Average Error: 0.3 → 0.4
Time: 5.5s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \left(\tan x \cdot \sin x\right) \cdot \frac{1}{\cos x}}{1 + \tan x \cdot \tan x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \left(\tan x \cdot \sin x\right) \cdot \frac{1}{\cos x}}{1 + \tan x \cdot \tan x}
double f(double x) {
        double r13105 = 1.0;
        double r13106 = x;
        double r13107 = tan(r13106);
        double r13108 = r13107 * r13107;
        double r13109 = r13105 - r13108;
        double r13110 = r13105 + r13108;
        double r13111 = r13109 / r13110;
        return r13111;
}
↓
double f(double x) {
        double r13112 = 1.0;
        double r13113 = x;
        double r13114 = tan(r13113);
        double r13115 = sin(r13113);
        double r13116 = r13114 * r13115;
        double r13117 = 1.0;
        double r13118 = cos(r13113);
        double r13119 = r13117 / r13118;
        double r13120 = r13116 * r13119;
        double r13121 = r13112 - r13120;
        double r13122 = r13114 * r13114;
        double r13123 = r13112 + r13122;
        double r13124 = r13121 / r13123;
        return r13124;
}

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

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

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))