sqrt times

Average Error: 0.76% → 0.76%

Time: 2.4s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

(FPCore (x) :precision binary64 (* (sqrt (- x 1.0)) (sqrt x)))

↓

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

C

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

↓

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x - 1.0d0)) * sqrt(x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + (-1.0d0))) * sqrt(x)
end function

Java

public static double code(double x) {
	return Math.sqrt((x - 1.0)) * Math.sqrt(x);
}

↓

public static double code(double x) {
	return Math.sqrt((x + -1.0)) * Math.sqrt(x);
}

Python

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

↓

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

Julia

function code(x)
	return Float64(sqrt(Float64(x - 1.0)) * sqrt(x))
end

↓

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

MATLAB

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

↓

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

Wolfram

code[x_] := N[(N[Sqrt[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[Sqrt[N[(x + -1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.76

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

2. Final simplification 0.76

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

Alternatives

Alternative 1
Error	0.42%
Cost	704

\[\left(x + -0.5\right) + \frac{-0.125 - \frac{0.0625}{x}}{x} \]

Alternative 2
Error	0.57%
Cost	448

\[\left(x + -0.5\right) + \frac{-0.125}{x} \]

Alternative 3
Error	0.91%
Cost	192

\[x + -0.5 \]

Alternative 4
Error	2.07%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023115 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))