Average Error: 0.3 → 0.5
Time: 5.2s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{\log \left(e^{{\left(\sin x\right)}^{2}}\right)}{{\left(\cos x\right)}^{2}}}{\frac{{\left(\sin x\right)}^{2}}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{2}\right)\right)} + 1}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \frac{\log \left(e^{{\left(\sin x\right)}^{2}}\right)}{{\left(\cos x\right)}^{2}}}{\frac{{\left(\sin x\right)}^{2}}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos x\right)}^{2}\right)\right)} + 1}
double f(double x) {
        double r9601 = 1.0;
        double r9602 = x;
        double r9603 = tan(r9602);
        double r9604 = r9603 * r9603;
        double r9605 = r9601 - r9604;
        double r9606 = r9601 + r9604;
        double r9607 = r9605 / r9606;
        return r9607;
}
↓
double f(double x) {
        double r9608 = 1.0;
        double r9609 = x;
        double r9610 = sin(r9609);
        double r9611 = 2.0;
        double r9612 = pow(r9610, r9611);
        double r9613 = exp(r9612);
        double r9614 = log(r9613);
        double r9615 = cos(r9609);
        double r9616 = pow(r9615, r9611);
        double r9617 = r9614 / r9616;
        double r9618 = r9608 - r9617;
        double r9619 = log1p(r9616);
        double r9620 = expm1(r9619);
        double r9621 = r9612 / r9620;
        double r9622 = r9621 + r9608;
        double r9623 = r9618 / r9622;
        return r9623;
}

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-log-exp0.5

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

  6. Applied expm1-log1p-u0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))