2isqrt (example 3.6)

Average Error: 19.4 → 0.2

Time: 10.1s

Precision: binary64

Cost: 13636

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 170000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + 0.5}}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x 170000.0)
   (- (pow x -0.5) (pow (+ x 1.0) -0.5))
   (/ (/ 1.0 (+ x 0.5)) (+ (sqrt x) (sqrt (+ x 1.0))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= 170000.0) {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	} else {
		tmp = (1.0 / (x + 0.5)) / (sqrt(x) + sqrt((x + 1.0)));
	}
	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 <= 170000.0d0) then
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    else
        tmp = (1.0d0 / (x + 0.5d0)) / (sqrt(x) + sqrt((x + 1.0d0)))
    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 <= 170000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	} else {
		tmp = (1.0 / (x + 0.5)) / (Math.sqrt(x) + Math.sqrt((x + 1.0)));
	}
	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 <= 170000.0:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	else:
		tmp = (1.0 / (x + 0.5)) / (math.sqrt(x) + math.sqrt((x + 1.0)))
	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 <= 170000.0)
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	else
		tmp = Float64(Float64(1.0 / Float64(x + 0.5)) / Float64(sqrt(x) + sqrt(Float64(x + 1.0))));
	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 <= 170000.0)
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	else
		tmp = (1.0 / (x + 0.5)) / (sqrt(x) + sqrt((x + 1.0)));
	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, 170000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x + 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	19.4
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 < 1.7e5

   1. Initial program 0.4

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

   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 1.7e5 < x

   1. Initial program 39.0

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

   2. Applied egg-rr 11.0

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

   3. Simplified 11.0

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

      [Start] 11.0	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      associate-/r* [=>] 11.0	\[ \color{blue}{\frac{\frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}}} \]
      +-commutative [=>] 11.0	\[ \frac{\frac{\color{blue}{\left(x - x\right) + 1}}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      +-inverses [=>] 11.0	\[ \frac{\frac{\color{blue}{0} + 1}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      metadata-eval [=>] 11.0	\[ \frac{\frac{\color{blue}{1}}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      +-commutative [=>] 11.0	\[ \frac{\frac{1}{\sqrt{x + x \cdot x}}}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} \]

   4. Taylor expanded in x around inf 0.3

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

   5. Simplified 0.3

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

      [Start] 0.3	\[ \frac{\frac{1}{0.5 + x}}{\sqrt{1 + x} + \sqrt{x}} \]
      +-commutative [=>] 0.3	\[ \frac{\frac{1}{\color{blue}{x + 0.5}}}{\sqrt{1 + x} + \sqrt{x}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 170000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + 0.5}}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	32704

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

Alternative 2
Error	0.4
Cost	26820

\[\begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{x} + t_0}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	13760

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

Alternative 4
Error	10.6
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 135000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 5
Error	11.1
Cost	13316

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

Alternative 6
Error	21.0
Cost	6848

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

Alternative 7
Error	29.9
Cost	6788

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

Alternative 8
Error	20.2
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.81:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \]

Alternative 9
Error	21.3
Cost	6784

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

Alternative 10
Error	31.5
Cost	6528

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

Alternative 11
Error	59.3
Cost	320

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

Alternative 12
Error	59.3
Cost	192

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

Alternative 13
Error	62.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023016 
(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)))))