Average Error: 31.1 → 1.1
Time: 5.0s
Precision: binary64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{\sqrt{1 + \cos x}}}{x} \cdot \frac{\frac{\sin x}{\sqrt{1 + \cos x}}}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{\sqrt{1 + \cos x}}}{x} \cdot \frac{\frac{\sin x}{\sqrt{1 + \cos x}}}{x}
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x)
 :precision binary64
 (*
  (/ (/ (sin x) (sqrt (+ 1.0 (cos x)))) x)
  (/ (/ (sin x) (sqrt (+ 1.0 (cos x)))) x)))
double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}
↓
double code(double x) {
	return ((sin(x) / sqrt(1.0 + cos(x))) / x) * ((sin(x) / sqrt(1.0 + cos(x))) / x);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.1

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

  3. Applied flip--_binary64_5231.2

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

  4. Simplified15.3

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

  6. Applied add-sqr-sqrt_binary64_9815.7

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

  7. Applied times-frac_binary64_8315.6

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

  8. Applied times-frac_binary64_831.1

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

  9. Final simplification1.1

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

Reproduce

herbie shell --seed 2020281 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))