Average Error: 0.3 → 0.5
Time: 5.2s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{1 + \tan x \cdot \tan x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{1 + \tan x \cdot \tan 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 - (pow(sin(x), 2.0) / pow(cos(x), 2.0))) / (1.0 + (tan(x) * tan(x))));
}

Error

Bits error versus x

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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 tan-quot0.4

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

  4. Applied tan-quot0.4

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

  5. Applied frac-times0.5

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

  6. Simplified0.5

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

  7. Simplified0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))