Average Error: 0.3 → 0.3
Time: 1.4min
Precision: binary64
Cost: 26176
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - {\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 (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 - 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

Alternatives

Alternative 1
Error25.6
Cost26626
\[\begin{array}{l} \mathbf{if}\;\tan x \leq -1.1011828055456616:\\ \;\;\;\;-1\\ \mathbf{elif}\;\tan x \leq 0.8892571677000385:\\ \;\;\;\;\frac{1}{1 + {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
Alternative 2
Error28.8
Cost64
\[1\]
Alternative 3
Error59.8
Cost64
\[-1\]

Error

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 pow1_binary64_11620.3

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

  4. Applied pow1_binary64_11620.3

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

  5. Applied pow-prod-down_binary64_11720.3

    \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \color{blue}{{\left(\tan x \cdot \tan x\right)}^{1}}}\]

  6. Simplified0.3

    \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\color{blue}{\left({\tan x}^{2}\right)}}^{1}}\]
  7. Using strategy rm

  8. Applied pow1_binary64_11620.3

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

  9. Applied pow1_binary64_11620.3

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

  10. Applied pow-prod-down_binary64_11720.3

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

  11. Simplified0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2021014 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))