sqrt E (should all be same)

Percentage Accurate: 54.3% → 100.0%

Time: 2.5s

Alternatives: 4

Speedup: 9.0×

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (sqrt (+ (pow x 2.0) (pow x 2.0))))

C

double code(double x) {
	return sqrt((pow(x, 2.0) + pow(x, 2.0)));
}

Fortran

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

Java

public static double code(double x) {
	return Math.sqrt((Math.pow(x, 2.0) + Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.sqrt((math.pow(x, 2.0) + math.pow(x, 2.0)))

Julia

function code(x)
	return sqrt(Float64((x ^ 2.0) + (x ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((x ^ 2.0) + (x ^ 2.0)));
end

Wolfram

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

Initial Program: 54.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (sqrt (+ (pow x 2.0) (pow x 2.0))))

C

double code(double x) {
	return sqrt((pow(x, 2.0) + pow(x, 2.0)));
}

Fortran

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

Java

public static double code(double x) {
	return Math.sqrt((Math.pow(x, 2.0) + Math.pow(x, 2.0)));
}

Python

def code(x):
	return math.sqrt((math.pow(x, 2.0) + math.pow(x, 2.0)))

Julia

function code(x)
	return sqrt(Float64((x ^ 2.0) + (x ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((x ^ 2.0) + (x ^ 2.0)));
end

Wolfram

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

Alternative 1: 100.0% accurate, 2.1× 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 56.1%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. 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: 99.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \sqrt{2}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (sqrt 2.0)))) (if (<= x -5e-310) (- t_0) t_0)))

C

double code(double x) {
	double t_0 = x * sqrt(2.0);
	double tmp;
	if (x <= -5e-310) {
		tmp = -t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * sqrt(2.0d0)
    if (x <= (-5d-310)) then
        tmp = -t_0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * Math.sqrt(2.0);
	double tmp;
	if (x <= -5e-310) {
		tmp = -t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * math.sqrt(2.0)
	tmp = 0
	if x <= -5e-310:
		tmp = -t_0
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(x * sqrt(2.0))
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(-t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * sqrt(2.0);
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e-310], (-t$95$0), t$95$0]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 59.3%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   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. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -4.999999999999985e-310 < x

   1. Initial program 53.1%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 3: 51.0% accurate, 13.5× 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 56.1%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
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 52.2

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

5. Applied rewrites 52.2%

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

Alternative 4: 3.8% accurate, 216.0× 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 56.1%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. 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. Step-by-step derivation

   1. lift-hypot.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. flip-+ N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. +-inverses N/A

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

   11. metadata-eval N/A

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

   12. flip-- N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval 3.9

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

6. Applied rewrites 3.9%

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

Reproduce

herbie shell --seed 2024233 
(FPCore (x)
  :name "sqrt E (should all be same)"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))