Average Error: 0.3 → 0.4
Time: 5.5s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{1 + {\tan x}^{2}} - \frac{{\tan x}^{2}}{1 + {\tan x}^{2}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1}{1 + {\tan x}^{2}} - \frac{{\tan x}^{2}}{1 + {\tan x}^{2}}
(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))
↓
(FPCore (x)
 :precision binary64
 (-
  (/ 1.0 (+ 1.0 (pow (tan x) 2.0)))
  (/ (pow (tan x) 2.0) (+ 1.0 (pow (tan x) 2.0)))))
double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) * tan(x)));
}
↓
double code(double x) {
	return (1.0 / (1.0 + pow(tan(x), 2.0))) - (pow(tan(x), 2.0) / (1.0 + pow(tan(x), 2.0)));
}

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 div-sub_binary640.4

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

  4. Simplified0.4

    \[\leadsto \color{blue}{\frac{1}{{\tan x}^{2} + 1}} - \frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]

  5. Simplified0.4

    \[\leadsto \frac{1}{{\tan x}^{2} + 1} - \color{blue}{\frac{{\tan x}^{2}}{{\tan x}^{2} + 1}}\]

  6. Final simplification0.4

    \[\leadsto \frac{1}{1 + {\tan x}^{2}} - \frac{{\tan x}^{2}}{1 + {\tan x}^{2}}\]

Reproduce

herbie shell --seed 2020280 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))