Average Error: 0.3 → 0.4
Time: 4.5s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\frac{1}{1 + \tan x \cdot \tan x}}{\frac{1}{1 - \tan x \cdot \tan x}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{\frac{1}{1 + \tan x \cdot \tan x}}{\frac{1}{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 ((1.0 / (1.0 + (tan(x) * tan(x)))) / (1.0 / (1.0 - (tan(x) * 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 clear-num0.4

    \[\leadsto \color{blue}{\frac{1}{\frac{1 + \tan x \cdot \tan x}{1 - \tan x \cdot \tan x}}}\]
  4. Using strategy rm

  5. Applied div-inv0.4

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

  6. Applied associate-/r*0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2020102 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))