2isqrt (example 3.6)

Average Error: 30.33% → 0.4%

Time: 12.7s

Precision: binary64

Cost: 13764

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} \cdot \left(x + \sqrt{x \cdot \left(1 + x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x 2e+64)
   (/ 1.0 (* (sqrt (+ 1.0 x)) (+ x (sqrt (* x (+ 1.0 x))))))
   (/ (/ 0.5 x) (sqrt 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 <= 2e+64) {
		tmp = 1.0 / (sqrt((1.0 + x)) * (x + sqrt((x * (1.0 + x)))));
	} else {
		tmp = (0.5 / x) / sqrt(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 <= 2d+64) then
        tmp = 1.0d0 / (sqrt((1.0d0 + x)) * (x + sqrt((x * (1.0d0 + x)))))
    else
        tmp = (0.5d0 / x) / sqrt(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 <= 2e+64) {
		tmp = 1.0 / (Math.sqrt((1.0 + x)) * (x + Math.sqrt((x * (1.0 + x)))));
	} else {
		tmp = (0.5 / x) / Math.sqrt(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 <= 2e+64:
		tmp = 1.0 / (math.sqrt((1.0 + x)) * (x + math.sqrt((x * (1.0 + x)))))
	else:
		tmp = (0.5 / x) / math.sqrt(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 <= 2e+64)
		tmp = Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) * Float64(x + sqrt(Float64(x * Float64(1.0 + x))))));
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(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 <= 2e+64)
		tmp = 1.0 / (sqrt((1.0 + x)) * (x + sqrt((x * (1.0 + x)))));
	else
		tmp = (0.5 / x) / sqrt(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, 2e+64], N[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] * N[(x + N[Sqrt[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Target

Original	30.33%
Target	1.08%
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 < 2.00000000000000004e64

   1. Initial program 13.4

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

   2. Applied egg-rr 0.46

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

   3. Simplified 0.46

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

      [Start] 0.46	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-commutative [=>] 0.46	\[ \frac{\color{blue}{\left(x - x\right) + 1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-inverses [=>] 0.46	\[ \frac{\color{blue}{0} + 1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      metadata-eval [=>] 0.46	\[ \frac{\color{blue}{1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      +-commutative [=>] 0.46	\[ \frac{1}{\sqrt{x + x \cdot x} \cdot \color{blue}{\left(\sqrt{1 + x} + \sqrt{x}\right)}} \]

   4. Applied egg-rr 0.57

      \[\leadsto \frac{1}{\color{blue}{\left({\left(x + 1\right)}^{0.25} \cdot \left({\left(x + 1\right)}^{0.25} \cdot \sqrt{x}\right)\right)} \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)} \]

   5. Applied egg-rr 0.45

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

   6. Simplified 0.45

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

      [Start] 0.45	\[ \frac{1}{\mathsf{fma}\left(\sqrt{\left(1 + x\right) \cdot x}, \sqrt{1 + x}, x \cdot \sqrt{1 + x}\right)} \]
      fma-udef [=>] 0.45	\[ \frac{1}{\color{blue}{\sqrt{\left(1 + x\right) \cdot x} \cdot \sqrt{1 + x} + x \cdot \sqrt{1 + x}}} \]
      distribute-rgt-out [=>] 0.45	\[ \frac{1}{\color{blue}{\sqrt{1 + x} \cdot \left(\sqrt{\left(1 + x\right) \cdot x} + x\right)}} \]
      +-commutative [=>] 0.45	\[ \frac{1}{\sqrt{\color{blue}{x + 1}} \cdot \left(\sqrt{\left(1 + x\right) \cdot x} + x\right)} \]
      *-commutative [=>] 0.45	\[ \frac{1}{\sqrt{x + 1} \cdot \left(\sqrt{\color{blue}{x \cdot \left(1 + x\right)}} + x\right)} \]
      +-commutative [=>] 0.45	\[ \frac{1}{\sqrt{x + 1} \cdot \left(\sqrt{x \cdot \color{blue}{\left(x + 1\right)}} + x\right)} \]

   if 2.00000000000000004e64 < x

   1. Initial program 55.21

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

   2. Applied egg-rr 55.21

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

   3. Simplified 55.21

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

      [Start] 55.21	\[ \frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{-\sqrt{1 + x}} \cdot -1 \]
      associate-*l/ [=>] 55.21	\[ \color{blue}{\frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}} \cdot -1}{-\sqrt{1 + x}}} \]
      associate-/l* [=>] 55.21	\[ \color{blue}{\frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\frac{-\sqrt{1 + x}}{-1}}} \]
      metadata-eval [<=] 55.21	\[ \frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\frac{-\sqrt{1 + x}}{\color{blue}{\frac{1}{-1}}}} \]
      associate-/l* [<=] 55.21	\[ \frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\color{blue}{\frac{\left(-\sqrt{1 + x}\right) \cdot -1}{1}}} \]
      *-commutative [<=] 55.21	\[ \frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\frac{\color{blue}{-1 \cdot \left(-\sqrt{1 + x}\right)}}{1}} \]
      mul-1-neg [=>] 55.21	\[ \frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\frac{\color{blue}{-\left(-\sqrt{1 + x}\right)}}{1}} \]
      remove-double-neg [=>] 55.21	\[ \frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\frac{\color{blue}{\sqrt{1 + x}}}{1}} \]
      /-rgt-identity [=>] 55.21	\[ \frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{x}}}{\color{blue}{\sqrt{1 + x}}} \]
      associate-/l/ [=>] 55.21	\[ \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x} \cdot \sqrt{x}}} \]
      associate-/r* [=>] 55.21	\[ \color{blue}{\frac{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}}}{\sqrt{x}}} \]
      div-sub [=>] 55.21	\[ \frac{\color{blue}{\frac{\sqrt{1 + x}}{\sqrt{1 + x}} - \frac{\sqrt{x}}{\sqrt{1 + x}}}}{\sqrt{x}} \]
      *-inverses [=>] 55.21	\[ \frac{\color{blue}{1} - \frac{\sqrt{x}}{\sqrt{1 + x}}}{\sqrt{x}} \]

   4. Taylor expanded in x around inf 0.32

      \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{\sqrt{1 + x} \cdot \left(x + \sqrt{x \cdot \left(1 + x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.34%
Cost	26948

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{x + 0.5}}{\sqrt{x} + t_0}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Error	1.08%
Cost	13632

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

Alternative 3
Error	0.36%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 160000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 + \frac{-0.375}{x}}{x}}{\sqrt{x}}\\ \end{array} \]

Alternative 4
Error	1.06%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 - x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 + \frac{-0.375}{x}}{x}}{\sqrt{x}}\\ \end{array} \]

Alternative 5
Error	1.08%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 + \frac{-0.375}{x}}{x}}{\sqrt{x}}\\ \end{array} \]

Alternative 6
Error	1.56%
Cost	7044

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

Alternative 7
Error	1.72%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.375}{x}}{{x}^{1.5}}\\ \end{array} \]

Alternative 8
Error	1.83%
Cost	6852

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

Alternative 9
Error	46.77%
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 10
Error	49.2%
Cost	6528

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

Alternative 11
Error	92.63%
Cost	320

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

Alternative 12
Error	98.07%
Cost	64

\[-1 \]

Alternative 13
Error	94.26%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023090 
(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)))))