2isqrt (example 3.6)

Average Accuracy: 68.9% → 99.3%

Time: 13.4s

Precision: binary64

Cost: 39424

Math

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

↓

\[\frac{\left(1 + \left(x - x\right)\right) \cdot {\left({\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-0.5}\right)}^{2}}{\sqrt{x} + \sqrt{1 + x}} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (/
  (* (+ 1.0 (- x x)) (pow (pow (hypot x (sqrt x)) -0.5) 2.0))
  (+ (sqrt x) (sqrt (+ 1.0 x)))))

C

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

↓

double code(double x) {
	return ((1.0 + (x - x)) * pow(pow(hypot(x, sqrt(x)), -0.5), 2.0)) / (sqrt(x) + sqrt((1.0 + x)));
}

Java

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

↓

public static double code(double x) {
	return ((1.0 + (x - x)) * Math.pow(Math.pow(Math.hypot(x, Math.sqrt(x)), -0.5), 2.0)) / (Math.sqrt(x) + Math.sqrt((1.0 + x)));
}

Python

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

↓

def code(x):
	return ((1.0 + (x - x)) * math.pow(math.pow(math.hypot(x, math.sqrt(x)), -0.5), 2.0)) / (math.sqrt(x) + math.sqrt((1.0 + x)))

Julia

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

↓

function code(x)
	return Float64(Float64(Float64(1.0 + Float64(x - x)) * ((hypot(x, sqrt(x)) ^ -0.5) ^ 2.0)) / Float64(sqrt(x) + sqrt(Float64(1.0 + x))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = ((1.0 + (x - x)) * ((hypot(x, sqrt(x)) ^ -0.5) ^ 2.0)) / (sqrt(x) + sqrt((1.0 + x)));
end

Wolfram

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

↓

code[x_] := N[(N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Sqrt[x ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision], -0.5], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	68.9%
Target	99.0%
Herbie	99.3%

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

Derivation

1. Initial program 68.9%

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

2. Applied egg-rr 91.1%

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

   [Start] 68.9	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   frac-sub [=>] 68.9	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
   div-inv [=>] 68.9	\[ \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
   *-un-lft-identity [<=] 68.9	\[ \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
   *-rgt-identity [=>] 68.9	\[ \left(\sqrt{x + 1} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
   flip-- [=>] 69.3	\[ \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
   metadata-eval [<=] 69.3	\[ \frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]
   frac-times [<=] 69.3	\[ \frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]
   associate-*l/ [=>] 69.3	\[ \color{blue}{\frac{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)}{\sqrt{x + 1} + \sqrt{x}}} \]

3. Applied egg-rr 99.3%

   \[\leadsto \frac{\left(1 + \left(x - x\right)\right) \cdot \color{blue}{{\left({\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-0.5}\right)}^{2}}}{\sqrt{x} + \sqrt{1 + x}} \]
   Proof

