Average Error: 0.3 → 0.4
Time: 5.5s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\left(\tan x + \sqrt{1}\right) \cdot \frac{\sqrt{1} - \tan x}{1 + {\left(\tan x\right)}^{2}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\left(\tan x + \sqrt{1}\right) \cdot \frac{\sqrt{1} - \tan x}{1 + {\left(\tan x\right)}^{2}}
(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))
↓
(FPCore (x)
 :precision binary64
 (*
  (+ (tan x) (sqrt 1.0))
  (/ (- (sqrt 1.0) (tan x)) (+ 1.0 (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 ((double) (((double) (((double) tan(x)) + ((double) sqrt(1.0)))) * (((double) (((double) sqrt(1.0)) - ((double) tan(x)))) / ((double) (1.0 + ((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 Error: 0.3 bits

    \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
  2. Using strategy rm

  3. Applied *-un-lft-identityError: 0.3 bits

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

  4. Applied add-sqr-sqrtError: 0.3 bits

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

  5. Applied difference-of-squaresError: 0.4 bits

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

  6. Applied times-fracError: 0.4 bits

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

  7. SimplifiedError: 0.4 bits

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

  8. SimplifiedError: 0.4 bits

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

  9. Final simplificationError: 0.4 bits

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

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))