sqrt C (should all be same)

Percentage Accurate: 54.2% → 99.3%

Time: 2.6s

Alternatives: 2

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

   \[\sqrt{2 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 45.9

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

5. Applied rewrites 45.9%

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

Alternative 2: 3.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 53.5%

   \[\sqrt{2 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. count-2-rev N/A

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

   5. lift-*.f64 N/A

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

   6. sqr-neg-rev N/A

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

   7. cancel-sub-sign N/A

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

   8. unpow1 N/A

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

   9. metadata-eval N/A

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

   10. sqrt-pow1 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. rem-square-sqrt N/A

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

   14. sqrt-prod N/A

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

   15. sqr-neg-rev N/A

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

   16. lift-*.f64 N/A

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

   17. rem-square-sqrt N/A

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

   18. +-inverses N/A

       \[\leadsto {\color{blue}{0}}^{\frac{1}{2}} \]

   19. metadata-eval 3.8

       \[\leadsto \color{blue}{0} \]

4. Applied rewrites 3.8%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(FPCore (x)
  :name "sqrt C (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (* x x))))