Average Error: 0.3 → 0.4
Time: 3.9s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{1 + \tan x \cdot \tan x} - \frac{\tan x \cdot \tan x}{1 + {\left(\tan x\right)}^{2}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1}{1 + \tan x \cdot \tan x} - \frac{\tan x \cdot \tan x}{1 + {\left(\tan x\right)}^{2}}
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) (1.0 / ((double) (1.0 + ((double) (((double) tan(x)) * ((double) tan(x)))))))) - ((double) (((double) (((double) tan(x)) * ((double) tan(x)))) / ((double) (1.0 + ((double) pow(((double) tan(x)), 2.0))))))));
}

Error

Bits error versus x

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Results

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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. Using strategy rm

  5. Applied pow10.4

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

  6. Applied pow10.4

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

  7. Applied pow-prod-up0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020152 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))