 = r11392 + r11395;
        double r11398 = r11396 / r11397;
        return r11398;
}
↓
double f(double x) {
        double r11399 = 1.0;
        double r11400 = 1.0;
        double r11401 = x;
        double r11402 = sin(r11401);
        double r11403 = 2.0;
        double r11404 = pow(r11402, r11403);
        double r11405 = cos(r11401);
        double r11406 = pow(r11405, r11403);
        double r11407 = r11404 / r11406;
        double r11408 = r11407 + r11399;
        double r11409 = r11400 / r11408;
        double r11410 = r11399 * r11409;
        double r11411 = r11408 * r11406;
        double r11412 = r11404 / r11411;
        double r11413 = r11410 - r11412;
        return r11413;
}

Error

Bits error versus x

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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-sub0.4

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

  4. Taylor expanded around inf 0.5

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

  5. Final simplification0.5

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

Reproduce

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