2isqrt (example 3.6)

Average Accuracy: 69.7% → 99.7%

Time: 12.1s

Precision: binary64

Cost: 13636

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 165000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + 0.5}}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x 165000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (/ (/ 1.0 (+ x 0.5)) (+ (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 <= 165000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = (1.0 / (x + 0.5)) / (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 <= 165000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = (1.0d0 / (x + 0.5d0)) / (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 <= 165000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = (1.0 / (x + 0.5)) / (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 <= 165000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = (1.0 / (x + 0.5)) / (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 <= 165000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(Float64(1.0 / Float64(x + 0.5)) / 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 <= 165000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = (1.0 / (x + 0.5)) / (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, 165000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x + 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	69.7%
Target	98.9%
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 x < 165000

   1. Initial program 99.4%

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

   2. Applied egg-rr 99.8%

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

   3. Simplified 99.8%

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

      [Start] 99.8	\[ {x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right) \]
      sub-neg [<=] 99.8	\[ \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

   if 165000 < x

   1. Initial program 38.5%

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

   2. Applied egg-rr 83.3%

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

   3. Simplified 83.3%

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

      [Start] 83.3	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-commutative [=>] 83.3	\[ \frac{\color{blue}{\left(x - x\right) + 1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-inverses [=>] 83.3	\[ \frac{\color{blue}{0} + 1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      metadata-eval [=>] 83.3	\[ \frac{\color{blue}{1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-commutative [=>] 83.3	\[ \frac{1}{\sqrt{x + x \cdot x} \cdot \color{blue}{\left(\sqrt{1 + x} + \sqrt{x}\right)}} \]

   4. Taylor expanded in x around inf 98.2%

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

   5. Simplified 98.2%

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

      [Start] 98.2	\[ \frac{1}{\left(0.5 + x\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} \]
      +-commutative [=>] 98.2	\[ \frac{1}{\color{blue}{\left(x + 0.5\right)} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} \]

   6. Applied egg-rr 37.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\left(x + 0.5\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)} - 1} \]

   7. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{0.5 + x}}{\sqrt{1 + x} + \sqrt{x}}} \]
      Proof

      [Start] 37.7	\[ e^{\mathsf{log1p}\left(\frac{1}{\left(x + 0.5\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)} - 1 \]
      expm1-def [=>] 98.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left(x + 0.5\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}\right)\right)} \]
      expm1-log1p [=>] 98.2	\[ \color{blue}{\frac{1}{\left(x + 0.5\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)}} \]
      associate-/r* [=>] 99.5	\[ \color{blue}{\frac{\frac{1}{x + 0.5}}{\sqrt{1 + x} + \sqrt{x}}} \]
      +-commutative [=>] 99.5	\[ \frac{\frac{1}{\color{blue}{0.5 + x}}}{\sqrt{1 + x} + \sqrt{x}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 165000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + 0.5}}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	26820

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{x} + t_0}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{x}}{x} \cdot \frac{0.5}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 3
Accuracy	98.9%
Cost	26368

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

Alternative 4
Accuracy	98.1%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 0.42:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{x} \cdot \frac{0.5}{1 + x}\\ \end{array} \]

Alternative 5
Accuracy	72.1%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 0.64:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(1 + \sqrt{x}\right)}\\ \end{array} \]

Alternative 6
Accuracy	67.7%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 - {x}^{-0.5}\right)\\ \end{array} \]

Alternative 7
Accuracy	53.5%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 0.5}\\ \end{array} \]

Alternative 8
Accuracy	7.4%
Cost	320

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

Alternative 9
Accuracy	7.4%
Cost	192

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

Alternative 10
Accuracy	5.8%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023129 
(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)))))