sqrt A (should all be same)

Percentage Accurate: 54.3% → 99.4%

Time: 791.0ms

Alternatives: 3

Speedup: 1.3×

Specification

Math

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

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)
use fmin_fmax_functions
    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: 54.3% accurate, 1.0× speedup

Math

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

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)
use fmin_fmax_functions
    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: 99.4% accurate, 0.7× speedup

Math

\[\sqrt{\left|x + x\right|} \cdot \sqrt{\left|x\right|} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

3. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\sqrt{\left|x + x\right|} \cdot \sqrt{\left|x\right|}} \]
4. Add Preprocessing

Alternative 2: 99.3% accurate, 0.8× speedup

Math

\[\frac{2 \cdot \left|x\right|}{\sqrt{2}} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ (* 2 (fabs x)) (sqrt 2)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\sqrt{x \cdot x + x \cdot x} \]
2. 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. count-2 N/A

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

   4. sqrt-prod N/A

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

   5. lift-*.f64 N/A

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

   6. fabs-sqr N/A

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

   7. rem-sqrt-square-rev N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-fabs.f64 99.3%

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

3. Applied rewrites 99.3%

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

   1. rem-square-sqrt N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lower-*.f64 99.1%

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

5. Applied rewrites 99.1%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. rem-square-sqrt 99.3%

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

   5. lift-sqrt.f64 N/A

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

   6. pow1/2 N/A

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

   7. metadata-eval N/A

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

   8. pow-div N/A

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

   9. metadata-eval N/A

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

   10. pow1/2 N/A

       \[\leadsto \frac{2}{\color{blue}{\sqrt{2}}} \cdot \left|x\right| \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \frac{2}{\color{blue}{\sqrt{2}}} \cdot \left|x\right| \]

   12. add-sound-/ N/A

       \[\leadsto \color{blue}{\frac{2}{\sqrt{2}}} \cdot \left|x\right| \]

   13. lower-/.f64 99.1%

       \[\leadsto \color{blue}{\frac{2}{\sqrt{2}}} \cdot \left|x\right| \]

7. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\frac{2}{\sqrt{2}}} \cdot \left|x\right| \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{\sqrt{2}} \cdot \left|x\right|} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{\sqrt{2}}} \cdot \left|x\right| \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \left|x\right|}{\sqrt{2}}} \]

   4. add-sound-/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \left|x\right|}{\sqrt{2}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot \left|x\right|}{\sqrt{2}}} \]

   6. lower-*.f64 99.3%

      \[\leadsto \frac{\color{blue}{2 \cdot \left|x\right|}}{\sqrt{2}} \]

9. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{2 \cdot \left|x\right|}{\sqrt{2}}} \]
10. Add Preprocessing

Alternative 3: 99.3% accurate, 1.3× speedup

Math

\[\sqrt{2} \cdot \left|x\right| \]

FPCore

(FPCore (x)
  :precision binary64
  (* (sqrt 2) (fabs x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[2], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\sqrt{x \cdot x + x \cdot x} \]
2. 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. count-2 N/A

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

   4. sqrt-prod N/A

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

   5. lift-*.f64 N/A

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

   6. fabs-sqr N/A

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

   7. rem-sqrt-square-rev N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-fabs.f64 99.3%

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

3. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left|x\right|} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2025326 -o generate:taylor -o generate:evaluate
(FPCore (x)
  :name "sqrt A (should all be same)"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))