Average Error: 0.3 → 0.5
Time: 5.7s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{{\left(\frac{1}{1 + \tan x \cdot \tan x}\right)}^{3} - {\left(\frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\right)}^{3}}{\frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \cdot \left(\frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x} + \frac{1}{1 + \tan x \cdot \tan x}\right) + \frac{1}{1 + \tan x \cdot \tan x} \cdot \frac{1}{1 + \tan x \cdot \tan x}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{{\left(\frac{1}{1 + \tan x \cdot \tan x}\right)}^{3} - {\left(\frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\right)}^{3}}{\frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \cdot \left(\frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x} + \frac{1}{1 + \tan x \cdot \tan x}\right) + \frac{1}{1 + \tan x \cdot \tan x} \cdot \frac{1}{1 + \tan x \cdot \tan x}}
double code(double x) {
	return ((double) (((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(x)))))) / ((double) (1.0 + ((double) (((double) tan(x)) * ((double) tan(x))))))));
}
↓
double code(double x) {
	return ((double) (((double) (((double) pow(((double) (1.0 / ((double) (1.0 + ((double) (((double) tan(x)) * ((double) tan(x)))))))), 3.0)) - ((double) pow(((double) (((double) (((double) tan(x)) * ((double) tan(x)))) / ((double) (1.0 + ((double) (((double) tan(x)) * ((double) tan(x)))))))), 3.0)))) / ((double) (((double) (((double) (((double) (((double) tan(x)) * ((double) tan(x)))) / ((double) (1.0 + ((double) (((double) tan(x)) * ((double) tan(x)))))))) * ((double) (((double) (((double) (((double) tan(x)) * ((double) tan(x)))) / ((double) (1.0 + ((double) (((double) tan(x)) * ((double) tan(x)))))))) + ((double) (1.0 / ((double) (1.0 + ((double) (((double) tan(x)) * ((double) tan(x)))))))))))) + ((double) (((double) (1.0 / ((double) (1.0 + ((double) (((double) tan(x)) * ((double) tan(x)))))))) * ((double) (1.0 / ((double) (1.0 + ((double) (((double) tan(x)) * ((double) tan(x))))))))))))));
}

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-sub0.4

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

  5. Applied flip3--0.5

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

  6. Simplified0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2020173 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))