2isqrt (example 3.6)

Average Error: 20.2 → 0.2

Time: 12.3s

Precision: binary64

Cost: 33476

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + t_0\right)}\\ \end{array} \]

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)) 0.0)
     (* 0.5 (pow x -1.5))
     (/ 1.0 (* (sqrt (+ x (* x x))) (+ (sqrt 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((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = 1.0 / (sqrt((x + (x * x))) * (sqrt(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((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = 1.0d0 / (sqrt((x + (x * x))) * (sqrt(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((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = 1.0 / (Math.sqrt((x + (x * x))) * (Math.sqrt(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((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = 1.0 / (math.sqrt((x + (x * x))) * (math.sqrt(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 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 0.0)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(x + Float64(x * x))) * Float64(sqrt(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((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 0.0)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = 1.0 / (sqrt((x + (x * x))) * (sqrt(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[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], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	20.2
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)))) < 0.0

   1. Initial program 39.6

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

   2. Applied egg-rr 39.6

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

   3. Simplified 39.6

      \[\leadsto \color{blue}{\frac{\frac{1}{x} + \frac{-1}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]
      Proof
      (/.f64 (+.f64 (/.f64 1 x) (/.f64 -1 (+.f64 1 x))) (+.f64 (pow.f64 x -1/2) (pow.f64 (+.f64 1 x) -1/2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 (/.f64 1 x) (/.f64 -1 (+.f64 1 x))) 1)) (+.f64 (pow.f64 x -1/2) (pow.f64 (+.f64 1 x) -1/2))): 3 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 (/.f64 1 x) (/.f64 -1 (+.f64 1 x))) (/.f64 1 (+.f64 (pow.f64 x -1/2) (pow.f64 (+.f64 1 x) -1/2))))): 3 points increase in error, 0 points decrease in error

   4. Taylor expanded in x around inf 21.0

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]

   5. Applied egg-rr 39.6

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-1.5}\right)} - 1\right)} \]

   6. Simplified 0.0

      \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]
      Proof
      (*.f64 1/2 (pow.f64 x -3/2)): 0 points increase in error, 0 points decrease in error
      (*.f64 1/2 (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (pow.f64 x -3/2))))): 0 points increase in error, 3 points decrease in error
      (*.f64 1/2 (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (pow.f64 x -3/2))) 1))): 1 points increase in error, 2 points decrease in error

   if 0.0 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 1.5

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

   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
      (/.f64 (+.f64 (/.f64 1 x) (/.f64 -1 (+.f64 1 x))) (+.f64 (pow.f64 x -1/2) (pow.f64 (+.f64 1 x) -1/2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 (/.f64 1 x) (/.f64 -1 (+.f64 1 x))) 1)) (+.f64 (pow.f64 x -1/2) (pow.f64 (+.f64 1 x) -1/2))): 3 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 (/.f64 1 x) (/.f64 -1 (+.f64 1 x))) (/.f64 1 (+.f64 (pow.f64 x -1/2) (pow.f64 (+.f64 1 x) -1/2))))): 3 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	27204

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + x \cdot x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	27204

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{1 + x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	26756

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{1 + x}}\\ \end{array} \]

Alternative 4
Error	0.3
Cost	26240

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

Alternative 5
Error	0.4
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 106000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 6
Error	1.0
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 7
Error	1.0
Cost	7044

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

Alternative 8
Error	1.2
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.65:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 9
Error	30.3
Cost	6656

\[0.5 \cdot {x}^{-1.5} \]

Alternative 10
Error	61.5
Cost	192

\[x \cdot 0.5 \]

Reproduce

herbie shell --seed 2022343 
(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)))))