sqrt times

Average Accuracy: 99.2% → 99.6%

Time: 2.2s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (+ (+ x -0.5) (/ (+ -0.125 (/ -0.0625 x)) x)))

C

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

↓

double code(double x) {
	return (x + -0.5) + ((-0.125 + (-0.0625 / x)) / 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) + ((-0.0625d0) / x)) / 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 + (-0.0625 / x)) / x);
}

Python

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

↓

def code(x):
	return (x + -0.5) + ((-0.125 + (-0.0625 / x)) / 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(Float64(-0.125 + Float64(-0.0625 / x)) / 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 + (-0.0625 / x)) / 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[(N[(-0.125 + N[(-0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around inf 99.6%

   \[\leadsto \color{blue}{x - \left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right)} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\left(x + -0.5\right) - \left(\frac{0.125}{x} + \frac{0.0625}{x \cdot x}\right)} \]
   Proof

   [Start] 99.6	\[ x - \left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right) \]
   associate--r+ [=>] 99.6	\[ \color{blue}{\left(x - 0.5\right) - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)} \]
   sub-neg [=>] 99.6	\[ \color{blue}{\left(x + \left(-0.5\right)\right)} - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right) \]
   metadata-eval [=>] 99.6	\[ \left(x + \color{blue}{-0.5}\right) - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right) \]
   +-commutative [=>] 99.6	\[ \left(x + -0.5\right) - \color{blue}{\left(0.125 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{2}}\right)} \]
   associate-*r/ [=>] 99.6	\[ \left(x + -0.5\right) - \left(\color{blue}{\frac{0.125 \cdot 1}{x}} + 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]
   metadata-eval [=>] 99.6	\[ \left(x + -0.5\right) - \left(\frac{\color{blue}{0.125}}{x} + 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]
   associate-*r/ [=>] 99.6	\[ \left(x + -0.5\right) - \left(\frac{0.125}{x} + \color{blue}{\frac{0.0625 \cdot 1}{{x}^{2}}}\right) \]
   metadata-eval [=>] 99.6	\[ \left(x + -0.5\right) - \left(\frac{0.125}{x} + \frac{\color{blue}{0.0625}}{{x}^{2}}\right) \]
   unpow2 [=>] 99.6	\[ \left(x + -0.5\right) - \left(\frac{0.125}{x} + \frac{0.0625}{\color{blue}{x \cdot x}}\right) \]

4. Taylor expanded in x around 0 99.6%

   \[\leadsto \left(x + -0.5\right) - \color{blue}{\left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)} \]

5. Simplified 99.6%

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

   [Start] 99.6	\[ \left(x + -0.5\right) - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right) \]
   associate-*r/ [=>] 99.6	\[ \left(x + -0.5\right) - \left(\color{blue}{\frac{0.0625 \cdot 1}{{x}^{2}}} + 0.125 \cdot \frac{1}{x}\right) \]
   metadata-eval [=>] 99.6	\[ \left(x + -0.5\right) - \left(\frac{\color{blue}{0.0625}}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right) \]
   unpow2 [=>] 99.6	\[ \left(x + -0.5\right) - \left(\frac{0.0625}{\color{blue}{x \cdot x}} + 0.125 \cdot \frac{1}{x}\right) \]
   associate-/r* [=>] 99.6	\[ \left(x + -0.5\right) - \left(\color{blue}{\frac{\frac{0.0625}{x}}{x}} + 0.125 \cdot \frac{1}{x}\right) \]
   *-lft-identity [<=] 99.6	\[ \left(x + -0.5\right) - \left(\frac{\color{blue}{1 \cdot \frac{0.0625}{x}}}{x} + 0.125 \cdot \frac{1}{x}\right) \]
   associate-*l/ [<=] 99.6	\[ \left(x + -0.5\right) - \left(\color{blue}{\frac{1}{x} \cdot \frac{0.0625}{x}} + 0.125 \cdot \frac{1}{x}\right) \]
   *-commutative [=>] 99.6	\[ \left(x + -0.5\right) - \left(\frac{1}{x} \cdot \frac{0.0625}{x} + \color{blue}{\frac{1}{x} \cdot 0.125}\right) \]
   distribute-lft-in [<=] 99.6	\[ \left(x + -0.5\right) - \color{blue}{\frac{1}{x} \cdot \left(\frac{0.0625}{x} + 0.125\right)} \]
   associate-*l/ [=>] 99.6	\[ \left(x + -0.5\right) - \color{blue}{\frac{1 \cdot \left(\frac{0.0625}{x} + 0.125\right)}{x}} \]
   *-lft-identity [=>] 99.6	\[ \left(x + -0.5\right) - \frac{\color{blue}{\frac{0.0625}{x} + 0.125}}{x} \]
   +-commutative [=>] 99.6	\[ \left(x + -0.5\right) - \frac{\color{blue}{0.125 + \frac{0.0625}{x}}}{x} \]

6. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	448

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

Alternative 2
Accuracy	99.1%
Cost	192

\[x + -0.5 \]

Alternative 3
Accuracy	98.1%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023129 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))