sqrt A (should all be same)

Percentage Accurate: 54.4% → 100.0%

Time: 3.0s

Alternatives: 5

Speedup: 0.2×

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: 54.4% 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: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (hypot x x))

C

double code(double x) {
	return hypot(x, x);
}

Java

public static double code(double x) {
	return Math.hypot(x, x);
}

Python

def code(x):
	return math.hypot(x, x)

Julia

function code(x)
	return hypot(x, x)
end

MATLAB

function tmp = code(x)
	tmp = hypot(x, x);
end

Wolfram

code[x_] := N[Sqrt[x ^ 2 + x ^ 2], $MachinePrecision]

Derivation

1. Initial program 59.3%

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

   1. lower-hypot.f64 100.0

      \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]
5. Add Preprocessing

Alternative 2: 49.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.3%

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

   1. distribute-lft-out N/A

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

   2. sqrt-prod N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-+.f64 52.4

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

4. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{x + x}} \]
5. Add Preprocessing

Alternative 3: 50.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 53.5

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

5. Applied rewrites 53.5%

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

Alternative 4: 3.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.3%

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 59.3

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

4. Applied rewrites 59.3%

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

   1. count-2 N/A

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

   2. associate-*r* N/A

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

   3. count-2 N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

      \[\leadsto \sqrt{\frac{\color{blue}{0}}{x \cdot x - x \cdot x}} \]

   6. +-inverses N/A

      \[\leadsto \sqrt{\frac{\color{blue}{x \cdot x - x \cdot x}}{x \cdot x - x \cdot x}} \]

   7. +-inverses N/A

      \[\leadsto \sqrt{\frac{x \cdot x - x \cdot x}{\color{blue}{0}}} \]

   8. +-inverses N/A

      \[\leadsto \sqrt{\frac{x \cdot x - x \cdot x}{\color{blue}{x - x}}} \]

   9. flip-+ N/A

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

   10. lift-+.f64 N/A

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

   11. lift-sqrt.f64 3.8

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

6. Applied rewrites 3.8%

   \[\leadsto \color{blue}{\sqrt{x + x}} \]
7. Add Preprocessing

Alternative 5: 3.8% accurate, 24.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 59.3%

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 59.3

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

4. Applied rewrites 59.3%

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

   1. count-2 N/A

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

   2. associate-*r* N/A

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

   3. count-2 N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

      \[\leadsto \sqrt{\frac{\color{blue}{0}}{x \cdot x - x \cdot x}} \]

   6. +-inverses N/A

      \[\leadsto \sqrt{\frac{\color{blue}{x \cdot x - x \cdot x}}{x \cdot x - x \cdot x}} \]

   7. +-inverses N/A

      \[\leadsto \sqrt{\frac{x \cdot x - x \cdot x}{\color{blue}{0}}} \]

   8. +-inverses N/A

      \[\leadsto \sqrt{\frac{x \cdot x - x \cdot x}{\color{blue}{x - x}}} \]

   9. flip-+ N/A

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

   10. lift-+.f64 N/A

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

   11. lift-sqrt.f64 3.8

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

6. Applied rewrites 3.8%

   \[\leadsto \color{blue}{\sqrt{x + x}} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift-+.f64 N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

      \[\leadsto {\left(\frac{\color{blue}{0}}{x - x}\right)}^{\frac{1}{2}} \]

   6. metadata-eval N/A

      \[\leadsto {\left(\frac{\color{blue}{0 - 0}}{x - x}\right)}^{\frac{1}{2}} \]

   7. metadata-eval N/A

      \[\leadsto {\left(\frac{\color{blue}{0 \cdot 0} - 0}{x - x}\right)}^{\frac{1}{2}} \]

   8. metadata-eval N/A

      \[\leadsto {\left(\frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{x - x}\right)}^{\frac{1}{2}} \]

   9. +-inverses N/A

      \[\leadsto {\left(\frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}}\right)}^{\frac{1}{2}} \]

   10. metadata-eval N/A

       \[\leadsto {\left(\frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}}\right)}^{\frac{1}{2}} \]

   11. flip-- N/A

       \[\leadsto {\color{blue}{\left(0 - 0\right)}}^{\frac{1}{2}} \]

   12. metadata-eval N/A

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

   13. metadata-eval 3.9

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

8. Applied rewrites 3.9%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (x)
  :name "sqrt A (should all be same)"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))