sqrt times

Average Accuracy: 99.2% → 99.3%

Time: 1.3s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (+ (+ x -0.5) (/ -0.125 x)))

C

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

↓

double code(double x) {
	return (x + -0.5) + (-0.125 / 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 = (x + (-0.5d0)) + ((-0.125d0) / 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 (x + -0.5) + (-0.125 / x);
}

Python

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

↓

def code(x):
	return (x + -0.5) + (-0.125 / x)

Julia

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

↓

function code(x)
	return Float64(Float64(x + -0.5) + Float64(-0.125 / x))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (x + -0.5) + (-0.125 / x);
end

Wolfram

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

↓

code[x_] := N[(N[(x + -0.5), $MachinePrecision] + N[(-0.125 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around inf 99.3%

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

3. Simplified 99.3%

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

   [Start] 99.3	\[ x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right) \]
   associate--r+ [=>] 99.3	\[ \color{blue}{\left(x - 0.5\right) - 0.125 \cdot \frac{1}{x}} \]
   sub-neg [=>] 99.3	\[ \color{blue}{\left(x + \left(-0.5\right)\right)} - 0.125 \cdot \frac{1}{x} \]
   metadata-eval [=>] 99.3	\[ \left(x + \color{blue}{-0.5}\right) - 0.125 \cdot \frac{1}{x} \]
   associate-*r/ [=>] 99.3	\[ \left(x + -0.5\right) - \color{blue}{\frac{0.125 \cdot 1}{x}} \]
   metadata-eval [=>] 99.3	\[ \left(x + -0.5\right) - \frac{\color{blue}{0.125}}{x} \]

4. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	99.0%
Cost	192

\[x + -0.5 \]

Alternative 2
Accuracy	97.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023136 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))