sqrt D (should all be same)

Percentage Accurate: 53.5% → 100.0%

Time: 1.8s

Alternatives: 4

Speedup: 4.9×

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: 53.5% 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: 100.0% accurate, 1.1× 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 56.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. count-2-rev N/A

      \[\leadsto \sqrt{\color{blue}{{x}^{2} + {x}^{2}}} \]

   4. lift-pow.f64 N/A

      \[\leadsto \sqrt{\color{blue}{{x}^{2}} + {x}^{2}} \]

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. 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: 99.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(-x\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + x} \cdot \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-310) (* (- x) (sqrt 2.0)) (* (sqrt (+ x x)) (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = -x * sqrt(2.0);
	} else {
		tmp = sqrt((x + x)) * sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-310)) then
        tmp = -x * sqrt(2.0d0)
    else
        tmp = sqrt((x + x)) * sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = -x * Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((x + x)) * Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1e-310:
		tmp = -x * math.sqrt(2.0)
	else:
		tmp = math.sqrt((x + x)) * math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-310)
		tmp = Float64(Float64(-x) * sqrt(2.0));
	else
		tmp = Float64(sqrt(Float64(x + x)) * sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e-310)
		tmp = -x * sqrt(2.0);
	else
		tmp = sqrt((x + x)) * sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1e-310], N[((-x) * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.999999999999969e-311

   1. Initial program 54.9%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sqrt.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -9.999999999999969e-311 < x

   1. Initial program 57.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. count-2-rev N/A

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

      8. pow1/2 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. count-2-rev N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. pow1/2 N/A

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

      19. lower-sqrt.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 99.6

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

   6. Applied rewrites 99.6%

      \[\leadsto \sqrt{\color{blue}{x + x}} \cdot \sqrt{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(-x\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-310) (* (- x) (sqrt 2.0)) (* (sqrt 2.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = -x * sqrt(2.0);
	} else {
		tmp = sqrt(2.0) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-310)) then
        tmp = -x * sqrt(2.0d0)
    else
        tmp = sqrt(2.0d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = -x * Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt(2.0) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1e-310:
		tmp = -x * math.sqrt(2.0)
	else:
		tmp = math.sqrt(2.0) * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-310)
		tmp = Float64(Float64(-x) * sqrt(2.0));
	else
		tmp = Float64(sqrt(2.0) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e-310)
		tmp = -x * sqrt(2.0);
	else
		tmp = sqrt(2.0) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1e-310], N[((-x) * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.999999999999969e-311

   1. Initial program 54.9%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-sqrt.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -9.999999999999969e-311 < x

   1. Initial program 57.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-pow.f64 N/A

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

      6. sqrt-pow1 N/A

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

      7. metadata-eval N/A

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

      8. unpow1 N/A

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

      9. lower-*.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 99.3

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

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 51.0% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-prod N/A

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

   4. pow1/2 N/A

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

   5. lift-pow.f64 N/A

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

   6. sqrt-pow1 N/A

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

   7. metadata-eval N/A

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

   8. unpow1 N/A

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

   9. lower-*.f64 N/A

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

   10. pow1/2 N/A

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

   11. lower-sqrt.f64 55.2

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

4. Applied rewrites 55.2%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))