Average Error: 0.3 → 0.4
Time: 42.4s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{\tan x \cdot \sin x}{\cos x}}{1 + \frac{1}{\frac{\cos x}{\tan x \cdot \sin x}}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \frac{\tan x \cdot \sin x}{\cos x}}{1 + \frac{1}{\frac{\cos x}{\tan x \cdot \sin x}}}
double f(double x) {
        double r1617495 = 1.0;
        double r1617496 = x;
        double r1617497 = tan(r1617496);
        double r1617498 = r1617497 * r1617497;
        double r1617499 = r1617495 - r1617498;
        double r1617500 = r1617495 + r1617498;
        double r1617501 = r1617499 / r1617500;
        return r1617501;
}
↓
double f(double x) {
        double r1617502 = 1.0;
        double r1617503 = x;
        double r1617504 = tan(r1617503);
        double r1617505 = sin(r1617503);
        double r1617506 = r1617504 * r1617505;
        double r1617507 = cos(r1617503);
        double r1617508 = r1617506 / r1617507;
        double r1617509 = r1617502 - r1617508;
        double r1617510 = r1617507 / r1617506;
        double r1617511 = r1617502 / r1617510;
        double r1617512 = r1617502 + r1617511;
        double r1617513 = r1617509 / r1617512;
        return r1617513;
}

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 tan-quot0.4

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

  7. Applied associate-*r/0.3

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

  9. Applied clear-num0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))