2isqrt (example 3.6)

Average Error: 19.6 → 0.2

Time: 12.0s

Precision: binary64

Cost: 14276

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 5500:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{x + \left(0.5 + \frac{-0.125}{x}\right)}}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x 5500.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (/
    (* (+ 1.0 (- x x)) (/ 1.0 (+ x (+ 0.5 (/ -0.125 x)))))
    (+ (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) {
	double tmp;
	if (x <= 5500.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = ((1.0 + (x - x)) * (1.0 / (x + (0.5 + (-0.125 / x))))) / (sqrt(x) + sqrt((1.0 + x)));
	}
	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 (x <= 5500.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = ((1.0d0 + (x - x)) * (1.0d0 / (x + (0.5d0 + ((-0.125d0) / x))))) / (sqrt(x) + sqrt((1.0d0 + x)))
    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 (x <= 5500.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = ((1.0 + (x - x)) * (1.0 / (x + (0.5 + (-0.125 / x))))) / (Math.sqrt(x) + Math.sqrt((1.0 + x)));
	}
	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 x <= 5500.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = ((1.0 + (x - x)) * (1.0 / (x + (0.5 + (-0.125 / x))))) / (math.sqrt(x) + math.sqrt((1.0 + x)))
	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 (x <= 5500.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x - x)) * Float64(1.0 / Float64(x + Float64(0.5 + Float64(-0.125 / x))))) / Float64(sqrt(x) + sqrt(Float64(1.0 + x))));
	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 (x <= 5500.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = ((1.0 + (x - x)) * (1.0 / (x + (0.5 + (-0.125 / x))))) / (sqrt(x) + sqrt((1.0 + x)));
	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[x, 5500.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + N[(0.5 + N[(-0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	19.6
Target	0.7
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if x < 5500

   1. Initial program 0.3

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

   2. Applied egg-rr 0.1

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

   3. Simplified 0.1

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

      [Start] 0.1	\[ {x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right) \]
      sub-neg [<=] 0.1	\[ \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

   if 5500 < x

   1. Initial program 38.8

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

   2. Applied egg-rr 10.8

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

   3. Taylor expanded in x around inf 0.3

      \[\leadsto \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{\color{blue}{\left(0.5 + x\right) - 0.125 \cdot \frac{1}{x}}}}{\sqrt{x} + \sqrt{1 + x}} \]

   4. Simplified 0.3

      \[\leadsto \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{\color{blue}{x + \left(0.5 - \frac{0.125}{x}\right)}}}{\sqrt{x} + \sqrt{1 + x}} \]
      Proof

      [Start] 0.3	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{\left(0.5 + x\right) - 0.125 \cdot \frac{1}{x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      +-commutative [=>] 0.3	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{\color{blue}{\left(x + 0.5\right)} - 0.125 \cdot \frac{1}{x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      associate--l+ [=>] 0.3	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{\color{blue}{x + \left(0.5 - 0.125 \cdot \frac{1}{x}\right)}}}{\sqrt{x} + \sqrt{1 + x}} \]
      associate-*r/ [=>] 0.3	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{x + \left(0.5 - \color{blue}{\frac{0.125 \cdot 1}{x}}\right)}}{\sqrt{x} + \sqrt{1 + x}} \]
      metadata-eval [=>] 0.3	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{x + \left(0.5 - \frac{\color{blue}{0.125}}{x}\right)}}{\sqrt{x} + \sqrt{1 + x}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5500:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{x + \left(0.5 + \frac{-0.125}{x}\right)}}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	27332

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{x + 0.5}}{\sqrt{x} + t_0}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	27204

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{x}}{\sqrt{x} + t_0}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{t_0}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	26240

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

Alternative 4
Error	0.7
Cost	26240

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

Alternative 5
Error	0.5
Cost	13636

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;x \leq 85000000:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + t_0} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 6
Error	9.8
Cost	13448

\[\begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{x + 0.5}}{1 + \sqrt{x}}\\ \end{array} \]

Alternative 7
Error	9.3
Cost	13448

\[\begin{array}{l} \mathbf{if}\;x \leq 120000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{x + 0.5}}{1 + \sqrt{x}}\\ \end{array} \]

Alternative 8
Error	9.3
Cost	13448

\[\begin{array}{l} \mathbf{if}\;x \leq 85000000:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{\sqrt{1 + x}}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{x + 0.5}}{1 + \sqrt{x}}\\ \end{array} \]

Alternative 9
Error	17.8
Cost	7492

\[\begin{array}{l} \mathbf{if}\;x \leq 0.49:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{x + 0.5}}{1 + \sqrt{x}}\\ \end{array} \]

Alternative 10
Error	19.9
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \sqrt{\frac{1}{x}}\right) \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 11
Error	19.9
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{x}}{x}}{1 + \sqrt{\frac{1}{x}}}\\ \end{array} \]

Alternative 12
Error	19.9
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 13
Error	20.0
Cost	7108

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

Alternative 14
Error	20.1
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x - x\right)\right) \cdot \left(\frac{1}{0.25 - x \cdot x} \cdot \left(0.5 - x\right)\right)\\ \end{array} \]

Alternative 15
Error	21.0
Cost	6660

\[\begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(x - x\right)\right) \cdot \left(\frac{1}{0.25 - x \cdot x} \cdot \left(0.5 - x\right)\right)\\ \end{array} \]

Alternative 16
Error	49.1
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 17
Error	59.2
Cost	704

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

Alternative 18
Error	59.3
Cost	576

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

Alternative 19
Error	62.8
Cost	64

\[-1 \]

Alternative 20
Error	60.3
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022349 
(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)))))