2isqrt (example 3.6)

Average Error: 19.8 → 0.2

Time: 11.4s

Precision: binary64

Cost: 27204

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{\left(x + 0.5\right) + \frac{-0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 5e-8)
     (/ (/ 1.0 (+ (sqrt x) t_0)) (+ (+ x 0.5) (/ -0.125 x)))
     (- (pow x -0.5) (pow (+ 1.0 x) -0.5)))))

C

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

↓

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 5e-8) {
		tmp = (1.0 / (sqrt(x) + t_0)) / ((x + 0.5) + (-0.125 / x));
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	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((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) - (1.0d0 / t_0)) <= 5d-8) then
        tmp = (1.0d0 / (sqrt(x) + t_0)) / ((x + 0.5d0) + ((-0.125d0) / x))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    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((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / t_0)) <= 5e-8) {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) / ((x + 0.5) + (-0.125 / x));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	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((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / t_0)) <= 5e-8:
		tmp = (1.0 / (math.sqrt(x) + t_0)) / ((x + 0.5) + (-0.125 / x))
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	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 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 5e-8)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) / Float64(Float64(x + 0.5) + Float64(-0.125 / x)));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	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((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 5e-8)
		tmp = (1.0 / (sqrt(x) + t_0)) / ((x + 0.5) + (-0.125 / x));
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	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[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 5e-8], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 0.5), $MachinePrecision] + N[(-0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Target

Original	19.8
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 (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 4.9999999999999998e-8

   1. Initial program 39.7

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

   2. Applied egg-rr 39.7

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

   3. Applied egg-rr 38.5

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

   4. Taylor expanded in x around 0 10.7

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

   5. Taylor expanded in x around inf 0.2

      \[\leadsto \frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\color{blue}{\left(0.5 + x\right) - 0.125 \cdot \frac{1}{x}}} \]

   6. Simplified 0.2

      \[\leadsto \frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\color{blue}{\left(x + 0.5\right) + \frac{-0.125}{x}}} \]
      Proof
      (+.f64 (+.f64 x 1/2) (/.f64 -1/8 x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 1/2 x)) (/.f64 -1/8 x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 1/2 x) (/.f64 (Rewrite<= metadata-eval (*.f64 1/8 -1)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 1/2 x) (/.f64 (*.f64 1/8 (Rewrite<= metadata-eval (/.f64 1 -1))) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 1/2 x) (/.f64 (*.f64 1/8 (/.f64 1 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))))) x)): 255 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 1/2 x) (/.f64 (*.f64 1/8 (/.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2)))) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 1/2 x) (Rewrite<= associate-*r/_binary64 (*.f64 1/8 (/.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2)) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 1/2 x) (*.f64 1/8 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 (pow.f64 (sqrt.f64 -1) 2) x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 1/8 (/.f64 1 (*.f64 (pow.f64 (sqrt.f64 -1) 2) x))) (+.f64 1/2 x))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> fma-def_binary64 (fma.f64 1/8 (/.f64 1 (*.f64 (pow.f64 (sqrt.f64 -1) 2) x)) (+.f64 1/2 x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/8 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2)) x)) (+.f64 1/2 x)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/8 (/.f64 (/.f64 1 (Rewrite=> unpow2_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1)))) x) (+.f64 1/2 x)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/8 (/.f64 (/.f64 1 (Rewrite=> rem-square-sqrt_binary64 -1)) x) (+.f64 1/2 x)): 0 points increase in error, 255 points decrease in error
      (fma.f64 1/8 (/.f64 (Rewrite=> metadata-eval -1) x) (+.f64 1/2 x)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/8 (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) x) (+.f64 1/2 x)): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/8 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1 x))) (+.f64 1/2 x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/8 (neg.f64 (/.f64 1 x))) (+.f64 1/2 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 1/8 (/.f64 1 x)))) (+.f64 1/2 x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1/2 x) (neg.f64 (*.f64 1/8 (/.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 (+.f64 1/2 x) (*.f64 1/8 (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error

   if 4.9999999999999998e-8 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 0.4

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

   2. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.1

      \[\leadsto \color{blue}{{x}^{-0.5}} - {\left(1 + x\right)}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\left(x + 0.5\right) + \frac{-0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	26948

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

Alternative 2
Error	0.4
Cost	26820

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 10^{-12}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{1 + x}}\\ \end{array} \]

Alternative 3
Error	0.8
Cost	26756

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 10^{-12}:\\ \;\;\;\;\frac{1}{2 \cdot {x}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{1 + x}}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	26432

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

Alternative 5
Error	0.8
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 25368878.874387436:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot {x}^{1.5}}\\ \end{array} \]

Alternative 6
Error	1.5
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 0.001837971661049503:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 - x \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot {x}^{1.5}}\\ \end{array} \]

Alternative 7
Error	1.7
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 0.001837971661049503:\\ \;\;\;\;\frac{1}{\sqrt{x}} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot {x}^{1.5}}\\ \end{array} \]

Alternative 8
Error	1.7
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 0.001837971661049503:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot {x}^{1.5}}\\ \end{array} \]

Alternative 9
Error	30.1
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.001837971661049503:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 0.5}\\ \end{array} \]

Alternative 10
Error	30.4
Cost	6720

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

Alternative 11
Error	31.7
Cost	6528

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

Alternative 12
Error	59.2
Cost	320

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

Alternative 13
Error	62.8
Cost	64

\[-1 \]

Alternative 14
Error	60.3
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022318 
(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)))))