Average Error: 30.0 → 0.2
Time: 14.6s
Precision: binary64
Cost: 13248
\[\sqrt{x + 1} - \sqrt{x}\]
↓
\[\frac{1}{\sqrt{1 + x} + \sqrt{x}}\]
\sqrt{x + 1} - \sqrt{x}
↓
\frac{1}{\sqrt{1 + x} + \sqrt{x}}
(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))
↓
(FPCore (x) :precision binary64 (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))))
double code(double x) {
	return sqrt(x + 1.0) - sqrt(x);
}
↓
double code(double x) {
	return 1.0 / (sqrt(1.0 + x) + sqrt(x));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.0
Target0.2
Herbie0.2
\[\frac{1}{\sqrt{x + 1} + \sqrt{x}}\]

Alternatives

Alternative 1
Error0.5
Cost97728
\[\frac{\sqrt{\frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}}{\sqrt[3]{\sqrt{1 + x} + \sqrt{x}}} \cdot \frac{\sqrt{\frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}}{\sqrt[3]{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}\]
Alternative 2
Error0.7
Cost97728
\[\frac{\sqrt[3]{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}}{\left|\sqrt[3]{\sqrt{1 + x} + \sqrt{x}}\right|} \cdot \frac{\sqrt[3]{\frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}}{\sqrt{\sqrt[3]{\sqrt{1 + x} + \sqrt{x}}}}\]
Alternative 3
Error0.6
Cost97728
\[\frac{\sqrt[3]{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{\sqrt{\sqrt{1 + x} + \sqrt{x}}}} \cdot \frac{\sqrt[3]{\frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}}{\sqrt{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}\]
Alternative 4
Error0.5
Cost91072
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\left|\sqrt[3]{\sqrt{1 + x} + \sqrt{x}}\right|} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{\sqrt[3]{\sqrt{1 + x} + \sqrt{x}}}}\]
Alternative 5
Error1.1
Cost84480
\[\frac{\frac{1}{\sqrt[3]{\sqrt{\sqrt{1 + x} + \sqrt{x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}}{{\left(\sqrt[3]{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\right)}^{4}}\]
Alternative 6
Error0.7
Cost78272
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{\sqrt{1 + x} + \sqrt{x}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 7
Error0.6
Cost77952
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\sqrt{\sqrt{1 + x} + \sqrt{x}}}} \cdot \frac{\sqrt[3]{1}}{{\left(\sqrt{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3}}\]
Alternative 8
Error0.5
Cost71744
\[\frac{1}{\left|\sqrt[3]{\sqrt{1 + x} + \sqrt{x}}\right|} \cdot \frac{\frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{\sqrt[3]{\sqrt{1 + x} + \sqrt{x}}}}\]
Alternative 9
Error0.4
Cost71744
\[\frac{\sqrt{\frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}}{\frac{\sqrt{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{\frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}}}\]
Alternative 10
Error0.5
Cost71680
\[\frac{1}{\left|\sqrt[3]{\sqrt{1 + x} + \sqrt{x}}\right|} \cdot \frac{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.5}}{\sqrt{\sqrt[3]{\sqrt{1 + x} + \sqrt{x}}}}\]
Alternative 11
Error0.4
Cost71680
\[\frac{1}{\sqrt{\sqrt{\sqrt{1 + x} + \sqrt{x}}}} \cdot \frac{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.5}}{\sqrt{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}\]
Alternative 12
Error0.4
Cost71616
\[\frac{{\left(\sqrt{\sqrt{1 + x} + \sqrt{x}}\right)}^{-0.5}}{\frac{\sqrt{\sqrt{1 + x} + \sqrt{x}}}{{\left(\sqrt{\sqrt{1 + x} + \sqrt{x}}\right)}^{-0.5}}}\]
Alternative 13
Error0.6
Cost65344
\[\frac{\sqrt[3]{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 14
Error10.6
Cost59328
\[\frac{1}{{\left(\sqrt{1 + x}\right)}^{3} + {x}^{1.5}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right)\]
Alternative 15
Error0.8
Cost59072
\[\sqrt[3]{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \cdot \left(\sqrt[3]{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}\right)\]
Alternative 16
Error0.7
Cost58944
\[\frac{1}{\sqrt[3]{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{\sqrt{1 + x} + \sqrt{x}}} \cdot \frac{1}{\sqrt[3]{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 17
Error0.4
Cost58816
\[\frac{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.25}}{\frac{\sqrt{\sqrt{1 + x} + \sqrt{x}}}{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.25}}}\]
Alternative 18
Error0.7
Cost58816
\[\frac{\frac{1}{\sqrt[3]{\sqrt{1 + x} + \sqrt{x}} \cdot \sqrt[3]{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt[3]{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 19
Error0.4
Cost58816
\[{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.25} \cdot \frac{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.25}}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 20
Error30.1
Cost58688
