sqrt E (should all be same)

Percentage Accurate: 53.8% → 100.0%

Time: 2.4s

Alternatives: 5

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: 53.8% 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 55.1%

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

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = sqrt(2.0) * -x;
	} 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 = sqrt(2.0d0) * -x
    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 = Math.sqrt(2.0) * -x;
	} else {
		tmp = Math.sqrt(2.0) * x;
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-310)
		tmp = Float64(sqrt(2.0) * Float64(-x));
	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 = sqrt(2.0) * -x;
	else
		tmp = sqrt(2.0) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1e-310], N[(N[Sqrt[2.0], $MachinePrecision] * (-x)), $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 53.4%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg 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 99.3

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

   5. Applied rewrites 99.3%

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

   if -9.999999999999969e-311 < x

   1. Initial program 57.0%

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

   3. Taylor expanded in x around 0

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

      1. *-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 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 99.3%

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

Alternative 3: 5.7% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-206}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 4e-206) 0.0 (sqrt (+ x x))))

C

double code(double x) {
	double tmp;
	if (x <= 4e-206) {
		tmp = 0.0;
	} else {
		tmp = sqrt((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4d-206) then
        tmp = 0.0d0
    else
        tmp = sqrt((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4e-206) {
		tmp = 0.0;
	} else {
		tmp = Math.sqrt((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4e-206:
		tmp = 0.0
	else:
		tmp = math.sqrt((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e-206)
		tmp = 0.0;
	else
		tmp = sqrt(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4e-206)
		tmp = 0.0;
	else
		tmp = sqrt((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4e-206], 0.0, N[Sqrt[N[(x + x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000011e-206

   1. Initial program 47.8%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg 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 88.1

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

   5. Applied rewrites 88.1%

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

   7. Applied rewrites 87.9%

      \[\leadsto \left(-x\right) \cdot e^{\log 2 \cdot 0.5} \]
   8. Step-by-step derivation

   9. Applied rewrites 4.1%

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

   if 4.00000000000000011e-206 < x

   1. Initial program 65.4%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. distribute-lft-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 65.4

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

   4. Applied rewrites 65.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in 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 7.4

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

   6. Applied rewrites 7.4%

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

Alternative 4: 51.7% accurate, 13.5× 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 55.1%

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

3. Taylor expanded in x around 0

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

   1. *-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 49.2

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

5. Applied rewrites 49.2%

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

Alternative 5: 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 55.1%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
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. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg 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 52.2

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

5. Applied rewrites 52.2%

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

7. Applied rewrites 52.1%

   \[\leadsto \left(-x\right) \cdot e^{\log 2 \cdot 0.5} \]
8. Step-by-step derivation

9. Applied rewrites 3.7%

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

Reproduce

herbie shell --seed 2024268 
(FPCore (x)
  :name "sqrt E (should all be same)"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))