sqrt A (should all be same)

Percentage Accurate: 53.8% → 99.5%

Time: 3.6s

Alternatives: 2

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out N/A

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

   6. sqrt-prod N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. count-2 N/A

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

   11. lower-*.f64 43.5

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

4. Applied rewrites 43.5%

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

   1. lift-*.f64 N/A

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

   2. count-2 N/A

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

   3. lower-+.f64 43.5

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

6. Applied rewrites 43.5%

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

7. Final simplification 43.5%

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

Alternative 2: 99.3% accurate, 1.5× 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 54.4%

   \[\sqrt{x \cdot x + x \cdot x} \]
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 44.8

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

5. Applied rewrites 44.8%

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

Reproduce

herbie shell --seed 2024249 
(FPCore (x)
  :name "sqrt A (should all be same)"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))