   [Start] 91.1	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
   metadata-eval [<=] 91.1	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{\color{blue}{1 + 0}}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
   +-inverses [<=] 91.1	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1 + \color{blue}{\left(x - x\right)}}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
   un-div-inv [<=] 91.1	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \color{blue}{\left(\left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{x + x \cdot x}}\right)}}{\sqrt{x} + \sqrt{1 + x}} \]
   add-sqr-sqrt [=>] 90.7	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \color{blue}{\left(\sqrt{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{x + x \cdot x}}} \cdot \sqrt{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{x + x \cdot x}}}\right)}}{\sqrt{x} + \sqrt{1 + x}} \]
   pow2 [=>] 90.7	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \color{blue}{{\left(\sqrt{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{x + x \cdot x}}}\right)}^{2}}}{\sqrt{x} + \sqrt{1 + x}} \]
   un-div-inv [=>] 90.7	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot {\left(\sqrt{\color{blue}{\frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x}}}}\right)}^{2}}{\sqrt{x} + \sqrt{1 + x}} \]
   +-inverses [=>] 90.7	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot {\left(\sqrt{\frac{1 + \color{blue}{0}}{\sqrt{x + x \cdot x}}}\right)}^{2}}{\sqrt{x} + \sqrt{1 + x}} \]
   metadata-eval [=>] 90.7	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot {\left(\sqrt{\frac{\color{blue}{1}}{\sqrt{x + x \cdot x}}}\right)}^{2}}{\sqrt{x} + \sqrt{1 + x}} \]
   inv-pow [=>] 90.7	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot {\left(\sqrt{\color{blue}{{\left(\sqrt{x + x \cdot x}\right)}^{-1}}}\right)}^{2}}{\sqrt{x} + \sqrt{1 + x}} \]
   metadata-eval [<=] 90.7	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot {\left(\sqrt{{\left(\sqrt{x + x \cdot x}\right)}^{\color{blue}{\left(-1\right)}}}\right)}^{2}}{\sqrt{x} + \sqrt{1 + x}} \]
   sqrt-pow1 [=>] 90.8	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot {\color{blue}{\left({\left(\sqrt{x + x \cdot x}\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{2}}{\sqrt{x} + \sqrt{1 + x}} \]
   +-commutative [=>] 90.8	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot {\left({\left(\sqrt{\color{blue}{x \cdot x + x}}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}}{\sqrt{x} + \sqrt{1 + x}} \]
   add-sqr-sqrt [=>] 90.8	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot {\left({\left(\sqrt{x \cdot x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2}}{\sqrt{x} + \sqrt{1 + x}} \]
   hypot-def [=>] 99.3	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot {\left({\color{blue}{\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{2}}{\sqrt{x} + \sqrt{1 + x}} \]
   metadata-eval [=>] 99.3	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot {\left({\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)}\right)}^{2}}{\sqrt{x} + \sqrt{1 + x}} \]
   metadata-eval [=>] 99.3	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot {\left({\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{\color{blue}{-0.5}}\right)}^{2}}{\sqrt{x} + \sqrt{1 + x}} \]

4. Final simplification 99.3%

   \[\leadsto \frac{\left(1 + \left(x - x\right)\right) \cdot {\left({\left(\mathsf{hypot}\left(x, \sqrt{x}\right)\right)}^{-0.5}\right)}^{2}}{\sqrt{x} + \sqrt{1 + x}} \]

Alternatives

Alternative 1
Accuracy	91.5%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\frac{-1}{1 + x}}{-x}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Accuracy	99.3%
Cost	13888

\[\frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}{-1 - x} \cdot \frac{-1}{x} \]

Alternative 3
Accuracy	98.7%
Cost	13760

\[\frac{1}{\left(1 + x\right) \cdot \left(x \cdot \left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)\right)} \]

Alternative 4
Accuracy	98.7%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;x \leq 106000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}\\ \end{array} \]

Alternative 5
Accuracy	90.7%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{1 + x}}{-x}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]

Alternative 6
Accuracy	89.9%
Cost	7364

\[\begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + x \cdot x}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]

Alternative 7
Accuracy	90.7%
Cost	7364

\[\begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - x}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]

Alternative 8
Accuracy	68.3%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.5}{-1 - x} + \frac{0.5}{x}\right)\\ \end{array} \]

Alternative 9
Accuracy	89.9%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq 1.68:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot x}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]

Alternative 10
Accuracy	68.3%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 5.8:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x - x\right)\right) \cdot \left(\frac{1}{0.25 - x \cdot x} \cdot \left(0.5 - x\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	68.2%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 1.35:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x - x\right)\right) \cdot \left(\frac{1}{0.25 - x \cdot x} \cdot \left(0.5 - x\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	67.9%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.59:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x - x\right)\right) \cdot \left(\frac{1}{0.25 - x \cdot x} \cdot \left(0.5 - x\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	66.4%
Cost	6660

\[\begin{array}{l} \mathbf{if}\;x \leq 0.092:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x - x\right)\right) \cdot \left(\frac{1}{0.25 - x \cdot x} \cdot \left(0.5 - x\right)\right)\\ \end{array} \]

Alternative 14
Accuracy	22.5%
Cost	1088

\[\left(1 + \left(x - x\right)\right) \cdot \left(\frac{1}{0.25 - x \cdot x} \cdot \left(0.5 - x\right)\right) \]

Alternative 15
Accuracy	7.4%
Cost	704

\[\left(1 + \left(x - x\right)\right) \cdot \frac{1}{x + 0.5} \]

Alternative 16
Accuracy	7.4%
Cost	192

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

Alternative 17
Accuracy	5.8%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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