Average Error: 0.3 → 0.4
Time: 6.0s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{{\left(\tan x\right)}^{2} + 1} \cdot \left(1 - {\left(\tan x\right)}^{2}\right)\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1}{{\left(\tan x\right)}^{2} + 1} \cdot \left(1 - {\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 (+ (pow (tan x) 2.0) 1.0)) (- 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) ((1.0 / ((double) (((double) pow(((double) tan(x)), 2.0)) + 1.0))) * ((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 0.3

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

  3. Applied add-sqr-sqrt0.3

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

  4. Applied difference-of-squares0.4

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

  5. Applied associate-/l*0.4

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

  6. Simplified0.4

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

  8. Applied div-inv0.4

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

  9. Applied *-un-lft-identity0.4

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

  10. Applied times-frac0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020198 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))