sqrt times

Percentage Accurate: 99.2% → 99.6%

Time: 2.7s

Alternatives: 5

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, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around inf 99.5%

   \[\leadsto \color{blue}{x - \left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right)} \]
3. Step-by-step derivation

   1. associate--r+ 99.5%

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

   2. sub-neg 99.5%

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

   3. metadata-eval 99.5%

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

   4. +-commutative 99.5%

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

   5. associate-*r/ 99.5%

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

   6. metadata-eval 99.5%

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

   7. associate-*r/ 99.5%

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

   8. metadata-eval 99.5%

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

   9. unpow2 99.5%

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

4. Simplified 99.5%

   \[\leadsto \color{blue}{\left(x + -0.5\right) - \left(\frac{0.125}{x} + \frac{0.0625}{x \cdot x}\right)} \]
5. Step-by-step derivation

   1. associate--r+ 99.5%

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

   2. sub-neg 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.6% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in x around inf 99.5%

   \[\leadsto \color{blue}{x - \left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right)} \]
3. Step-by-step derivation

   1. associate--r+ 99.5%

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

   2. sub-neg 99.5%

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

   3. metadata-eval 99.5%

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

   4. +-commutative 99.5%

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

   5. associate-*r/ 99.5%

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

   6. metadata-eval 99.5%

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

   7. associate-*r/ 99.5%

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

   8. metadata-eval 99.5%

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

   9. unpow2 99.5%

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

4. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 29.3× speedup

Math

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

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(Float64(x + -0.5) - Float64(0.125 / x))
end

MATLAB

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

Wolfram

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 99.4%

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

   1. sub-neg 99.4%

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

   2. +-commutative 99.4%

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

   3. neg-sub0 99.4%

      \[\leadsto \color{blue}{\left(0 - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)\right)} + x \]

   4. associate-+l- 99.4%

      \[\leadsto \color{blue}{0 - \left(\left(0.5 + 0.125 \cdot \frac{1}{x}\right) - x\right)} \]

   5. sub-neg 99.4%

      \[\leadsto 0 - \color{blue}{\left(\left(0.5 + 0.125 \cdot \frac{1}{x}\right) + \left(-x\right)\right)} \]

   6. mul-1-neg 99.4%

      \[\leadsto 0 - \left(\left(0.5 + 0.125 \cdot \frac{1}{x}\right) + \color{blue}{-1 \cdot x}\right) \]

   7. *-commutative 99.4%

      \[\leadsto 0 - \left(\left(0.5 + 0.125 \cdot \frac{1}{x}\right) + \color{blue}{x \cdot -1}\right) \]

   8. associate--l- 99.4%

      \[\leadsto \color{blue}{\left(0 - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)\right) - x \cdot -1} \]

   9. neg-sub0 99.4%

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

   10. +-commutative 99.4%

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

   11. distribute-neg-in 99.4%

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

   12. metadata-eval 99.4%

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

   13. metadata-eval 99.4%

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

   14. rem-square-sqrt 0.0%

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

   15. unpow2 0.0%

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

   16. associate-+r- 0.0%

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

   17. rem-square-sqrt 0.0%

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

   18. unpow2 0.0%

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

   19. *-commutative 0.0%

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

4. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 4: 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 99.2%

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

2. Taylor expanded in x around inf 99.2%

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

3. Final simplification 99.2%

   \[\leadsto x - 0.5 \]

Alternative 5: 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 99.2%

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

2. Taylor expanded in x around inf 97.9%

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

3. Final simplification 97.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023175 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))