2isqrt (example 3.6)

Average Error: 19.3 → 0.2

Time: 9.7s

Precision: binary64

Cost: 13764

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 85000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{\frac{1}{x}} \cdot 1.5 + \sqrt{x} \cdot 2}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x 85000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (/ (/ 1.0 x) (+ (* (sqrt (/ 1.0 x)) 1.5) (* (sqrt x) 2.0)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= 85000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = (1.0 / x) / ((sqrt((1.0 / x)) * 1.5) + (sqrt(x) * 2.0));
	}
	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 <= 85000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = (1.0d0 / x) / ((sqrt((1.0d0 / x)) * 1.5d0) + (sqrt(x) * 2.0d0))
    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 <= 85000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = (1.0 / x) / ((Math.sqrt((1.0 / x)) * 1.5) + (Math.sqrt(x) * 2.0));
	}
	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 <= 85000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = (1.0 / x) / ((math.sqrt((1.0 / x)) * 1.5) + (math.sqrt(x) * 2.0))
	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 <= 85000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(sqrt(Float64(1.0 / x)) * 1.5) + Float64(sqrt(x) * 2.0)));
	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 <= 85000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = (1.0 / x) / ((sqrt((1.0 / x)) * 1.5) + (sqrt(x) * 2.0));
	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, 85000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], 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]]

Target

Original	19.3
Target	0.6
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 < 85000

   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 85000 < x

   1. Initial program 39.2

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

   2. Applied egg-rr 39.1

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

   3. Simplified 39.1

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

      [Start] 39.1	\[ \left(\frac{1}{x} + \frac{-1}{1 + x}\right) \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      associate-*r/ [=>] 39.1	\[ \color{blue}{\frac{\left(\frac{1}{x} + \frac{-1}{1 + x}\right) \cdot 1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]
      *-rgt-identity [=>] 39.1	\[ \frac{\color{blue}{\frac{1}{x} + \frac{-1}{1 + x}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

   4. Applied egg-rr 11.1

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

   5. Simplified 11.1

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

      [Start] 11.1	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{x + x \cdot x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      associate-*r/ [=>] 11.1	\[ \frac{\color{blue}{\frac{\left(1 + \left(x - x\right)\right) \cdot 1}{x + x \cdot x}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      *-rgt-identity [=>] 11.1	\[ \frac{\frac{\color{blue}{1 + \left(x - x\right)}}{x + x \cdot x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      +-commutative [=>] 11.1	\[ \frac{\frac{\color{blue}{\left(x - x\right) + 1}}{x + x \cdot x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      +-inverses [=>] 11.1	\[ \frac{\frac{\color{blue}{0} + 1}{x + x \cdot x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
      metadata-eval [=>] 11.1	\[ \frac{\frac{\color{blue}{1}}{x + x \cdot x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

   6. Applied egg-rr 0.3

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

   7. Simplified 0.3

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

      [Start] 0.3	\[ \frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}{1 + x} \cdot \frac{1}{x} \]
      associate-/l/ [=>] 0.3	\[ \color{blue}{\frac{1}{\left(1 + x\right) \cdot \left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)}} \cdot \frac{1}{x} \]
      associate-*l/ [=>] 0.3	\[ \color{blue}{\frac{1 \cdot \frac{1}{x}}{\left(1 + x\right) \cdot \left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)}} \]
      *-lft-identity [=>] 0.3	\[ \frac{\color{blue}{\frac{1}{x}}}{\left(1 + x\right) \cdot \left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)} \]

   8. Taylor expanded in x around inf 0.3

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

   9. Simplified 0.3

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

      [Start] 0.3	\[ \frac{\frac{1}{x}}{-0.5 \cdot \sqrt{\frac{1}{x}} + \left(2 \cdot \sqrt{x} + 2 \cdot \sqrt{\frac{1}{x}}\right)} \]
      +-commutative [=>] 0.3	\[ \frac{\frac{1}{x}}{-0.5 \cdot \sqrt{\frac{1}{x}} + \color{blue}{\left(2 \cdot \sqrt{\frac{1}{x}} + 2 \cdot \sqrt{x}\right)}} \]
      associate-+r+ [=>] 0.3	\[ \frac{\frac{1}{x}}{\color{blue}{\left(-0.5 \cdot \sqrt{\frac{1}{x}} + 2 \cdot \sqrt{\frac{1}{x}}\right) + 2 \cdot \sqrt{x}}} \]
      distribute-rgt-out [=>] 0.3	\[ \frac{\frac{1}{x}}{\color{blue}{\sqrt{\frac{1}{x}} \cdot \left(-0.5 + 2\right)} + 2 \cdot \sqrt{x}} \]
      metadata-eval [=>] 0.3	\[ \frac{\frac{1}{x}}{\sqrt{\frac{1}{x}} \cdot \color{blue}{1.5} + 2 \cdot \sqrt{x}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 85000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{\frac{1}{x}} \cdot 1.5 + \sqrt{x} \cdot 2}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 10^{-15}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	26240

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

Alternative 3
Error	0.4
Cost	13696

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

Alternative 4
Error	0.9
Cost	7044

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

Alternative 5
Error	2.0
Cost	6788

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

Alternative 6
Error	1.1
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 7
Error	31.2
Cost	6528

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

Alternative 8
Error	61.5
Cost	192

\[x \cdot 0.5 \]

Alternative 9
Error	62.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023027 
(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)))))