sqrt times

Percentage Accurate: 99.2% → 99.2%

Time: 4.1s

Alternatives: 7

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 99.3%

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

3. Final simplification 99.3%

   \[\leadsto \sqrt{x + -1} \cdot \sqrt{x} \]
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 99.3%

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

   1. add-log-exp 6.6%

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

   2. *-un-lft-identity 6.6%

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

   3. log-prod 6.6%

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

   4. metadata-eval 6.6%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{x - 1} \cdot \sqrt{x}}\right) \]

   5. add-log-exp 99.3%

      \[\leadsto 0 + \color{blue}{\sqrt{x - 1} \cdot \sqrt{x}} \]

   6. *-commutative 99.3%

      \[\leadsto 0 + \color{blue}{\sqrt{x} \cdot \sqrt{x - 1}} \]

   7. sqrt-unprod 52.1%

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

   8. sub-neg 52.1%

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

   9. metadata-eval 52.1%

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

4. Applied egg-rr 52.1%

   \[\leadsto \color{blue}{0 + \sqrt{x \cdot \left(x + -1\right)}} \]
5. Step-by-step derivation

   1. +-lft-identity 52.1%

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

6. Simplified 52.1%

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

7. Taylor expanded in x around inf 99.2%

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

   1. associate--l+ 99.2%

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

   2. *-commutative 99.2%

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

   3. cancel-sign-sub-inv 99.2%

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

   4. associate-*r/ 99.2%

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

   5. unpow2 99.2%

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

   6. times-frac 99.2%

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

   7. metadata-eval 99.2%

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

   8. distribute-neg-frac 99.2%

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

   9. distribute-lft-out 99.2%

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

   10. distribute-neg-frac 99.2%

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

   11. metadata-eval 99.2%

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

   12. associate-*r/ 99.2%

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

   13. metadata-eval 99.2%

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

9. Simplified 99.2%

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

10. Taylor expanded in x around inf 99.2%

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

12. Simplified 99.2%

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

Alternative 3: 99.6% accurate, 18.6× speedup

Math

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

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(x + Float64(-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[(x + N[(-0.5 + N[(N[(-0.125 + N[(-0.0625 / x), $MachinePrecision]), $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 99.2%

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

5. Simplified 99.2%

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

Alternative 4: 99.5% 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 99.0%

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

5. Simplified 99.0%

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

Alternative 5: 99.2% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(1.0 - N[(0.5 / 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 98.5%

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

   1. associate-*r/ 98.5%

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

   2. metadata-eval 98.5%

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

5. Simplified 98.5%

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

Alternative 6: 99.2% 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 98.5%

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

   1. associate-*r/ 98.5%

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

   2. metadata-eval 98.5%

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

   3. metadata-eval 98.5%

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

   4. associate-*r/ 98.5%

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

   5. rem-square-sqrt 0.0%

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

   6. unpow2 0.0%

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

   7. distribute-lft-out-- 0.0%

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

   8. *-rgt-identity 0.0%

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

   9. *-commutative 0.0%

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

   10. associate-*l* 0.0%

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

   11. unpow2 0.0%

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

   12. rem-square-sqrt 98.5%

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

   13. metadata-eval 98.5%

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

   14. distribute-neg-frac 98.5%

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

   15. distribute-lft-neg-out 98.5%

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

   16. lft-mult-inverse 98.5%

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

   17. distribute-rgt-neg-in 98.5%

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

   18. distribute-lft-neg-in 98.5%

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

   19. metadata-eval 98.5%

       \[\leadsto x - \color{blue}{0.5} \cdot 1 \]

   20. cancel-sign-sub-inv 98.5%

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

   21. metadata-eval 98.5%

       \[\leadsto x + \color{blue}{-0.5} \cdot 1 \]

   22. metadata-eval 98.5%

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

5. Simplified 98.5%

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

Alternative 7: 98.1% 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 97.0%

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

Reproduce

herbie shell --seed 2024095 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))