Average Error: 0.3 → 0.3
Time: 9.9s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\left(1 + \tan x\right) \cdot \left(1 - \tan x\right)}{1 + \tan x \cdot \tan x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{\left(1 + \tan x\right) \cdot \left(1 - \tan x\right)}{1 + \tan x \cdot \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 (tan x))) (+ 1.0 (* (tan x) (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 - tan(x))) / (1.0 + (tan(x) * 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-sqrt_binary64_7820.3

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

  4. Applied difference-of-squares_binary64_7290.3

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

  5. Simplified0.3

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2021028 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))