Average Error: 0.3 → 0.3
Time: 9.3s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \tan x \cdot \tan x}{1 + {\tan x}^{2}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \tan x \cdot \tan x}{1 + {\tan x}^{2}}
(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))
↓
(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (pow (tan x) 2.0))))
double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) * tan(x)));
}
↓
double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (1.0 + pow(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 pow1_binary640.3

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

  4. Applied pow1_binary640.3

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

  5. Applied pow-prod-down_binary640.3

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

  6. Simplified0.3

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

  7. Final simplification0.3

    \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + {\tan x}^{2}}\]

Reproduce

herbie shell --seed 2021139 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))