Average Error: 31.9 → 0.1
Time: 10.5s
Precision: binary64
Cost: 13376
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}
(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))
↓
(FPCore (x) :precision binary64 (* (/ (tan (/ x 2.0)) x) (/ (sin x) x)))
double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}
↓
double code(double x) {
	return (tan(x / 2.0) / x) * (sin(x) / x);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error32.2
Cost39872
\[\sqrt[3]{\frac{1 - \cos x}{x \cdot x}} \cdot \left(\sqrt[3]{\frac{1 - \cos x}{x \cdot x}} \cdot \sqrt[3]{\frac{1 - \cos x}{x \cdot x}}\right)\]
Alternative 2
Error0.7
Cost39616
\[\sqrt{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}} \cdot \sqrt{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}}\]
Alternative 3
Error31.4
Cost39360
\[\frac{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}{x} \cdot \frac{\sqrt[3]{1 - \cos x}}{x}\]
Alternative 4
Error15.9
Cost39360
\[\frac{\sqrt{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x} \cdot \frac{\sqrt{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x}\]
Alternative 5
Error31.6
Cost39360
\[\frac{\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}}{\frac{x}{\frac{\sqrt[3]{1 - \cos x}}{x}}}\]
Alternative 6
Error32.0
Cost26624
\[\frac{1 - {\cos x}^{3}}{\left(x \cdot x\right) \cdot \left(1 + \cos x \cdot \left(1 + \cos x\right)\right)}\]
Alternative 7
Error32.0
Cost26624
\[\frac{\frac{1 - {\cos x}^{3}}{1 + \cos x \cdot \left(1 + \cos x\right)}}{x \cdot x}\]
Alternative 8
Error32.0
Cost26560
\[\sqrt{\frac{1 - \cos x}{x \cdot x}} \cdot \sqrt{\frac{1 - \cos x}{x \cdot x}}\]
Alternative 9
Error47.2
Cost26304
\[\frac{1 + \sqrt{\cos x}}{x} \cdot \frac{1 - \sqrt{\cos x}}{x}\]
Alternative 10
Error31.2
Cost26304
\[\frac{\sqrt{1 - \cos x}}{x} \cdot \frac{\sqrt{1 - \cos x}}{x}\]
Alternative 11
Error32.0
Cost26304
\[\frac{\sqrt{1 - \cos x} \cdot \sqrt{1 - \cos x}}{x \cdot x}\]
Alternative 12
Error47.3
Cost26304
\[\frac{1 + \sqrt{\cos x}}{\frac{x}{\frac{1 - \sqrt{\cos x}}{x}}}\]
Alternative 13
Error31.5
Cost26304
\[\frac{\sqrt{1 - \cos x}}{\frac{x}{\frac{\sqrt{1 - \cos x}}{x}}}\]
Alternative 14
Error32.1
Cost20032
\[\frac{1 - \cos x \cdot \cos x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
Alternative 15
Error16.3
Cost19904
\[\frac{\frac{\sin x \cdot \sin x}{1 + \cos x}}{x \cdot x}\]
Alternative 16
Error41.5
Cost19712
\[\sqrt[3]{{\left(\frac{1 - \cos x}{x \cdot x}\right)}^{3}}\]
Alternative 17
Error32.0
Cost19712
\[\frac{\sqrt[3]{{\left(1 - \cos x\right)}^{3}}}{x \cdot x}\]
Alternative 18
Error32.0
Cost19648
\[\frac{\log \left(e^{1 - \cos x}\right)}{x \cdot x}\]
Alternative 19
Error0.1
Cost13504
\[\frac{1}{\frac{x}{\sin x}} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}\]
Alternative 20
Error0.2
Cost13504
\[\frac{1}{x} \cdot \left(\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x}\right)\]
Alternative 21
Error16.1
Cost13504
\[\frac{\tan \left(\frac{x}{2}\right) \cdot \left(-\sin x\right)}{-x \cdot x}\]
Alternative 22
Error0.4
Cost13376
\[\frac{\tan \left(\frac{x}{2}\right)}{x \cdot \frac{x}{\sin x}}\]
Alternative 23
Error16.1
Cost13376
\[\frac{\tan \left(\frac{x}{2}\right) \cdot \sin x}{x \cdot x}\]
Alternative 24
Error0.1
Cost13376
\[\frac{\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x}}{x}\]
Alternative 25
Error31.9
Cost7104
\[\frac{1}{x \cdot x} - \frac{\cos x}{x \cdot x}\]
Alternative 26
Error32.0
Cost6976
\[\left(1 - \cos x\right) \cdot \frac{1}{x \cdot x}\]
Alternative 27
Error31.5
Cost6976
\[\frac{1}{\frac{x}{\frac{1 - \cos x}{x}}}\]
Alternative 28
Error31.2
Cost6976
\[\frac{1 - \cos x}{x} \cdot \frac{1}{x}\]
Alternative 29
Error31.9
Cost6848
\[\frac{1 - \cos x}{x \cdot x}\]
Alternative 30
Error31.2
Cost6848
\[\frac{\frac{1 - \cos x}{x}}{x}\]
Alternative 31
Error30.5
Cost64
\[0.5\]
Alternative 32
Error56.6
Cost64
\[1\]
Alternative 33
Error46.8
Cost64
\[0\]
Alternative 34
Error62.6
Cost64
\[-1\]

Error

Derivation

  1. Initial program 31.9

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

  3. Applied flip--_binary64_4632.1

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

  4. Simplified16.3

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

  6. Applied *-un-lft-identity_binary64_7116.3

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

  7. Applied times-frac_binary64_7716.3

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

  8. Applied times-frac_binary64_770.3

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

  9. Simplified0.3

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

  10. Simplified0.1

    \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{x}}\]

  11. Simplified0.1

    \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}}\]

  12. Final simplification0.1

    \[\leadsto \frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))