Average Error: 0.3 → 0.4
Time: 4.1s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{{\left(1 - \sqrt[3]{{\left(\tan x\right)}^{6}}\right)}^{1}}{1 + \sqrt[3]{{\left(\tan x\right)}^{6}}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{{\left(1 - \sqrt[3]{{\left(\tan x\right)}^{6}}\right)}^{1}}{1 + \sqrt[3]{{\left(\tan x\right)}^{6}}}
double code(double x) {
	return ((1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) * tan(x))));
}
↓
double code(double x) {
	return (pow((1.0 - cbrt(pow(tan(x), 6.0))), 1.0) / (1.0 + cbrt(pow(tan(x), 6.0))));
}

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 pow10.3

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

  5. Applied add-cbrt-cube0.5

    \[\leadsto \frac{{\left(1 - \tan x \cdot \color{blue}{\sqrt[3]{\left(\tan x \cdot \tan x\right) \cdot \tan x}}\right)}^{1}}{1 + \tan x \cdot \tan x}\]

  6. Applied add-cbrt-cube0.6

    \[\leadsto \frac{{\left(1 - \color{blue}{\sqrt[3]{\left(\tan x \cdot \tan x\right) \cdot \tan x}} \cdot \sqrt[3]{\left(\tan x \cdot \tan x\right) \cdot \tan x}\right)}^{1}}{1 + \tan x \cdot \tan x}\]

  7. Applied cbrt-unprod0.5

    \[\leadsto \frac{{\left(1 - \color{blue}{\sqrt[3]{\left(\left(\tan x \cdot \tan x\right) \cdot \tan x\right) \cdot \left(\left(\tan x \cdot \tan x\right) \cdot \tan x\right)}}\right)}^{1}}{1 + \tan x \cdot \tan x}\]

  8. Simplified0.5

    \[\leadsto \frac{{\left(1 - \sqrt[3]{\color{blue}{{\left(\tan x\right)}^{6}}}\right)}^{1}}{1 + \tan x \cdot \tan x}\]
  9. Using strategy rm

  10. Applied add-cbrt-cube0.5

    \[\leadsto \frac{{\left(1 - \sqrt[3]{{\left(\tan x\right)}^{6}}\right)}^{1}}{1 + \tan x \cdot \color{blue}{\sqrt[3]{\left(\tan x \cdot \tan x\right) \cdot \tan x}}}\]

  11. Applied add-cbrt-cube0.5

    \[\leadsto \frac{{\left(1 - \sqrt[3]{{\left(\tan x\right)}^{6}}\right)}^{1}}{1 + \color{blue}{\sqrt[3]{\left(\tan x \cdot \tan x\right) \cdot \tan x}} \cdot \sqrt[3]{\left(\tan x \cdot \tan x\right) \cdot \tan x}}\]

  12. Applied cbrt-unprod0.4

    \[\leadsto \frac{{\left(1 - \sqrt[3]{{\left(\tan x\right)}^{6}}\right)}^{1}}{1 + \color{blue}{\sqrt[3]{\left(\left(\tan x \cdot \tan x\right) \cdot \tan x\right) \cdot \left(\left(\tan x \cdot \tan x\right) \cdot \tan x\right)}}}\]

  13. Simplified0.4

    \[\leadsto \frac{{\left(1 - \sqrt[3]{{\left(\tan x\right)}^{6}}\right)}^{1}}{1 + \sqrt[3]{\color{blue}{{\left(\tan x\right)}^{6}}}}\]

  14. Final simplification0.4

    \[\leadsto \frac{{\left(1 - \sqrt[3]{{\left(\tan x\right)}^{6}}\right)}^{1}}{1 + \sqrt[3]{{\left(\tan x\right)}^{6}}}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))