Average Error: 0.3 → 0.4
Time: 3.5s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)}{\frac{\left(\left(1 - \tan x \cdot \tan x\right) \cdot \mathsf{fma}\left(\tan x, \tan x, 1\right)\right) \cdot \left(1 - \tan x \cdot \tan x\right)}{1 + \tan x \cdot \tan x}}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)}{\frac{\left(\left(1 - \tan x \cdot \tan x\right) \cdot \mathsf{fma}\left(\tan x, \tan x, 1\right)\right) \cdot \left(1 - \tan x \cdot \tan x\right)}{1 + \tan x \cdot \tan 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 * 1.0) - ((tan(x) * tan(x)) * (tan(x) * tan(x)))) / ((((1.0 - (tan(x) * tan(x))) * fma(tan(x), tan(x), 1.0)) * (1.0 - (tan(x) * tan(x)))) / (1.0 + (tan(x) * 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 clear-num0.4

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

  5. Applied flip-+0.4

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

  6. Applied associate-/l/0.4

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

  8. Applied flip--0.4

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

  9. Applied associate-*r/0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

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)))))