2isqrt (example 3.6)

Percentage Accurate: 69.5% → 99.8%

Time: 10.0s

Precision: binary64

Cost: 19776

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 Math.pow(x, -0.5) / (Math.hypot(x, Math.sqrt(x)) + (x + 1.0));
}

Python

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

↓

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

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((x ^ -0.5) / Float64(hypot(x, sqrt(x)) + Float64(x + 1.0)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (x ^ -0.5) / (hypot(x, sqrt(x)) + (x + 1.0));
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[Power[x, -0.5], $MachinePrecision] / N[(N[Sqrt[x ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	69.5%
Target	99.0%
Herbie	99.8%

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

Derivation

1. Initial program 70.9%

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

2. Applied egg-rr 71.2%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}}} \]
   Step-by-step derivation

   [Start] 70.9%	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   frac-sub [=>] 71.0%	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
   div-inv [=>] 71.0%	\[ \color{blue}{\left(1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
   *-un-lft-identity [<=] 71.0%	\[ \left(\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
   +-commutative [=>] 71.0%	\[ \left(\sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot 1\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
   *-rgt-identity [=>] 71.0%	\[ \left(\sqrt{1 + x} - \color{blue}{\sqrt{x}}\right) \cdot \frac{1}{\sqrt{x} \cdot \sqrt{x + 1}} \]
   metadata-eval [<=] 71.0%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\sqrt{x} \cdot \sqrt{x + 1}} \]
   frac-times [<=] 71.0%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{x}} \cdot \frac{1}{\sqrt{x + 1}}\right)} \]
   un-div-inv [=>] 71.0%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \color{blue}{\frac{\frac{1}{\sqrt{x}}}{\sqrt{x + 1}}} \]
   pow1/2 [=>] 71.0%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\frac{1}{\color{blue}{{x}^{0.5}}}}{\sqrt{x + 1}} \]
   pow-flip [=>] 71.2%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{\color{blue}{{x}^{\left(-0.5\right)}}}{\sqrt{x + 1}} \]
   metadata-eval [=>] 71.2%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{x + 1}} \]
   +-commutative [=>] 71.2%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{\color{blue}{1 + x}}} \]

3. Simplified 71.2%

   \[\leadsto \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}}} \]
   Step-by-step derivation

   [Start] 71.2%	\[ \left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}} \]
   associate-*r/ [=>] 71.2%	\[ \color{blue}{\frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}}} \]

4. Applied egg-rr 72.0%

   \[\leadsto \frac{\color{blue}{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
   Step-by-step derivation

   [Start] 71.2%	\[ \frac{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
   flip-- [=>] 71.3%	\[ \frac{\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
   add-sqr-sqrt [<=] 61.2%	\[ \frac{\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
   add-sqr-sqrt [<=] 72.0%	\[ \frac{\frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

5. Simplified 99.7%

   \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
   Step-by-step derivation

   [Start] 72.0%	\[ \frac{\frac{\left(1 + x\right) - x}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
   associate--l+ [=>] 99.7%	\[ \frac{\frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
   +-inverses [=>] 99.7%	\[ \frac{\frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
   metadata-eval [=>] 99.7%	\[ \frac{\frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]

6. Applied egg-rr 67.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-0.5}}{\sqrt{1 + x} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)} - 1} \]
   Step-by-step derivation

   [Start] 99.7%	\[ \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}} \]
   expm1-log1p-u [=>] 96.0%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}}\right)\right)} \]
   expm1-udef [=>] 66.9%	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}} \cdot {x}^{-0.5}}{\sqrt{1 + x}}\right)} - 1} \]
   associate-*l/ [=>] 66.9%	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{1 \cdot {x}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}}}{\sqrt{1 + x}}\right)} - 1 \]
   *-un-lft-identity [<=] 66.9%	\[ e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{{x}^{-0.5}}}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{1 + x}}\right)} - 1 \]
   associate-/l/ [=>] 67.0%	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{{x}^{-0.5}}{\sqrt{1 + x} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}}\right)} - 1 \]

7. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}} \]
   Step-by-step derivation

   [Start] 67.0%	\[ e^{\mathsf{log1p}\left(\frac{{x}^{-0.5}}{\sqrt{1 + x} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)} - 1 \]
   expm1-def [=>] 96.0%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-0.5}}{\sqrt{1 + x} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)\right)} \]
   expm1-log1p [=>] 99.7%	\[ \color{blue}{\frac{{x}^{-0.5}}{\sqrt{1 + x} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}} \]
   distribute-rgt-in [=>] 99.7%	\[ \frac{{x}^{-0.5}}{\color{blue}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \sqrt{x} \cdot \sqrt{1 + x}}} \]
   rem-square-sqrt [=>] 99.8%	\[ \frac{{x}^{-0.5}}{\color{blue}{\left(1 + x\right)} + \sqrt{x} \cdot \sqrt{1 + x}} \]
   +-commutative [=>] 99.8%	\[ \frac{{x}^{-0.5}}{\color{blue}{\left(x + 1\right)} + \sqrt{x} \cdot \sqrt{1 + x}} \]
   +-commutative [=>] 99.8%	\[ \frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{\color{blue}{x + 1}}} \]

