2isqrt (example 3.6)

Average Error: 19.4 → 19.4

Time: 19.6s

Precision: binary64

Cost: 13376

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
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) {
	return (1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)));
}

Python

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

↓

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

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

↓

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end

↓

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
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_] := N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	19.4
Target	0.6
Herbie	19.4

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

Derivation

1. Initial program 19.4

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

2. Final simplification 19.4

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

Alternatives

Alternative 1
Error	20.3
Cost	7232

\[\left(\frac{1}{\sqrt{x}} + \left(1 - \frac{2}{x + 2}\right)\right) + -1 \]

Alternative 2
Error	20.3
Cost	7232

\[\left(1 - \frac{2}{x + 2}\right) + \left(\frac{1}{\sqrt{x}} - 1\right) \]

Alternative 3
Error	20.3
Cost	7108

\[\begin{array}{l} t_0 := \frac{1}{\sqrt{x}}\\ \mathbf{if}\;x \leq 8 \cdot 10^{+76}:\\ \;\;\;\;t_0 - \left(-0.5 \cdot x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_0 - 1\right)\\ \end{array} \]

Alternative 4
Error	20.1
Cost	7108

\[\begin{array}{l} t_0 := \frac{1}{\sqrt{x}}\\ \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;t_0 - \frac{2}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_0 - 1\right)\\ \end{array} \]

Alternative 5
Error	20.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sqrt{x}} - 1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	21.4
Cost	6848

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

Alternative 7
Error	61.5
Cost	192

\[0.5 \cdot x \]

Alternative 8
Error	51.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023075 
(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)))))