2isqrt (example 3.6)

Average Error: 69.8% → 99.7%

Time: 11.3s

Precision: binary64

Cost: 27076.00

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{\frac{1}{x}} \cdot 1.5 + \sqrt{x} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-10)
   (/ (/ 1.0 x) (+ (* (sqrt (/ 1.0 x)) 1.5) (* (sqrt x) 2.0)))
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

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

↓

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-10) {
		tmp = (1.0 / x) / ((sqrt((1.0 / x)) * 1.5) + (sqrt(x) * 2.0));
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 2d-10) then
        tmp = (1.0d0 / x) / ((sqrt((1.0d0 / x)) * 1.5d0) + (sqrt(x) * 2.0d0))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

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) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 2e-10) {
		tmp = (1.0 / x) / ((Math.sqrt((1.0 / x)) * 1.5) + (Math.sqrt(x) * 2.0));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 2e-10:
		tmp = (1.0 / x) / ((math.sqrt((1.0 / x)) * 1.5) + (math.sqrt(x) * 2.0))
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-10)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(sqrt(Float64(1.0 / x)) * 1.5) + Float64(sqrt(x) * 2.0)));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-10)
		tmp = (1.0 / x) / ((sqrt((1.0 / x)) * 1.5) + (sqrt(x) * 2.0));
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
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_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-10], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 1.5), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Target

Original	69.8%
Target	99.0%
Herbie	99.7%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 2.00000000000000007e-10

   1. Initial program 37.6

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

   2. Applied egg-rr 37.7

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

   3. Applied egg-rr 83.3

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

   4. Simplified 99.5

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

      [Start] 83.3	\[ \frac{1}{\frac{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \left(x + x \cdot x\right)}{1 + \left(x - x\right)}} \]
      associate-*r/ [<=] 83.3	\[ \frac{1}{\color{blue}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \frac{x + x \cdot x}{1 + \left(x - x\right)}}} \]
      associate-/r* [=>] 83.3	\[ \color{blue}{\frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}{\frac{x + x \cdot x}{1 + \left(x - x\right)}}} \]
      associate-/l* [<=] 83.3	\[ \color{blue}{\frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \cdot \left(1 + \left(x - x\right)\right)}{x + x \cdot x}} \]
      *-commutative [<=] 83.3	\[ \frac{\color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}}{x + x \cdot x} \]
      associate-/l* [=>] 83.2	\[ \color{blue}{\frac{1 + \left(x - x\right)}{\frac{x + x \cdot x}{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}}} \]
      associate-/r/ [=>] 83.2	\[ \color{blue}{\frac{1 + \left(x - x\right)}{x + x \cdot x} \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]
      times-frac [<=] 83.3	\[ \color{blue}{\frac{\left(1 + \left(x - x\right)\right) \cdot 1}{\left(x + x \cdot x\right) \cdot \left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)}} \]
      associate-/r* [=>] 83.3	\[ \color{blue}{\frac{\frac{\left(1 + \left(x - x\right)\right) \cdot 1}{x + x \cdot x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]

   5. Taylor expanded in x around inf 99.6

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

   6. Simplified 99.6

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\sqrt{\frac{1}{x}} \cdot 1.5 + 2 \cdot \sqrt{x}}} \]
      Proof

      [Start] 99.6	\[ \frac{\frac{1}{x}}{-0.5 \cdot \sqrt{\frac{1}{x}} + \left(2 \cdot \sqrt{x} + 2 \cdot \sqrt{\frac{1}{x}}\right)} \]
      +-commutative [=>] 99.6	\[ \frac{\frac{1}{x}}{-0.5 \cdot \sqrt{\frac{1}{x}} + \color{blue}{\left(2 \cdot \sqrt{\frac{1}{x}} + 2 \cdot \sqrt{x}\right)}} \]
      associate-+r+ [=>] 99.6	\[ \frac{\frac{1}{x}}{\color{blue}{\left(-0.5 \cdot \sqrt{\frac{1}{x}} + 2 \cdot \sqrt{\frac{1}{x}}\right) + 2 \cdot \sqrt{x}}} \]
      distribute-rgt-out [=>] 99.6	\[ \frac{\frac{1}{x}}{\color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-0.5 + 2\right)} + 2 \cdot \sqrt{x}} \]
      metadata-eval [=>] 99.6	\[ \frac{\frac{1}{x}}{\sqrt{\frac{1}{x}} \cdot \color{blue}{1.5} + 2 \cdot \sqrt{x}} \]

   if 2.00000000000000007e-10 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.3

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

   2. Applied egg-rr 99.7

      \[\leadsto \color{blue}{{x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)} \]

   3. Simplified 99.7

      \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]
      Proof

      [Start] 99.7	\[ {x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right) \]
      sub-neg [<=] 99.7	\[ \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{\frac{1}{x}} \cdot 1.5 + \sqrt{x} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternatives

Alternative 1
Error	99.3%
Cost	26692.00

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-12}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + x\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Error	99.4%
Cost	13696.00

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

Alternative 3
Error	98.5%
Cost	7364.00

\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + \left(-1 + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + x\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot 2\right)}\\ \end{array} \]

Alternative 4
Error	98.5%
Cost	7044.00

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

Alternative 5
Error	98.2%
Cost	6916.00

\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{{x}^{-0.5}}{x}\\ \end{array} \]

Alternative 6
Error	98.2%
Cost	6852.00

\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\sqrt{x}}}{x}\\ \end{array} \]

Alternative 7
Error	54.3%
Cost	6788.00

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

Alternative 8
Error	97.6%
Cost	6788.00

\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{{x}^{1.5}}\\ \end{array} \]

Alternative 9
Error	51.8%
Cost	6528.00

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

Alternative 10
Error	7.4%
Cost	192.00

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

Reproduce

herbie shell --seed 2023097 
(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)))))