sqrt D (should all be same)

Percentage Accurate: 54.2% → 99.6%

Time: 8.0s

Alternatives: 3

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot {x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))

C

double code(double x) {
	return sqrt((2.0 * pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((2.0d0 * (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt((2.0 * Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.sqrt((2.0 * math.pow(x, 2.0)))

Julia

function code(x)
	return sqrt(Float64(2.0 * (x ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((2.0 * (x ^ 2.0)));
end

Wolfram

code[x_] := N[Sqrt[N[(2.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 54.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot {x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))

C

double code(double x) {
	return sqrt((2.0 * pow(x, 2.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((2.0d0 * (x ** 2.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt((2.0 * Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.sqrt((2.0 * math.pow(x, 2.0)))

Julia

function code(x)
	return sqrt(Float64(2.0 * (x ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((2.0 * (x ^ 2.0)));
end

Wolfram

code[x_] := N[Sqrt[N[(2.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {2}^{0.125} \cdot \left({8}^{0.125} \cdot x\_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (pow 2.0 0.125) (* (pow 8.0 0.125) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(2.0, 0.125) * (pow(8.0, 0.125) * x_m);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (2.0d0 ** 0.125d0) * ((8.0d0 ** 0.125d0) * x_m)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(2.0, 0.125) * (Math.pow(8.0, 0.125) * x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(2.0, 0.125) * (math.pow(8.0, 0.125) * x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64((2.0 ^ 0.125) * Float64((8.0 ^ 0.125) * x_m))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (2.0 ^ 0.125) * ((8.0 ^ 0.125) * x_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[2.0, 0.125], $MachinePrecision] * N[(N[Power[8.0, 0.125], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

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

   1. expm1-log1p-u 54.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot {x}^{2}}\right)\right)} \]

   2. expm1-udef 32.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2 \cdot {x}^{2}}\right)} - 1} \]

   3. log1p-udef 32.5%

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

   4. rem-exp-log 34.5%

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

   5. *-commutative 34.5%

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

   6. sqrt-prod 34.3%

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

   7. unpow2 34.3%

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

   8. sqrt-prod 27.3%

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

   9. add-sqr-sqrt 28.8%

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

4. Applied egg-rr 28.8%

   \[\leadsto \color{blue}{\left(1 + x \cdot \sqrt{2}\right) - 1} \]
5. Step-by-step derivation

   1. add-exp-log 26.5%

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

   2. log1p-udef 26.5%

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(x \cdot \sqrt{2}\right)}} - 1 \]

   3. expm1-udef 50.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{2}\right)\right)} \]

   4. expm1-log1p-u 53.0%

      \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]

   5. *-commutative 53.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

   6. pow1/2 53.0%

      \[\leadsto \color{blue}{{2}^{0.5}} \cdot x \]

   7. metadata-eval 53.0%

      \[\leadsto {2}^{\color{blue}{\left(0.25 + 0.25\right)}} \cdot x \]

   8. pow-prod-up 52.9%

      \[\leadsto \color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right)} \cdot x \]

   9. associate-*r* 53.0%

      \[\leadsto \color{blue}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot x\right)} \]

   10. add-sqr-sqrt 53.0%

       \[\leadsto \color{blue}{\left(\sqrt{{2}^{0.25}} \cdot \sqrt{{2}^{0.25}}\right)} \cdot \left({2}^{0.25} \cdot x\right) \]

   11. associate-*l* 53.1%

       \[\leadsto \color{blue}{\sqrt{{2}^{0.25}} \cdot \left(\sqrt{{2}^{0.25}} \cdot \left({2}^{0.25} \cdot x\right)\right)} \]

   12. sqrt-pow1 53.1%

       \[\leadsto \color{blue}{{2}^{\left(\frac{0.25}{2}\right)}} \cdot \left(\sqrt{{2}^{0.25}} \cdot \left({2}^{0.25} \cdot x\right)\right) \]

   13. metadata-eval 53.1%

       \[\leadsto {2}^{\color{blue}{0.125}} \cdot \left(\sqrt{{2}^{0.25}} \cdot \left({2}^{0.25} \cdot x\right)\right) \]

   14. sqrt-pow1 53.1%

       \[\leadsto {2}^{0.125} \cdot \left(\color{blue}{{2}^{\left(\frac{0.25}{2}\right)}} \cdot \left({2}^{0.25} \cdot x\right)\right) \]

   15. metadata-eval 53.1%

       \[\leadsto {2}^{0.125} \cdot \left({2}^{\color{blue}{0.125}} \cdot \left({2}^{0.25} \cdot x\right)\right) \]

6. Applied egg-rr 53.1%

   \[\leadsto \color{blue}{{2}^{0.125} \cdot \left({2}^{0.125} \cdot \left({2}^{0.25} \cdot x\right)\right)} \]

7. Taylor expanded in x around 0 53.1%

   \[\leadsto {2}^{0.125} \cdot \color{blue}{\left({8}^{0.125} \cdot x\right)} \]

8. Final simplification 53.1%

   \[\leadsto {2}^{0.125} \cdot \left({8}^{0.125} \cdot x\right) \]
9. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \sqrt{2 \cdot x\_m} \cdot \sqrt{x\_m} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (sqrt (* 2.0 x_m)) (sqrt x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return sqrt((2.0 * x_m)) * sqrt(x_m);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = sqrt((2.0d0 * x_m)) * sqrt(x_m)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.sqrt((2.0 * x_m)) * Math.sqrt(x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.sqrt((2.0 * x_m)) * math.sqrt(x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(sqrt(Float64(2.0 * x_m)) * sqrt(x_m))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = sqrt((2.0 * x_m)) * sqrt(x_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Sqrt[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

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

   1. unpow2 56.5%

      \[\leadsto \sqrt{2 \cdot \color{blue}{\left(x \cdot x\right)}} \]

   2. associate-*r* 56.5%

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot x\right) \cdot x}} \]

   3. sqrt-prod 52.0%

      \[\leadsto \color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{x}} \]

4. Applied egg-rr 52.0%

   \[\leadsto \color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{x}} \]

5. Final simplification 52.0%

   \[\leadsto \sqrt{2 \cdot x} \cdot \sqrt{x} \]
6. Add Preprocessing

Alternative 3: 99.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot \sqrt{2} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* x_m (sqrt 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * sqrt(2.0);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = x_m * sqrt(2.0d0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * Math.sqrt(2.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * math.sqrt(2.0)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * sqrt(2.0))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m * sqrt(2.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

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

   1. sqrt-prod 56.2%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{{x}^{2}}} \]

   2. unpow2 56.2%

      \[\leadsto \sqrt{2} \cdot \sqrt{\color{blue}{x \cdot x}} \]

   3. sqrt-prod 51.8%

      \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]

   4. add-sqr-sqrt 53.0%

      \[\leadsto \sqrt{2} \cdot \color{blue}{x} \]

4. Applied egg-rr 53.0%

   \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

5. Final simplification 53.0%

   \[\leadsto x \cdot \sqrt{2} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))