\[\sqrt[3]{\sqrt{1 + x} - \sqrt{x}} \cdot \left(\sqrt[3]{\sqrt{1 + x} - \sqrt{x}} \cdot \sqrt[3]{\sqrt{1 + x} - \sqrt{x}}\right)\]
Alternative 21
Error0.4
Cost58688
\[\frac{\sqrt[3]{1}}{\sqrt{\sqrt{1 + x} + \sqrt{x}}} \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 22
Error0.6
Cost58624
\[\frac{1}{\sqrt{\sqrt{\sqrt{1 + x} + \sqrt{x}}}} \cdot \frac{1}{{\left(\sqrt{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3}}\]
Alternative 23
Error0.6
Cost58496
\[\frac{\frac{1}{{\left(\sqrt{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3}}}{\sqrt{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}\]
Alternative 24
Error1.1
Cost58496
\[\frac{\frac{1}{{\left(\sqrt[3]{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\right)}^{5}}}{\sqrt[3]{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}\]
Alternative 25
Error0.6
Cost58496
\[\frac{\frac{1}{\sqrt{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}}{{\left(\sqrt{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3}}\]
Alternative 26
Error1.0
Cost52224
\[\sqrt[3]{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \cdot {\left(\sqrt[3]{\frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}\right)}^{4}\]
Alternative 27
Error0.4
Cost52096
\[\frac{\frac{1}{\sqrt[3]{{\left(\sqrt{\sqrt{1 + x} + \sqrt{x}}\right)}^{3}}}}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 28
Error0.5
Cost52096
\[\frac{\sqrt[3]{{\left(\frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\right)}^{3}}}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 29
Error0.4
Cost52032
\[\frac{\sqrt[3]{{\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.5}\right)}^{3}}}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 30
Error0.5
Cost45760
\[\frac{1}{\sqrt{x} + \sqrt[3]{\sqrt{1 + x}} \cdot \left(\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}\right)}\]
Alternative 31
Error0.4
Cost45632
\[\frac{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.5}}{\sqrt[3]{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{1.5}}}\]
Alternative 32
Error10.2
Cost45632
\[\sqrt[3]{\frac{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.5}}{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{2.5}}}\]
Alternative 33
Error0.4
Cost39360
\[\frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}}} \cdot \frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 34
Error0.3
Cost39360
\[\sqrt{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \cdot \sqrt{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 35
Error0.3
Cost39296
\[{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.5} \cdot \frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 36
Error0.3
Cost39232
\[\frac{\frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 37
Error0.2
Cost39168
\[\frac{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-0.5}}{\sqrt{\sqrt{1 + x} + \sqrt{x}}}\]
Alternative 38
Error30.0
Cost39104
\[\sqrt{\sqrt{1 + x} - \sqrt{x}} \cdot \sqrt{\sqrt{1 + x} - \sqrt{x}}\]
Alternative 39
Error30.5
Cost33216
\[\frac{{\left(\sqrt{1 + x}\right)}^{3} - {x}^{1.5}}{\left(1 + x\right) + \left(x + \sqrt{1 + x} \cdot \sqrt{x}\right)}\]
Alternative 40
Error0.3
Cost32704
\[\frac{1}{\sqrt{x} + \sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}}\]
Alternative 41
Error30.9
Cost26496
\[\frac{\sqrt{1 + {x}^{3}}}{\sqrt{x \cdot x + \left(1 - x\right)}} - \sqrt{x}\]
Alternative 42
Error10.7
Cost26112
\[\sqrt[3]{\frac{1}{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{3}}}\]
Alternative 43
Error10.7
Cost26112
\[\frac{1}{\sqrt[3]{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{3}}}\]
Alternative 44
Error10.2
Cost26112
\[\sqrt[3]{{\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}}\right)}^{3}}\]
Alternative 45
Error2.4
Cost25984
\[e^{-\log \left(\sqrt{1 + x} + \sqrt{x}\right)}\]
Alternative 46
Error30.0
Cost25920
\[e^{\log \left(\sqrt{1 + x} - \sqrt{x}\right)}\]
Alternative 47
Error30.0
Cost25920
\[\log \left(e^{\sqrt{1 + x} - \sqrt{x}}\right)\]
Alternative 48
Error47.4
Cost20032
\[\frac{1}{\sqrt{x} + \frac{\sqrt{x \cdot x + -1}}{\sqrt{x + -1}}}\]
Alternative 49
Error30.0
Cost13120
\[\sqrt{1 + x} - \sqrt{x}\]
Alternative 50
Error32.2
Cost7104
\[\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\]
Alternative 51
Error31.1
Cost6848
\[\left(1 + x \cdot 0.5\right) - \sqrt{x}\]
Alternative 52
Error32.3
Cost6592
\[1 - \sqrt{x}\]
Alternative 53
Error31.3
Cost64
\[1\]
Alternative 54
Error61.8
Cost64
\[0\]
Alternative 55
Error62.9
Cost64
\[-1\]

Error

Derivation

  1. Initial program 30.0

    \[\sqrt{x + 1} - \sqrt{x}\]
  2. Using strategy rm

  3. Applied flip--_binary64_304329.8

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))