Average Error: 0.3 → 0.4
Time: 4.2s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\left(1 \cdot 1 + \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) - 1 \cdot \left(\tan x \cdot \tan x\right)\right)\right) \cdot \left(\frac{1}{{1}^{3} + {\left(\tan x \cdot \tan x\right)}^{3}} - \frac{\tan x \cdot \tan x}{{1}^{3} + {\left(\tan x \cdot \tan x\right)}^{3}}\right)\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\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) \cdot \left(\frac{1}{{1}^{3} + {\left(\tan x \cdot \tan x\right)}^{3}} - \frac{\tan x \cdot \tan x}{{1}^{3} + {\left(\tan x \cdot \tan x\right)}^{3}}\right)
double f(double x) {
        double r11122 = 1.0;
        double r11123 = x;
        double r11124 = tan(r11123);
        double r11125 = r11124 * r11124;
        double r11126 = r11122 - r11125;
        double r11127 = r11122 + r11125;
        double r11128 = r11126 / r11127;
        return r11128;
}
↓
double f(double x) {
        double r11129 = 1.0;
        double r11130 = r11129 * r11129;
        double r11131 = x;
        double r11132 = tan(r11131);
        double r11133 = r11132 * r11132;
        double r11134 = r11133 * r11133;
        double r11135 = r11129 * r11133;
        double r11136 = r11134 - r11135;
        double r11137 = r11130 + r11136;
        double r11138 = 3.0;
        double r11139 = pow(r11129, r11138);
        double r11140 = pow(r11133, r11138);
        double r11141 = r11139 + r11140;
        double r11142 = r11129 / r11141;
        double r11143 = r11133 / r11141;
        double r11144 = r11142 - r11143;
        double r11145 = r11137 * r11144;
        return r11145;
}

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-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. Using strategy rm

  5. Applied flip3-+0.5

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

  6. Applied associate-/r/0.5

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

  7. Applied flip3-+0.5

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

  8. Applied associate-/r/0.5

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

  9. Applied distribute-rgt-out--0.4

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

  10. Final simplification0.4

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

Reproduce

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