Average Error: 0.3 → 0.5
Time: 5.1s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[1 \cdot \frac{1}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1} - \frac{{\left(\sin x\right)}^{2}}{\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot {\left(\cos x\right)}^{2}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
1 \cdot \frac{1}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1} - \frac{{\left(\sin x\right)}^{2}}{\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot {\left(\cos x\right)}^{2}}
double code(double x) {
	return ((1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) * tan(x))));
}
↓
double code(double x) {
	return ((1.0 * (1.0 / ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0))) - (pow(sin(x), 2.0) / (((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + 1.0) * 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. 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. Taylor expanded around inf 0.5

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

  5. Final simplification0.5

    \[\leadsto 1 \cdot \frac{1}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1} - \frac{{\left(\sin x\right)}^{2}}{\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot {\left(\cos x\right)}^{2}}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))