2isqrt (example 3.6)

Average Error: 20.2 → 0.1

Time: 7.1s

Precision: binary64

Cost: 26692

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x} \cdot \left(\frac{-0.375}{x} + \left(0.5 + \frac{0.3125}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

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)))) 5e-7)
   (* (/ (pow x -0.5) x) (+ (/ -0.375 x) (+ 0.5 (/ 0.3125 (* x x)))))
   (- (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)))) <= 5e-7) {
		tmp = (pow(x, -0.5) / x) * ((-0.375 / x) + (0.5 + (0.3125 / (x * x))));
	} 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)))) <= 5d-7) then
        tmp = ((x ** (-0.5d0)) / x) * (((-0.375d0) / x) + (0.5d0 + (0.3125d0 / (x * x))))
    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)))) <= 5e-7) {
		tmp = (Math.pow(x, -0.5) / x) * ((-0.375 / x) + (0.5 + (0.3125 / (x * x))));
	} 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)))) <= 5e-7:
		tmp = (math.pow(x, -0.5) / x) * ((-0.375 / x) + (0.5 + (0.3125 / (x * x))))
	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)))) <= 5e-7)
		tmp = Float64(Float64((x ^ -0.5) / x) * Float64(Float64(-0.375 / x) + Float64(0.5 + Float64(0.3125 / Float64(x * x)))));
	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)))) <= 5e-7)
		tmp = ((x ^ -0.5) / x) * ((-0.375 / x) + (0.5 + (0.3125 / (x * x))));
	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], 5e-7], N[(N[(N[Power[x, -0.5], $MachinePrecision] / x), $MachinePrecision] * N[(N[(-0.375 / x), $MachinePrecision] + N[(0.5 + N[(0.3125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Target

Original	20.2
Target	0.7
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 4.99999999999999977e-7

   1. Initial program 40.2

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

   2. Applied egg-rr 40.2

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

   3. Taylor expanded in x around -inf 64.0

      \[\leadsto \color{blue}{0.3125 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{3}} + \left(0.5 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{x} + -0.375 \cdot \frac{e^{-0.5 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log -1\right)}}{{x}^{2}}\right)} \]

   4. Simplified 0.2

      \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{x} \cdot \left(\frac{-0.375}{x} + \left(0.5 + \frac{0.3125}{x \cdot x}\right)\right)} \]

   if 4.99999999999999977e-7 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 0.3

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

   2. Applied egg-rr 0.6

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

   3. Applied egg-rr 0.1

      \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x} \cdot \left(\frac{-0.375}{x} + \left(0.5 + \frac{0.3125}{x \cdot x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	7556

\[\begin{array}{l} \mathbf{if}\;x \leq 0.001605866299076687:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x} \cdot \left(\frac{-0.375}{x} + \left(0.5 + \frac{0.3125}{x \cdot x}\right)\right)\\ \end{array} \]

Alternative 2
Error	1.0
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 0.001605866299076687:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 3
Error	0.8
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 0.001605866299076687:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x} \cdot \left(\frac{-0.375}{x} + 0.5\right)\\ \end{array} \]

Alternative 4
Error	1.0
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 0.001605866299076687:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 5
Error	1.7
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.001605866299076687:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{{x}^{1.5}}\\ \end{array} \]

Alternative 6
Error	1.2
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.001605866299076687:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 7
Error	31.9
Cost	6528

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

Alternative 8
Error	62.5
Cost	320

\[x \cdot 0.5 + -1 \]

Alternative 9
Error	62.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022217 
(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)))))