sqrt D (should all be same)

Percentage Accurate: 53.1% → 99.6%

Time: 6.5s

Alternatives: 4

Speedup: 7.3×

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: 53.1% 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, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {2}^{0.375} \cdot \left(x\_m \cdot {2}^{0.125}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (pow 2.0 0.375) (* x_m (pow 2.0 0.125))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(2.0, 0.375) * (x_m * pow(2.0, 0.125));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (2.0d0 ** 0.375d0) * (x_m * (2.0d0 ** 0.125d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(2.0, 0.375) * (x_m * Math.pow(2.0, 0.125));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(2.0, 0.375) * (x_m * math.pow(2.0, 0.125))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((2.0 ^ 0.375) * Float64(x_m * (2.0 ^ 0.125)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (2.0 ^ 0.375) * (x_m * (2.0 ^ 0.125));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[2.0, 0.375], $MachinePrecision] * N[(x$95$m * N[Power[2.0, 0.125], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \sqrt{\color{blue}{2 \cdot {x}^{2}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 55.4

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

5. Simplified 55.4%

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

   1. associate-*r* N/A

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

   2. sqrt-prod N/A

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

   3. pow1/2 N/A

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

   4. *-commutative N/A

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

   5. pow1/2 N/A

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

   6. sqr-pow N/A

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

   7. pow1/2 N/A

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

   8. sqrt-prod N/A

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

   9. rem-square-sqrt N/A

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

   10. sqr-pow N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 N/A

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

7. Applied egg-rr 52.1%

   \[\leadsto \color{blue}{{2}^{0.375} \cdot \left(x \cdot {2}^{0.125}\right)} \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 3.8× 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 55.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \sqrt{\color{blue}{2 \cdot {x}^{2}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 55.4

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

5. Simplified 55.4%

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

   1. pow1/2 N/A

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

   2. *-commutative N/A

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

   3. unpow-prod-down N/A

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

   4. *-lowering-*.f64 N/A

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

   5. pow1/2 N/A

      \[\leadsto \color{blue}{\sqrt{x \cdot 2}} \cdot {x}^{\frac{1}{2}} \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{x \cdot 2}} \cdot {x}^{\frac{1}{2}} \]

   7. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{2 \cdot x}} \cdot {x}^{\frac{1}{2}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{2 \cdot x}} \cdot {x}^{\frac{1}{2}} \]

   9. pow1/2 N/A

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

   10. sqrt-lowering-sqrt.f64 50.9

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

7. Applied egg-rr 50.9%

   \[\leadsto \color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{x}} \]
8. Add Preprocessing

Alternative 3: 99.3% accurate, 7.3× 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 55.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 51.9

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

5. Simplified 51.9%

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

Alternative 4: 20.3% accurate, 117.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

x_m = abs(x)
function code(x_m)
	return x_m
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := x$95$m

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 51.9

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

5. Simplified 51.9%

   \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. pow1/2 N/A

      \[\leadsto x \cdot \color{blue}{{2}^{\frac{1}{2}}} \]

   2. sqr-pow N/A

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

   3. pow2 N/A

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

   4. pow-lowering-pow.f64 N/A

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

   5. pow-lowering-pow.f64 N/A

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

   6. metadata-eval 51.8

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

7. Applied egg-rr 51.8%

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

9. Applied egg-rr 11.5%

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

10. Final simplification 11.5%

    \[\leadsto x \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024203 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))