8. Applied egg-rr 67.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-0.5}}{x + \left(1 + \mathsf{hypot}\left(x, \sqrt{x}\right)\right)}\right)} - 1} \]
   Step-by-step derivation

   [Start] 99.8%	\[ \frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}} \]
   expm1-log1p-u [=>] 96.1%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}\right)\right)} \]
   expm1-udef [=>] 67.0%	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{-0.5}}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}}\right)} - 1} \]
   sqrt-prod [<=] 67.0%	\[ e^{\mathsf{log1p}\left(\frac{{x}^{-0.5}}{\left(x + 1\right) + \color{blue}{\sqrt{x \cdot \left(x + 1\right)}}}\right)} - 1 \]
   associate-+l+ [=>] 67.0%	\[ e^{\mathsf{log1p}\left(\frac{{x}^{-0.5}}{\color{blue}{x + \left(1 + \sqrt{x \cdot \left(x + 1\right)}\right)}}\right)} - 1 \]
   distribute-rgt-in [=>] 67.0%	\[ e^{\mathsf{log1p}\left(\frac{{x}^{-0.5}}{x + \left(1 + \sqrt{\color{blue}{x \cdot x + 1 \cdot x}}\right)}\right)} - 1 \]
   *-un-lft-identity [<=] 67.0%	\[ e^{\mathsf{log1p}\left(\frac{{x}^{-0.5}}{x + \left(1 + \sqrt{x \cdot x + \color{blue}{x}}\right)}\right)} - 1 \]
   add-sqr-sqrt [=>] 67.0%	\[ e^{\mathsf{log1p}\left(\frac{{x}^{-0.5}}{x + \left(1 + \sqrt{x \cdot x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}\right)} - 1 \]
   hypot-def [=>] 67.0%	\[ e^{\mathsf{log1p}\left(\frac{{x}^{-0.5}}{x + \left(1 + \color{blue}{\mathsf{hypot}\left(x, \sqrt{x}\right)}\right)}\right)} - 1 \]

9. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{\mathsf{hypot}\left(x, \sqrt{x}\right) + \left(x + 1\right)}} \]
   Step-by-step derivation

   [Start] 67.0%	\[ e^{\mathsf{log1p}\left(\frac{{x}^{-0.5}}{x + \left(1 + \mathsf{hypot}\left(x, \sqrt{x}\right)\right)}\right)} - 1 \]
   expm1-def [=>] 96.1%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-0.5}}{x + \left(1 + \mathsf{hypot}\left(x, \sqrt{x}\right)\right)}\right)\right)} \]
   expm1-log1p [=>] 99.9%	\[ \color{blue}{\frac{{x}^{-0.5}}{x + \left(1 + \mathsf{hypot}\left(x, \sqrt{x}\right)\right)}} \]
   +-commutative [=>] 99.9%	\[ \frac{{x}^{-0.5}}{\color{blue}{\left(1 + \mathsf{hypot}\left(x, \sqrt{x}\right)\right) + x}} \]
   +-commutative [=>] 99.9%	\[ \frac{{x}^{-0.5}}{\color{blue}{\left(\mathsf{hypot}\left(x, \sqrt{x}\right) + 1\right)} + x} \]
   associate-+r+ [<=] 99.9%	\[ \frac{{x}^{-0.5}}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{x}\right) + \left(1 + x\right)}} \]
   +-commutative [<=] 99.9%	\[ \frac{{x}^{-0.5}}{\mathsf{hypot}\left(x, \sqrt{x}\right) + \color{blue}{\left(x + 1\right)}} \]

10. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	19776

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

Alternative 2
Accuracy	99.8%
Cost	26756

\[\begin{array}{l} t_0 := \frac{-1}{\sqrt{x + 1}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 10^{-7}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \left(\frac{0.0625}{x \cdot x} + \left(x + \left(0.5 + \frac{-0.125}{x}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_0\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 820:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \left(\frac{0.0625}{x \cdot x} + \left(x + \left(0.5 + \frac{-0.125}{x}\right)\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	99.2%
Cost	7812

\[\begin{array}{l} \mathbf{if}\;x \leq 0.44:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{\left(x + 1\right) + \left(\frac{0.0625}{x \cdot x} + \left(x + \left(0.5 + \frac{-0.125}{x}\right)\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	99.2%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;x \leq 0.43:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}\\ \end{array} \]

Alternative 6
Accuracy	98.5%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 0.72:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \]

Alternative 7
Accuracy	99.0%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 0.41:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1.5 + x \cdot 2}\\ \end{array} \]

Alternative 8
Accuracy	98.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \]

Alternative 9
Accuracy	98.2%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x \cdot 2}\\ \end{array} \]

Alternative 10
Accuracy	68.6%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{1 + x \cdot 0.5}\\ \end{array} \]

Alternative 11
Accuracy	67.2%
Cost	6660

\[\begin{array}{l} \mathbf{if}\;x \leq 0.205:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{1 + x \cdot 0.5}\\ \end{array} \]

Alternative 12
Accuracy	22.3%
Cost	576

\[\frac{\frac{0.5}{x}}{1 + x \cdot 0.5} \]

Alternative 13
Accuracy	7.4%
Cost	192

\[\frac{0.5}{x} \]

Reproduce

herbie shell --seed 2023165 
(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)))))