sqrt times

Percentage Accurate: 99.2% → 99.2%

Time: 4.9s

Alternatives: 6

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: 99.2% 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]

Derivation

1. Initial program 99.2%

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

3. Final simplification 99.2%

   \[\leadsto \sqrt{x + -1} \cdot \sqrt{x} \]
4. Add Preprocessing

Alternative 2: 99.6% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \frac{-0.5 - \frac{0.125 + \frac{0.0625}{x}}{x}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 1.0 (/ (- -0.5 (/ (+ 0.125 (/ 0.0625 x)) x)) x))))

C

double code(double x) {
	return x * (1.0 + ((-0.5 - ((0.125 + (0.0625 / x)) / x)) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (((-0.5d0) - ((0.125d0 + (0.0625d0 / x)) / x)) / x))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((-0.5 - ((0.125 + (0.0625 / x)) / x)) / x));
}

Python

def code(x):
	return x * (1.0 + ((-0.5 - ((0.125 + (0.0625 / x)) / x)) / x))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(-0.5 - Float64(Float64(0.125 + Float64(0.0625 / x)) / x)) / x)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((-0.5 - ((0.125 + (0.0625 / x)) / x)) / x));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(-0.5 - N[(N[(0.125 + N[(0.0625 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\sqrt{x - 1} \cdot \sqrt{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 99.2%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{x - 1}} \]

   2. sqrt-unprod 58.6%

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

   3. sub-neg 58.6%

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

   4. metadata-eval 58.6%

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

4. Applied egg-rr 58.6%

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

5. Taylor expanded in x around inf 99.0%

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

   1. associate--l+ 99.0%

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

   2. associate-*r/ 99.0%

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

   3. metadata-eval 99.0%

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

   4. unpow2 99.0%

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

   5. associate-/r* 99.0%

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

   6. associate-/l* 99.0%

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

   7. metadata-eval 99.0%

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

   8. associate-*r/ 99.0%

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

   9. associate-*r/ 99.0%

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

   10. metadata-eval 99.0%

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

   11. div-sub 99.0%

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

7. Simplified 99.0%

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

Alternative 3: 99.6% accurate, 18.6× speedup

Math

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

FPCore

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

C

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 = (x - 0.5d0) - ((0.125d0 + (0.0625d0 / x)) / x)
end function

Java

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

Python

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

Julia

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

Wolfram

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. Add Preprocessing

3. Taylor expanded in x around inf 99.0%

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

5. Simplified 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. associate--r+ 99.0%

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

7. Applied egg-rr 99.0%

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

Alternative 4: 99.4% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around inf 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in x around inf 98.9%

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

   1. *-un-lft-identity 98.9%

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

   2. +-commutative 98.9%

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

   3. associate--r+ 98.9%

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

8. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\left(x - \frac{0.125}{x}\right) - 0.5} \]
9. Add Preprocessing

Alternative 5: 99.0% 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. Add Preprocessing

3. Taylor expanded in x around inf 98.7%

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

   1. sub-neg 98.7%

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

   2. distribute-lft-in 98.7%

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

   3. *-rgt-identity 98.7%

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

   4. distribute-rgt-neg-in 98.7%

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

   5. distribute-neg-frac 98.7%

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

   6. metadata-eval 98.7%

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

   7. rem-square-sqrt 0.0%

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

   8. unpow2 0.0%

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

   9. metadata-eval 0.0%

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

   10. distribute-lft-neg-in 0.0%

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

   11. distribute-rgt-neg-in 0.0%

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

   12. *-commutative 0.0%

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

   13. associate-*r* 0.0%

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

   14. distribute-lft-neg-in 0.0%

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

5. Simplified 98.7%

   \[\leadsto \color{blue}{x + -0.5} \]
6. Add Preprocessing

Alternative 6: 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. Add Preprocessing

3. Taylor expanded in x around inf 97.9%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))