2isqrt (example 3.6)

Average Error: 20.1 → 0.3

Time: 12.5s

Precision: binary64

Cost: 20164

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{x} + \sqrt{1 + x}\\ \mathbf{if}\;x \leq 5 \cdot 10^{+40}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x} \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (sqrt x) (sqrt (+ 1.0 x)))))
   (if (<= x 5e+40) (/ 1.0 (* (sqrt (+ x (* x x))) t_0)) (/ (/ 1.0 x) t_0))))

C

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

↓

double code(double x) {
	double t_0 = sqrt(x) + sqrt((1.0 + x));
	double tmp;
	if (x <= 5e+40) {
		tmp = 1.0 / (sqrt((x + (x * x))) * t_0);
	} else {
		tmp = (1.0 / x) / t_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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) + sqrt((1.0d0 + x))
    if (x <= 5d+40) then
        tmp = 1.0d0 / (sqrt((x + (x * x))) * t_0)
    else
        tmp = (1.0d0 / x) / t_0
    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 t_0 = Math.sqrt(x) + Math.sqrt((1.0 + x));
	double tmp;
	if (x <= 5e+40) {
		tmp = 1.0 / (Math.sqrt((x + (x * x))) * t_0);
	} else {
		tmp = (1.0 / x) / t_0;
	}
	return tmp;
}

Python

def code(x):
	return (1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))

↓

def code(x):
	t_0 = math.sqrt(x) + math.sqrt((1.0 + x))
	tmp = 0
	if x <= 5e+40:
		tmp = 1.0 / (math.sqrt((x + (x * x))) * t_0)
	else:
		tmp = (1.0 / x) / t_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)
	t_0 = Float64(sqrt(x) + sqrt(Float64(1.0 + x)))
	tmp = 0.0
	if (x <= 5e+40)
		tmp = Float64(1.0 / Float64(sqrt(Float64(x + Float64(x * x))) * t_0));
	else
		tmp = Float64(Float64(1.0 / x) / t_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)
	t_0 = sqrt(x) + sqrt((1.0 + x));
	tmp = 0.0;
	if (x <= 5e+40)
		tmp = 1.0 / (sqrt((x + (x * x))) * t_0);
	else
		tmp = (1.0 / x) / t_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_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e+40], N[(1.0 / N[(N[Sqrt[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Target

Original	20.1
Target	0.7
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000003e40

   1. Initial program 5.7

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

   2. Applied egg-rr 0.3

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

   3. Simplified 0.3

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

      [Start] 0.3	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-commutative [=>] 0.3	\[ \frac{\color{blue}{\left(x - x\right) + 1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-inverses [=>] 0.3	\[ \frac{\color{blue}{0} + 1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      metadata-eval [=>] 0.3	\[ \frac{\color{blue}{1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-commutative [=>] 0.3	\[ \frac{1}{\sqrt{x + x \cdot x} \cdot \color{blue}{\left(\sqrt{1 + x} + \sqrt{x}\right)}} \]

   if 5.00000000000000003e40 < x

   1. Initial program 38.8

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

   2. Applied egg-rr 12.7

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

   3. Simplified 12.7

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

      [Start] 12.7	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-commutative [=>] 12.7	\[ \frac{\color{blue}{\left(x - x\right) + 1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-inverses [=>] 12.7	\[ \frac{\color{blue}{0} + 1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      metadata-eval [=>] 12.7	\[ \frac{\color{blue}{1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-commutative [=>] 12.7	\[ \frac{1}{\sqrt{x + x \cdot x} \cdot \color{blue}{\left(\sqrt{1 + x} + \sqrt{x}\right)}} \]

   4. Taylor expanded in x around inf 1.2

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

   5. Applied egg-rr 0.3

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

   6. Simplified 0.2

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

      [Start] 0.3	\[ \frac{1}{x} \cdot \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
      associate-*r/ [=>] 0.2	\[ \color{blue}{\frac{\frac{1}{x} \cdot 1}{\sqrt{1 + x} + \sqrt{x}}} \]
      associate-/l* [=>] 0.2	\[ \color{blue}{\frac{\frac{1}{x}}{\frac{\sqrt{1 + x} + \sqrt{x}}{1}}} \]
      /-rgt-identity [=>] 0.2	\[ \frac{\frac{1}{x}}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+40}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	26820

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

Alternative 2
Error	0.3
Cost	26240

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

Alternative 3
Error	0.4
Cost	13508

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

Alternative 4
Error	11.0
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 110000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 5
Error	11.6
Cost	13316

\[\begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 6
Error	20.4
Cost	7236

\[\begin{array}{l} t_0 := 1 + x \cdot 0.5\\ \mathbf{if}\;x \leq 1.35:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt{x} + t_0\right)}\\ \end{array} \]

Alternative 7
Error	20.4
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 6:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x}}\\ \end{array} \]

Alternative 8
Error	20.5
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;{x}^{-0.5} - \left(1 + x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x}}\\ \end{array} \]

Alternative 9
Error	21.7
Cost	6848

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

Alternative 10
Error	29.9
Cost	6788

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

Alternative 11
Error	22.0
Cost	6784

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

Alternative 12
Error	30.3
Cost	6720

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

Alternative 13
Error	31.5
Cost	6528

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

Alternative 14
Error	59.3
Cost	192

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

Reproduce

herbie shell --seed 2023034 
(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)))))