Average Error: 0.3 → 0.4
Time: 6.7s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{e^{\log \left({\left(\sin x\right)}^{2}\right)}}{{\left(\cos x\right)}^{2}}}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \frac{e^{\log \left({\left(\sin x\right)}^{2}\right)}}{{\left(\cos x\right)}^{2}}}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1}
double f(double x) {
        double r18505 = 1.0;
        double r18506 = x;
        double r18507 = tan(r18506);
        double r18508 = r18507 * r18507;
        double r18509 = r18505 - r18508;
        double r18510 = r18505 + r18508;
        double r18511 = r18509 / r18510;
        return r18511;
}
↓
double f(double x) {
        double r18512 = 1.0;
        double r18513 = x;
        double r18514 = sin(r18513);
        double r18515 = 2.0;
        double r18516 = pow(r18514, r18515);
        double r18517 = log(r18516);
        double r18518 = exp(r18517);
        double r18519 = cos(r18513);
        double r18520 = pow(r18519, r18515);
        double r18521 = r18518 / r18520;
        double r18522 = r18512 - r18521;
        double r18523 = r18516 / r18520;
        double r18524 = r18523 + r18512;
        double r18525 = r18522 / r18524;
        return r18525;
}

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 add-exp-log32.3

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

  5. Applied pow-exp32.3

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019322 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))