2isqrt (example 3.6)

Average Error: 19.9 → 19.9

Time: 13.0s

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	19.9
Target	0.7
Herbie	19.9

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

Derivation

1. Initial program 19.9

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

2. Final simplification 19.9

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

Alternatives

Alternative 1
Error	30.7
Cost	7872

\[\begin{array}{l} t_0 := \frac{1}{1 + x \cdot 0.5}\\ \left(\frac{1}{\sqrt{x}} + t_0 \cdot 2\right) + t_0 \cdot -3 \end{array} \]

Alternative 2
Error	30.7
Cost	7744

\[\begin{array}{l} t_0 := \frac{1}{1 + x \cdot 0.5}\\ \left(\frac{1}{\sqrt{x}} + t_0\right) + t_0 \cdot -2 \end{array} \]

Alternative 3
Error	30.7
Cost	7104

\[\frac{1}{\sqrt{x}} - \frac{1}{0.5 \cdot x + 1} \]

Alternative 4
Error	31.5
Cost	6976

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

Alternative 5
Error	31.9
Cost	6720

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

Alternative 6
Error	62.2
Cost	192

\[\frac{-2}{x} \]

Alternative 7
Error	62.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023096 
(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)))))