2sqrt (example 3.1)

Average Error: 53.7% → 99.7%

Time: 12.3s

Precision: binary64

Cost: 19584.00

Math

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

↓

\[\frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} \]

FPCore

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

↓

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

C

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

↓

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

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 / (Math.sqrt(x) + Math.hypot(1.0, Math.sqrt(x)));
}

Python

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

↓

def code(x):
	return 1.0 / (math.sqrt(x) + math.hypot(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(sqrt(x) + hypot(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 / (sqrt(x) + hypot(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[Sqrt[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	53.7%
Target	99.7%
Herbie	99.7%

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

Derivation

1. Initial program 53.7

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

2. Applied egg-rr 54.7

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

3. Simplified 99.7

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

   [Start] 54.7	\[ \left(x + \left(1 - x\right)\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
   associate-*r/ [=>] 54.7	\[ \color{blue}{\frac{\left(x + \left(1 - x\right)\right) \cdot 1}{\sqrt{x + 1} + \sqrt{x}}} \]
   *-rgt-identity [=>] 54.7	\[ \frac{\color{blue}{x + \left(1 - x\right)}}{\sqrt{x + 1} + \sqrt{x}} \]
   +-commutative [=>] 54.7	\[ \frac{\color{blue}{\left(1 - x\right) + x}}{\sqrt{x + 1} + \sqrt{x}} \]
   associate-+l- [=>] 99.7	\[ \frac{\color{blue}{1 - \left(x - x\right)}}{\sqrt{x + 1} + \sqrt{x}} \]
   +-inverses [=>] 99.7	\[ \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]
   metadata-eval [=>] 99.7	\[ \frac{\color{blue}{1}}{\sqrt{x + 1} + \sqrt{x}} \]
   +-commutative [=>] 99.7	\[ \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} \]

4. Applied egg-rr 99.7

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

5. Final simplification 99.7

   \[\leadsto \frac{1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} \]

Alternatives

Alternative 1
Error	99.5%
Cost	13252.00

\[\begin{array}{l} \mathbf{if}\;x \leq 44000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 2
Error	99.7%
Cost	13248.00

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

Alternative 3
Error	98.5%
Cost	6980.00

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

Alternative 4
Error	97.9%
Cost	6852.00

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

Alternative 5
Error	96.7%
Cost	6788.00

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

Alternative 6
Error	51.5%
Cost	64.00

\[1 \]

Reproduce

herbie shell --seed 2023093 
(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)))