Average Error: 0.3 → 0.4
Time: 4.6s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\frac{\mathsf{fma}\left(1, \sqrt{1}, {\left(\tan x\right)}^{3}\right)}{\mathsf{fma}\left(\tan x, \tan x - \sqrt{1}, 1\right)} \cdot \left(\sqrt{1} - \tan x\right)}{1 + \tan x \cdot \tan x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{\frac{\mathsf{fma}\left(1, \sqrt{1}, {\left(\tan x\right)}^{3}\right)}{\mathsf{fma}\left(\tan x, \tan x - \sqrt{1}, 1\right)} \cdot \left(\sqrt{1} - \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 (((fma(1.0, sqrt(1.0), pow(tan(x), 3.0)) / fma(tan(x), (tan(x) - sqrt(1.0)), 1.0)) * (sqrt(1.0) - 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 add-sqr-sqrt0.3

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

  4. Applied difference-of-squares0.4

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

  6. Applied flip3-+0.4

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))