2isqrt (example 3.6)

Average Accuracy: 69.4% → 99.6%

Time: 13.0s

Precision: binary64

Cost: 26240

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

def code(x):
	return (1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) / math.hypot(x, math.sqrt(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(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) / hypot(x, sqrt(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 / (sqrt((1.0 + x)) + sqrt(x))) / hypot(x, sqrt(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[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Target

Original	69.4%
Target	99.0%
Herbie	99.6%

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

Derivation

1. Initial program 69.4%

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

2. Applied egg-rr 69.4%

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

   [Start] 69.4	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   frac-sub [=>] 69.4	\[ \color{blue}{\frac{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}{\sqrt{x} \cdot \sqrt{x + 1}}} \]
   clear-num [=>] 69.4	\[ \color{blue}{\frac{1}{\frac{\sqrt{x} \cdot \sqrt{x + 1}}{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}}} \]
   sqrt-unprod [=>] 69.4	\[ \frac{1}{\frac{\color{blue}{\sqrt{x \cdot \left(x + 1\right)}}}{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}} \]
   +-commutative [=>] 69.4	\[ \frac{1}{\frac{\sqrt{x \cdot \color{blue}{\left(1 + x\right)}}}{1 \cdot \sqrt{x + 1} - \sqrt{x} \cdot 1}} \]
   *-un-lft-identity [<=] 69.4	\[ \frac{1}{\frac{\sqrt{x \cdot \left(1 + x\right)}}{\color{blue}{\sqrt{x + 1}} - \sqrt{x} \cdot 1}} \]
   *-rgt-identity [=>] 69.4	\[ \frac{1}{\frac{\sqrt{x \cdot \left(1 + x\right)}}{\sqrt{x + 1} - \color{blue}{\sqrt{x}}}} \]
   +-commutative [=>] 69.4	\[ \frac{1}{\frac{\sqrt{x \cdot \left(1 + x\right)}}{\sqrt{\color{blue}{1 + x}} - \sqrt{x}}} \]

3. Simplified 69.4%

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

   [Start] 69.4	\[ \frac{1}{\frac{\sqrt{x \cdot \left(1 + x\right)}}{\sqrt{1 + x} - \sqrt{x}}} \]
   associate-/r/ [=>] 69.4	\[ \color{blue}{\frac{1}{\sqrt{x \cdot \left(1 + x\right)}} \cdot \left(\sqrt{1 + x} - \sqrt{x}\right)} \]
   associate-*l/ [=>] 69.4	\[ \color{blue}{\frac{1 \cdot \left(\sqrt{1 + x} - \sqrt{x}\right)}{\sqrt{x \cdot \left(1 + x\right)}}} \]
   *-lft-identity [=>] 69.4	\[ \frac{\color{blue}{\sqrt{1 + x} - \sqrt{x}}}{\sqrt{x \cdot \left(1 + x\right)}} \]
   +-commutative [=>] 69.4	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot \color{blue}{\left(x + 1\right)}}} \]
   distribute-lft-in [=>] 69.4	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{\color{blue}{x \cdot x + x \cdot 1}}} \]
   *-rgt-identity [=>] 69.4	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot x + \color{blue}{x}}} \]
   unpow1 [<=] 69.4	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot x + \color{blue}{{x}^{1}}}} \]
   sqr-pow [=>] 69.4	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot x + \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}} \]
   metadata-eval [=>] 69.4	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot x + {x}^{\color{blue}{0.5}} \cdot {x}^{\left(\frac{1}{2}\right)}}} \]
   exp-to-pow [<=] 66.6	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot x + \color{blue}{e^{\log x \cdot 0.5}} \cdot {x}^{\left(\frac{1}{2}\right)}}} \]
   metadata-eval [=>] 66.6	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot x + e^{\log x \cdot 0.5} \cdot {x}^{\color{blue}{0.5}}}} \]
   exp-to-pow [<=] 65.9	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x \cdot x + e^{\log x \cdot 0.5} \cdot \color{blue}{e^{\log x \cdot 0.5}}}} \]
   hypot-def [=>] 65.9	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\color{blue}{\mathsf{hypot}\left(x, e^{\log x \cdot 0.5}\right)}} \]
   exp-to-pow [=>] 69.4	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\mathsf{hypot}\left(x, \color{blue}{{x}^{0.5}}\right)} \]
   unpow1/2 [=>] 69.4	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{x}}\right)} \]

4. Applied egg-rr 99.6%

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

   [Start] 69.4	\[ \frac{\sqrt{1 + x} - \sqrt{x}}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \]
   flip-- [=>] 69.7	\[ \frac{\color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}}}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \]
   div-inv [=>] 69.7	\[ \frac{\color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \]
   add-sqr-sqrt [<=] 61.8	\[ \frac{\left(\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \]
   add-sqr-sqrt [<=] 70.4	\[ \frac{\left(\left(1 + x\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \]
   associate--l+ [=>] 99.6	\[ \frac{\color{blue}{\left(1 + \left(x - x\right)\right)} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \]

5. Simplified 99.6%

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

   [Start] 99.6	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \]
   +-inverses [=>] 99.6	\[ \frac{\left(1 + \color{blue}{0}\right) \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \]
   metadata-eval [=>] 99.6	\[ \frac{\color{blue}{1} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \]
   *-lft-identity [=>] 99.6	\[ \frac{\color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \]

6. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	27460

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 0:\\ \;\;\;\;\frac{\frac{1}{t_0 + \sqrt{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1 + \left(x - x\right)}{x}}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	26820

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 10^{-19}:\\ \;\;\;\;\frac{\frac{1}{t_0 + \sqrt{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 3
Accuracy	83.0%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-19}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 4
Accuracy	99.0%
Cost	26240

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

Alternative 5
Accuracy	82.4%
Cost	13316

\[\begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 6
Accuracy	67.5%
Cost	7296

\[\frac{\frac{1}{x} - \frac{1}{1 + x}}{1 + {x}^{-0.5}} \]

Alternative 7
Accuracy	51.2%
Cost	6912

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

Alternative 8
Accuracy	52.5%
Cost	6912

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

Alternative 9
Accuracy	52.5%
Cost	6912

\[\frac{\frac{1}{x}}{1 + {x}^{-0.5}} \]

Alternative 10
Accuracy	52.1%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5}\\ \end{array} \]

Alternative 11
Accuracy	50.7%
Cost	6528

\[{x}^{-0.5} \]

Alternative 12
Accuracy	1.9%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023147 
(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)))))