2isqrt (example 3.6)

Average Error: 19.6 → 0.4

Time: 10.8s

Precision: binary64

Cost: 13508

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x 85000000.0)
   (- (pow x -0.5) (pow (+ x 1.0) -0.5))
   (/ (/ 1.0 x) (+ (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 <= 85000000.0) {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	} else {
		tmp = (1.0 / x) / (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 <= 85000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    else
        tmp = (1.0d0 / x) / (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 <= 85000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	} else {
		tmp = (1.0 / x) / (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 <= 85000000.0:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	else:
		tmp = (1.0 / x) / (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 <= 85000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	else
		tmp = Float64(Float64(1.0 / x) / 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 <= 85000000.0)
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	else
		tmp = (1.0 / x) / (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, 85000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	19.6
Target	0.6
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if x < 8.5e7

   1. Initial program 0.6

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

   2. Applied egg-rr 0.3

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

   3. Simplified 0.3

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

      [Start] 0.3	\[ {x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right) \]
      sub-neg [<=] 0.3	\[ \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

   if 8.5e7 < x

   1. Initial program 39.4

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

   2. Applied egg-rr 10.5

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

   3. Taylor expanded in x around inf 0.5

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

   4. Applied egg-rr 1.3

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

   5. Simplified 0.5

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

      [Start] 1.3	\[ {\left(x \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)\right)}^{-1} \]
      unpow-1 [=>] 1.3	\[ \color{blue}{\frac{1}{x \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} \]
      associate-/r* [=>] 0.5	\[ \color{blue}{\frac{\frac{1}{x}}{\sqrt{x} + \sqrt{1 + x}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	13888

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

Alternative 2
Error	0.8
Cost	13760

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

Alternative 3
Error	0.8
Cost	13508

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

Alternative 4
Error	5.0
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 70000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - x}}{\sqrt{\frac{1}{x}} \cdot 2}\\ \end{array} \]

Alternative 5
Error	5.6
Cost	7364

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;-1 + \left(t_0 + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - x}}{t_0 \cdot 2}\\ \end{array} \]

Alternative 6
Error	17.9
Cost	7108

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

Alternative 7
Error	18.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{1 + \sqrt{x}}\\ \end{array} \]

Alternative 8
Error	20.6
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left({x}^{-0.5} + 1\right)\\ \end{array} \]

Alternative 9
Error	30.1
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.82:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 10
Error	30.5
Cost	6720

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

Alternative 11
Error	31.8
Cost	6528

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

Alternative 12
Error	59.3
Cost	192

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

Reproduce

herbie shell --seed 2023018 
(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)))))