Average Error: 0.3 → 0.3
Time: 6.1s
Precision: binary64
Cost: 39360
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{\sin x \cdot \tan x}{\cos x}}{1 + \frac{\sin x \cdot \tan x}{\cos x}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \frac{\sin x \cdot \tan x}{\cos x}}{1 + \frac{\sin x \cdot \tan x}{\cos x}}
(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))
↓
(FPCore (x)
 :precision binary64
 (/
  (- 1.0 (/ (* (sin x) (tan x)) (cos x)))
  (+ 1.0 (/ (* (sin x) (tan x)) (cos x)))))
double code(double x) {
	return (1.0 - (tan(x) * tan(x))) / (1.0 + (tan(x) * tan(x)));
}
↓
double code(double x) {
	return (1.0 - ((sin(x) * tan(x)) / cos(x))) / (1.0 + ((sin(x) * tan(x)) / cos(x)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.3
Cost26176
\[\frac{{\tan x}^{2} + -1}{-1 - {\tan x}^{2}}\]
Alternative 2
Error25.7
Cost26690
\[\begin{array}{l} \mathbf{if}\;\tan x \leq -1.0516589239156482:\\ \;\;\;\;-1\\ \mathbf{elif}\;\tan x \leq 0.9962876849726159:\\ \;\;\;\;\frac{1}{1 + \tan x \cdot \tan x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
Alternative 3
Error29.1
Cost64
\[1\]

Error

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 tan-quot_binary64_9190.4

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

  4. Applied associate-*l/_binary64_7030.4

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

  5. Simplified0.4

    \[\leadsto \frac{1 - \frac{\color{blue}{\tan x \cdot \sin x}}{\cos x}}{1 + \tan x \cdot \tan x}\]
  6. Using strategy rm

  7. Applied tan-quot_binary64_9190.4

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

  8. Applied associate-*l/_binary64_7030.3

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

  9. Simplified0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

    \[\leadsto \frac{1 - \frac{\sin x \cdot \tan x}{\cos x}}{1 + \frac{\sin x \cdot \tan x}{\cos x}}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))