Average Error: 0.3 → 0.5
Time: 6.5s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\frac{1}{1}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} - \frac{1}{\frac{\mathsf{fma}\left(\tan x, \tan x, 1\right)}{\tan x}} \cdot \frac{\sin x}{\cos x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{\frac{1}{1}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} - \frac{1}{\frac{\mathsf{fma}\left(\tan x, \tan x, 1\right)}{\tan x}} \cdot \frac{\sin x}{\cos x}
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) / fma(tan(x), tan(x), 1.0)) - ((1.0 / (fma(tan(x), tan(x), 1.0) / tan(x))) * (sin(x) / cos(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 tan-quot0.4

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

  4. Applied associate-*r/0.4

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

  6. Applied div-sub0.5

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

  7. Simplified0.5

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} - \frac{\frac{\tan x \cdot \sin x}{\cos x}}{1 + \tan x \cdot \tan x}\]

  8. Simplified0.4

    \[\leadsto \frac{\frac{1}{1}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} - \color{blue}{\frac{\tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \cdot \frac{\sin x}{\cos x}}\]
  9. Using strategy rm

  10. Applied clear-num0.5

    \[\leadsto \frac{\frac{1}{1}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} - \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\tan x, \tan x, 1\right)}{\tan x}}} \cdot \frac{\sin x}{\cos x}\]

  11. Final simplification0.5

    \[\leadsto \frac{\frac{1}{1}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} - \frac{1}{\frac{\mathsf{fma}\left(\tan x, \tan x, 1\right)}{\tan x}} \cdot \frac{\sin x}{\cos x}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))