sqrt D (should all be same)

Percentage Accurate: 53.7% → 99.4%

Time: 4.8s

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: 53.7% 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: 99.4% 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 56.3%

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

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

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot {x}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({x}^{2}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(x \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 56.3%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

3. Simplified 56.3%

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

   1. pow1/2 N/A

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

   2. associate-*r* N/A

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

   3. unpow-prod-down N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left({\left(2 \cdot x\right)}^{\frac{1}{2}}\right), \color{blue}{\left({x}^{\frac{1}{2}}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot x\right), \frac{1}{2}\right), \left({\color{blue}{x}}^{\frac{1}{2}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, x\right), \frac{1}{2}\right), \left({x}^{\frac{1}{2}}\right)\right) \]

   7. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, x\right), \frac{1}{2}\right), \left(\sqrt{x}\right)\right) \]

   8. sqrt-lowering-sqrt.f64 49.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, x\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(x\right)\right) \]

6. Applied egg-rr 49.0%

   \[\leadsto \color{blue}{{\left(2 \cdot x\right)}^{0.5} \cdot \sqrt{x}} \]
7. Step-by-step derivation

   1. unpow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2 \cdot x}\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot x\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right) \]

   3. *-lowering-*.f64 49.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right) \]

8. Applied egg-rr 49.0%

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

Alternative 2: 99.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{x\_m}{{4}^{-0.25}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ x_m (pow 4.0 -0.25)))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m / pow(4.0, -0.25);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = x_m / (4.0d0 ** (-0.25d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m / Math.pow(4.0, -0.25);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m / math.pow(4.0, -0.25)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m / (4.0 ^ -0.25))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m / (4.0 ^ -0.25);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m / N[Power[4.0, -0.25], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.3%

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

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

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot {x}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({x}^{2}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(x \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 56.3%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

3. Simplified 56.3%

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

   1. pow1/2 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

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

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(2 \cdot \left(x \cdot x\right)\right) \cdot \left(2 \cdot \left(x \cdot x\right)\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(x \cdot x\right) \cdot 2\right) \cdot \left(2 \cdot \left(x \cdot x\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(x \cdot x\right) \cdot \left(2 \cdot \left(2 \cdot \left(x \cdot x\right)\right)\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(x \cdot x\right) \cdot \left(\left(2 \cdot 2\right) \cdot \left(x \cdot x\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   8. swap-sqr N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(x \cdot x\right) \cdot \left(\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

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

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left(\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

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

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(2 \cdot x\right) \cdot \left(2 \cdot x\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot 2\right) \cdot \left(2 \cdot x\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot 2\right) \cdot \left(x \cdot 2\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   13. swap-sqr N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \left(2 \cdot 2\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

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

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(2 \cdot 2\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

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

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(2 \cdot 2\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 4\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   17. metadata-eval 31.7%

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 4\right)\right), \frac{1}{4}\right) \]

6. Applied egg-rr 31.7%

   \[\leadsto \color{blue}{{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 4\right)\right)}^{0.25}} \]

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{4}^{\frac{1}{4}} \cdot x} \]
8. Step-by-step derivation

   1. /-rgt-identity N/A

      \[\leadsto \frac{{4}^{\frac{1}{4}} \cdot x}{\color{blue}{1}} \]

   2. associate-*r/ N/A

      \[\leadsto {4}^{\frac{1}{4}} \cdot \color{blue}{\frac{x}{1}} \]

   3. exp-to-pow N/A

      \[\leadsto e^{\log 4 \cdot \frac{1}{4}} \cdot \frac{\color{blue}{x}}{1} \]

   4. *-commutative N/A

      \[\leadsto e^{\frac{1}{4} \cdot \log 4} \cdot \frac{x}{1} \]

   5. remove-double-neg N/A

      \[\leadsto e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot \log 4\right)\right)\right)} \cdot \frac{x}{1} \]

   6. exp-neg N/A

      \[\leadsto \frac{1}{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \log 4\right)}} \cdot \frac{\color{blue}{x}}{1} \]

   7. times-frac N/A

      \[\leadsto \frac{1 \cdot x}{\color{blue}{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \log 4\right)} \cdot 1}} \]

   8. *-lft-identity N/A

      \[\leadsto \frac{x}{\color{blue}{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \log 4\right)}} \cdot 1} \]

   9. *-rgt-identity N/A

      \[\leadsto \frac{x}{e^{\mathsf{neg}\left(\frac{1}{4} \cdot \log 4\right)}} \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(e^{\mathsf{neg}\left(\frac{1}{4} \cdot \log 4\right)}\right)}\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(e^{\mathsf{neg}\left(\log 4 \cdot \frac{1}{4}\right)}\right)\right) \]

   12. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(e^{\log 4 \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(e^{\log 4 \cdot \frac{-1}{4}}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(e^{\log 4 \cdot \left(\frac{1}{4} \cdot -1\right)}\right)\right) \]

   15. exp-to-pow N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left({4}^{\color{blue}{\left(\frac{1}{4} \cdot -1\right)}}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{pow.f64}\left(4, \color{blue}{\left(\frac{1}{4} \cdot -1\right)}\right)\right) \]

   17. metadata-eval 49.9%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{pow.f64}\left(4, \frac{-1}{4}\right)\right) \]

9. Simplified 49.9%

   \[\leadsto \color{blue}{\frac{x}{{4}^{-0.25}}} \]
10. 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 56.3%

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

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

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot {x}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({x}^{2}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(x \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 56.3%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

3. Simplified 56.3%

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

   1. sqrt-prod N/A

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

   2. pow1/2 N/A

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left({2}^{\frac{1}{2}}\right), \color{blue}{x}\right) \]

   6. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2}\right), x\right) \]

   7. sqrt-lowering-sqrt.f64 49.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), x\right) \]

6. Applied egg-rr 49.9%

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

7. Final simplification 49.9%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))