Average Error: 0.3 → 0.3
Time: 7.4s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
↓
\[\begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{1 - t_0}{1 + t_0} \end{array} \]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\begin{array}{l}
t_0 := {\tan x}^{2}\\
\frac{1 - t_0}{1 + t_0}
\end{array}

(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))) (/ (- 1.0 t_0) (+ 1.0 t_0))))
double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) * tan(x)));
}
↓
double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	return (1.0 - t_0) / (1.0 + t_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

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

  5. Simplified0.4

    \[\leadsto \frac{1}{1 + {\tan x}^{2}} - \color{blue}{\frac{{\tan x}^{2}}{1 + {\tan x}^{2}}} \]
  6. Using strategy rm

  7. Applied sub-div_binary640.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2021198 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))