sqrt times

Percentage Accurate: 99.2% → 99.6%

Time: 3.0s

Alternatives: 4

Speedup: 29.3×

Specification

Math

\[\begin{array}{l} \\ \sqrt{x - 1} \cdot \sqrt{x} \end{array} \]

FPCore

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

C

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

Java

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)

Julia

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

Wolfram

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

Initial Program: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x - 1} \cdot \sqrt{x} \end{array} \]

FPCore

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

C

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

Java

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)

Julia

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

Wolfram

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

Alternative 1: 99.6% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((-0.5d0) + ((((-0.0625d0) / x) - 0.125d0) / x))
end function

Java

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

Python

def code(x):
	return x + (-0.5 + (((-0.0625 / x) - 0.125) / x))

Julia

function code(x)
	return Float64(x + Float64(-0.5 + Float64(Float64(Float64(-0.0625 / x) - 0.125) / x)))
end

MATLAB

function tmp = code(x)
	tmp = x + (-0.5 + (((-0.0625 / x) - 0.125) / x));
end

Wolfram

code[x_] := N[(x + N[(-0.5 + N[(N[(N[(-0.0625 / x), $MachinePrecision] - 0.125), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\sqrt{x - 1} \cdot \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 99.6%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ x + \left(-0.5 + \frac{-0.125}{x}\right) \end{array} \]

FPCore

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

C

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

Fortran

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 x + (-0.5 + (-0.125 / x));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

   \[\sqrt{x - 1} \cdot \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 99.5%

   \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{0.5 + 0.125 \cdot \frac{1}{x}}{x}\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in 99.5%

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

   2. *-lft-identity 99.5%

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

   3. associate-*r/ 99.5%

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

   4. associate-*l/ 99.5%

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

   5. associate-/l* 99.5%

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

   6. mul-1-neg 99.5%

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

   7. *-inverses 99.5%

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

   8. distribute-lft-neg-in 99.5%

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

   9. distribute-rgt-neg-in 99.5%

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

   10. metadata-eval 99.5%

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

   11. *-commutative 99.5%

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

   12. distribute-rgt-in 99.5%

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

   13. metadata-eval 99.5%

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

   14. associate-*r/ 99.5%

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

   15. metadata-eval 99.5%

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

   16. associate-*l/ 99.5%

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

   17. metadata-eval 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 99.1% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ x + -0.5 \end{array} \]

FPCore

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

C

double code(double x) {
	return x + -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + (-0.5d0)
end function

Java

public static double code(double x) {
	return x + -0.5;
}

Python

def code(x):
	return x + -0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

   \[\sqrt{x - 1} \cdot \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 99.3%

   \[\leadsto \color{blue}{x \cdot \left(1 - 0.5 \cdot \frac{1}{x}\right)} \]
4. Step-by-step derivation

   1. sub-neg 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. distribute-rgt-neg-in 99.3%

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

   4. metadata-eval 99.3%

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

   5. distribute-neg-frac 99.3%

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

   6. metadata-eval 99.3%

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

   7. rem-square-sqrt 0.0%

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

   8. unpow2 0.0%

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

   9. distribute-lft-neg-in 0.0%

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

   10. distribute-rgt-neg-in 0.0%

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

   11. distribute-lft-neg-in 0.0%

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

   12. mul-1-neg 0.0%

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

   13. *-rgt-identity 0.0%

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

   14. cancel-sign-sub 0.0%

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

   15. mul-1-neg 0.0%

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

   16. remove-double-neg 0.0%

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

   17. associate-*r/ 0.0%

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

   18. associate-*r/ 0.0%

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

5. Simplified 99.3%

   \[\leadsto \color{blue}{x + -0.5} \]

6. Final simplification 99.3%

   \[\leadsto x + -0.5 \]
7. Add Preprocessing

Alternative 4: 97.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 99.3%

   \[\sqrt{x - 1} \cdot \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 98.5%

   \[\leadsto \color{blue}{x} \]

4. Final simplification 98.5%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))