Average Error: 0.3 → 0.4
Time: 10.1s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\log \left(e^{\tan x + 1}\right) \cdot \left(1 - \tan x\right)}{1 + \tan x \cdot \tan x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{\log \left(e^{\tan x + 1}\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
 (/
  (* (log (exp (+ (tan x) 1.0))) (- 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 (log(exp(tan(x) + 1.0)) * (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 *-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. Simplified0.4

    \[\leadsto \frac{\color{blue}{\left(\tan x + 1\right)} \cdot \left(1 - \tan x\right)}{1 + \tan x \cdot \tan x}\]
  6. Using strategy rm

  7. Applied add-log-exp_binary640.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2020288 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))