Average Error: 0.3 → 0.4
Time: 6.1s
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 r14910 = 1.0;
        double r14911 = x;
        double r14912 = tan(r14911);
        double r14913 = r14912 * r14912;
        double r14914 = r14910 - r14913;
        double r14915 = r14910 + r14913;
        double r14916 = r14914 / r14915;
        return r14916;
}
↓
double f(double x) {
        double r14917 = 1.0;
        double r14918 = 1.0;
        double r14919 = x;
        double r14920 = sin(r14919);
        double r14921 = 2.0;
        double r14922 = pow(r14920, r14921);
        double r14923 = cos(r14919);
        double r14924 = pow(r14923, r14921);
        double r14925 = r14922 / r14924;
        double r14926 = r14925 + r14917;
        double r14927 = r14918 / r14926;
        double r14928 = r14917 * r14927;
        double r14929 = r14926 * r14924;
        double r14930 = r14922 / r14929;
        double r14931 = r14928 - r14930;
        return r14931;
}

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 tan-quot0.4

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

  4. Applied associate-*r/0.4

    \[\leadsto \frac{1 - \color{blue}{\frac{\tan x \cdot \sin x}{\cos x}}}{1 + \tan x \cdot \tan x}\]
  5. Using strategy rm

  6. Applied expm1-log1p-u0.4

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

  7. Taylor expanded around inf 0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))