Average Error: 0.3 → 1.0
Time: 4.5s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{\frac{1 + {\left(\tan x\right)}^{2}}{1 - \log \left(e^{{\left(\tan x\right)}^{2}}\right)}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1}{\frac{1 + {\left(\tan x\right)}^{2}}{1 - \log \left(e^{{\left(\tan x\right)}^{2}}\right)}}
(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))
↓
(FPCore (x)
 :precision binary64
 (/ 1.0 (/ (+ 1.0 (pow (tan x) 2.0)) (- 1.0 (log (exp (pow (tan x) 2.0)))))))
double code(double x) {
	return (((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(x)))))) / ((double) (1.0 + ((double) (((double) tan(x)) * ((double) tan(x)))))));
}
↓
double code(double x) {
	return (1.0 / (((double) (1.0 + ((double) pow(((double) tan(x)), 2.0)))) / ((double) (1.0 - ((double) log(((double) exp(((double) pow(((double) tan(x)), 2.0))))))))));
}

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 clear-num_binary640.4

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

  4. Simplified0.4

    \[\leadsto \frac{1}{\color{blue}{\frac{1 + {\left(\tan x\right)}^{2}}{1 - {\left(\tan x\right)}^{2}}}}\]
  5. Using strategy rm

  6. Applied add-log-exp_binary641.0

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

  7. Final simplification1.0

    \[\leadsto \frac{1}{\frac{1 + {\left(\tan x\right)}^{2}}{1 - \log \left(e^{{\left(\tan x\right)}^{2}}\right)}}\]

Reproduce

herbie shell --seed 2020219 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))