2sqrt (example 3.1)

Average Accuracy: 53.4% → 99.7%

Time: 9.7s

Precision: binary64

Cost: 13504

Math

\[\sqrt{x + 1} - \sqrt{x} \]

↓

\[\frac{1}{\left(\left(1 + \sqrt{1 + x}\right) + -1\right) + \sqrt{x}} \]

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))

↓

(FPCore (x)
 :precision binary64
 (/ 1.0 (+ (+ (+ 1.0 (sqrt (+ 1.0 x))) -1.0) (sqrt x))))

C

double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}

↓

double code(double x) {
	return 1.0 / (((1.0 + sqrt((1.0 + x))) + -1.0) + sqrt(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + 1.0d0)) - sqrt(x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (((1.0d0 + sqrt((1.0d0 + x))) + (-1.0d0)) + sqrt(x))
end function

Java

public static double code(double x) {
	return Math.sqrt((x + 1.0)) - Math.sqrt(x);
}

↓

public static double code(double x) {
	return 1.0 / (((1.0 + Math.sqrt((1.0 + x))) + -1.0) + Math.sqrt(x));
}

Python

def code(x):
	return math.sqrt((x + 1.0)) - math.sqrt(x)

↓

def code(x):
	return 1.0 / (((1.0 + math.sqrt((1.0 + x))) + -1.0) + math.sqrt(x))

Julia

function code(x)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end

↓

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(1.0 + sqrt(Float64(1.0 + x))) + -1.0) + sqrt(x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x + 1.0)) - sqrt(x);
end

↓

function tmp = code(x)
	tmp = 1.0 / (((1.0 + sqrt((1.0 + x))) + -1.0) + sqrt(x));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(1.0 / N[(N[(N[(1.0 + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	53.4%
Target	99.8%
Herbie	99.7%

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

Derivation

1. Initial program 53.4%

   \[\sqrt{x + 1} - \sqrt{x} \]

2. Applied egg-rr 54.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{x + 1} + \sqrt{x}}{\left(x + 1\right) - x}}} \]
   Proof

   [Start] 53.4	\[ \sqrt{x + 1} - \sqrt{x} \]
   flip-- [=>] 53.7	\[ \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} \]
   clear-num [=>] 53.7	\[ \color{blue}{\frac{1}{\frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}}} \]
   add-sqr-sqrt [<=] 53.9	\[ \frac{1}{\frac{\sqrt{x + 1} + \sqrt{x}}{\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}}} \]
   add-sqr-sqrt [<=] 54.4	\[ \frac{1}{\frac{\sqrt{x + 1} + \sqrt{x}}{\left(x + 1\right) - \color{blue}{x}}} \]

3. Applied egg-rr 54.4%

   \[\leadsto \frac{1}{\frac{\color{blue}{\left(\left(1 + \sqrt{x + 1}\right) - 1\right)} + \sqrt{x}}{\left(x + 1\right) - x}} \]
   Proof

   [Start] 54.4	\[ \frac{1}{\frac{\sqrt{x + 1} + \sqrt{x}}{\left(x + 1\right) - x}} \]
   expm1-log1p-u [=>] 54.4	\[ \frac{1}{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1}\right)\right)} + \sqrt{x}}{\left(x + 1\right) - x}} \]
   expm1-udef [=>] 54.4	\[ \frac{1}{\frac{\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{x + 1}\right)} - 1\right)} + \sqrt{x}}{\left(x + 1\right) - x}} \]
   log1p-udef [=>] 54.4	\[ \frac{1}{\frac{\left(e^{\color{blue}{\log \left(1 + \sqrt{x + 1}\right)}} - 1\right) + \sqrt{x}}{\left(x + 1\right) - x}} \]
   add-exp-log [<=] 54.4	\[ \frac{1}{\frac{\left(\color{blue}{\left(1 + \sqrt{x + 1}\right)} - 1\right) + \sqrt{x}}{\left(x + 1\right) - x}} \]

4. Taylor expanded in x around 0 99.7%

   \[\leadsto \frac{1}{\frac{\left(\left(1 + \sqrt{x + 1}\right) - 1\right) + \sqrt{x}}{\color{blue}{1}}} \]

5. Final simplification 99.7%

   \[\leadsto \frac{1}{\left(\left(1 + \sqrt{1 + x}\right) + -1\right) + \sqrt{x}} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq 66000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	13248

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

Alternative 3
Accuracy	98.5%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + \left(x \cdot 0.5 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 4
Accuracy	98.0%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1}{1 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 5
Accuracy	96.9%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 6
Accuracy	51.3%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023147 
(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)))