Average Error: 0.3 → 0.4
Time: 4.0s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 + \tan x}{\frac{1 + {\tan x}^{2}}{1 - \tan x}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 + \tan x}{\frac{1 + {\tan x}^{2}}{1 - \tan x}}
(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))
↓
(FPCore (x)
 :precision binary64
 (/ (+ 1.0 (tan x)) (/ (+ 1.0 (pow (tan x) 2.0)) (- 1.0 (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 + tan(x)) / ((1.0 + pow(tan(x), 2.0)) / (1.0 - tan(x)));
}

Error

Bits error versus x

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Your Program's Arguments

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 *-un-lft-identity_binary640.3

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

  4. Applied difference-of-squares_binary640.4

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

  5. Applied associate-/l*_binary640.4

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2020220 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))