Average Error: 0.3 → 0.4
Time: 5.8s
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{1 + {\left(\tan x\right)}^{2}} - \frac{{\left(\tan x\right)}^{2}}{1 + {\left(\tan x\right)}^{2}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1}{1 + {\left(\tan x\right)}^{2}} - \frac{{\left(\tan x\right)}^{2}}{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
 (-
  (/ 1.0 (+ 1.0 (pow (tan x) 2.0)))
  (/ (pow (tan x) 2.0) (+ 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) (1.0 + ((double) pow(((double) tan(x)), 2.0))))) - (((double) pow(((double) tan(x)), 2.0)) / ((double) (1.0 + ((double) pow(((double) tan(x)), 2.0)))))));
}

Error

Bits error versus x

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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 div-subError: 0.4 bits

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

  4. SimplifiedError: 0.4 bits

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

  5. SimplifiedError: 0.4 bits

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

  6. Final simplificationError: 0.4 bits

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

Reproduce

herbie shell --seed 2020203 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))