Average Error: 0.3 → 0.4
Time: 20.1s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{1}{\cos x} \cdot \left(\tan x \cdot \sin x\right)}{1 + \tan x \cdot \tan x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \frac{1}{\cos x} \cdot \left(\tan x \cdot \sin x\right)}{1 + \tan x \cdot \tan x}
double f(double x) {
        double r446625 = 1.0;
        double r446626 = x;
        double r446627 = tan(r446626);
        double r446628 = r446627 * r446627;
        double r446629 = r446625 - r446628;
        double r446630 = r446625 + r446628;
        double r446631 = r446629 / r446630;
        return r446631;
}
↓
double f(double x) {
        double r446632 = 1.0;
        double r446633 = x;
        double r446634 = cos(r446633);
        double r446635 = r446632 / r446634;
        double r446636 = tan(r446633);
        double r446637 = sin(r446633);
        double r446638 = r446636 * r446637;
        double r446639 = r446635 * r446638;
        double r446640 = r446632 - r446639;
        double r446641 = r446636 * r446636;
        double r446642 = r446632 + r446641;
        double r446643 = r446640 / r446642;
        return r446643;
}

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 div-inv0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))