sqrt E (should all be same)

Percentage Accurate: 53.8% → 99.4%

Time: 3.5s

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: 99.4% accurate, 5.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310) (* (sqrt 2.0) (- x)) (* (sqrt x) (sqrt (* 2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = sqrt(2.0) * -x;
	} else {
		tmp = sqrt(x) * sqrt((2.0 * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = sqrt(2.0d0) * -x
    else
        tmp = sqrt(x) * sqrt((2.0d0 * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = Math.sqrt(2.0) * -x;
	} else {
		tmp = Math.sqrt(x) * Math.sqrt((2.0 * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = math.sqrt(2.0) * -x
	else:
		tmp = math.sqrt(x) * math.sqrt((2.0 * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(sqrt(2.0) * Float64(-x));
	else
		tmp = Float64(sqrt(x) * sqrt(Float64(2.0 * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = sqrt(2.0) * -x;
	else
		tmp = sqrt(x) * sqrt((2.0 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[Sqrt[2.0], $MachinePrecision] * (-x)), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[Sqrt[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 55.4%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   if -1.999999999999994e-310 < x

   1. Initial program 53.2%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   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 53.2

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

   4. Applied rewrites 53.2%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow-prod-down N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 99.5%

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

4. Final simplification 99.4%

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

Alternative 2: 99.3% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310) (* (sqrt 2.0) (- x)) (* (sqrt 2.0) x)))

C

double code(double x) {
	double tmp;
	if (x <= -2e-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 <= (-2d-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 <= -2e-310) {
		tmp = Math.sqrt(2.0) * -x;
	} else {
		tmp = Math.sqrt(2.0) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2e-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 <= -2e-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 <= -2e-310)
		tmp = sqrt(2.0) * -x;
	else
		tmp = sqrt(2.0) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-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 < -1.999999999999994e-310

   1. Initial program 55.4%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

   if -1.999999999999994e-310 < x

   1. Initial program 53.2%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 99.4%

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

Alternative 3: 5.6% 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 51.7%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   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 92.2

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

   5. Applied rewrites 92.2%

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

   7. Applied rewrites 92.0%

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

   9. Applied rewrites 4.2%

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

   if 4.00000000000000011e-206 < x

   1. Initial program 58.6%

      \[\sqrt{{x}^{2} + {x}^{2}} \]
   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 58.6

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

   4. Applied rewrites 58.6%

      \[\leadsto \sqrt{\color{blue}{x \cdot \left(x + x\right)}} \]
   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. flip-+ N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. +-inverses N/A

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

      7. +-inverses N/A

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

      8. associate-*r/ N/A

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

      9. +-inverses N/A

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

      10. distribute-lft-out-- N/A

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

      11. +-inverses N/A

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

      12. flip-+ N/A

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

      13. lift-+.f64 7.2

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

   6. Applied rewrites 7.2%

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

Alternative 4: 50.4% 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 54.4%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
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 44.8

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

5. Applied rewrites 44.8%

   \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
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 54.4%

   \[\sqrt{{x}^{2} + {x}^{2}} \]
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 56.7

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

5. Applied rewrites 56.7%

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

7. Applied rewrites 56.6%

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

9. Applied rewrites 3.7%

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

Reproduce

herbie shell --seed 2024249 
(FPCore (x)
  :name "sqrt E (should all be same)"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))