Average Error: 0.3 → 0.4
Time: 4.8s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \frac{{\sin x}^{2}}{{\cos x}^{2}}}{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}
(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))
↓
(FPCore (x)
 :precision binary64
 (/
  (- 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))
  (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos 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 - (pow(sin(x), 2.0) / pow(cos(x), 2.0))) / (1.0 + (pow(sin(x), 2.0) / pow(cos(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. Taylor expanded around inf 0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2020253 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))