Average Error: 0.3 → 0.3
Time: 5.3s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\left(\left|\tan x\right| + \sqrt{1}\right) \cdot \left(\sqrt{1} + \left(-\left|\tan x\right|\right)\right)}{1 + \tan x \cdot \tan x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{\left(\left|\tan x\right| + \sqrt{1}\right) \cdot \left(\sqrt{1} + \left(-\left|\tan x\right|\right)\right)}{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) (((double) fabs(((double) tan(x)))) + ((double) sqrt(1.0)))) * ((double) (((double) sqrt(1.0)) + ((double) -(((double) fabs(((double) tan(x)))))))))) / ((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 add-sqr-sqrt0.3

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

  4. Applied add-sqr-sqrt0.3

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

  5. Applied difference-of-squares0.3

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

  6. Simplified0.3

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

  7. Simplified0.3

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

  9. Applied *-un-lft-identity0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020113 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))