sqrt A (should all be same)

Percentage Accurate: 54.3% → 100.0%

Time: 5.2s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

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)
    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

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

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)
    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: 100.0% accurate, 1.0× 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 53.5%

   \[\sqrt{x \cdot x + x \cdot x} \]
2. Step-by-step derivation

   1. hypot-def 100.0%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

4. Final simplification 100.0%

   \[\leadsto \mathsf{hypot}\left(x, x\right) \]

Alternative 2: 20.3% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-x\right) - x\\ \mathbf{else}:\\ \;\;\;\;x + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -5e-310) (- (- x) x) (+ x x)))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -x - x;
	} else {
		tmp = x + x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = -x - x
    else
        tmp = x + x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -x - x;
	} else {
		tmp = x + x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = -x - x
	else:
		tmp = x + x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(-x) - x);
	else
		tmp = Float64(x + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -x - x;
	else
		tmp = x + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[((-x) - x), $MachinePrecision], N[(x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 56.6%

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

   2. Taylor expanded in x around 0 2.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-prod 56.2%

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

      3. sqrt-prod 56.6%

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

   4. Applied egg-rr 56.6%

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

   6. Applied egg-rr 2.4%

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

   8. Applied egg-rr 20.4%

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

   if -4.999999999999985e-310 < x

   1. Initial program 50.2%

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

   2. Taylor expanded in x around 0 99.2%

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

   4. Simplified 20.3%

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

4. Final simplification 20.3%

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

Alternative 3: 10.3% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.9:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 3.9) (* x x) (+ x -2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 3.9) {
		tmp = x * x;
	} else {
		tmp = x + -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.9d0) then
        tmp = x * x
    else
        tmp = x + (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.9) {
		tmp = x * x;
	} else {
		tmp = x + -2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.9:
		tmp = x * x
	else:
		tmp = x + -2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.9)
		tmp = Float64(x * x);
	else
		tmp = Float64(x + -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.9)
		tmp = x * x;
	else
		tmp = x + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.9], N[(x * x), $MachinePrecision], N[(x + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.89999999999999991

   1. Initial program 54.9%

      \[\sqrt{x \cdot x + x \cdot x} \]
   2. Step-by-step derivation

      1. flip-+ 0.0%

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

      2. difference-of-squares 0.0%

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

      3. associate-*r/ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. flip-+ 6.7%

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

      7. sqrt-unprod 7.0%

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

      8. add-sqr-sqrt 7.0%

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

      9. *-un-lft-identity 7.0%

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

      10. fma-def 7.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(1, x \cdot x, x \cdot x\right)} \]

   3. Applied egg-rr 7.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, x \cdot x, x \cdot x\right)} \]

   5. Simplified 7.0%

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

   if 3.89999999999999991 < x

   1. Initial program 49.6%

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

   2. Taylor expanded in x around 0 99.2%

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

      1. add-sqr-sqrt 98.7%

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

      2. sqrt-prod 49.3%

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

      3. sqrt-prod 49.6%

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

   4. Applied egg-rr 49.6%

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

   6. Applied egg-rr 20.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, x, x\right)} \]

   8. Simplified 20.3%

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

4. Final simplification 10.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + -2\\ \end{array} \]

Alternative 4: 13.8% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -5e-310) (* x x) (+ x x)))

C

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

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = x * x;
	} else {
		tmp = x + x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e-310:
		tmp = x * x
	else:
		tmp = x + x
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(x * x), $MachinePrecision], N[(x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 56.6%

      \[\sqrt{x \cdot x + x \cdot x} \]
   2. Step-by-step derivation

      1. flip-+ 0.0%

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

      2. difference-of-squares 0.0%

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

      3. associate-*r/ 0.0%

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

      4. +-inverses 0.0%

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

      5. +-inverses 0.0%

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

      6. flip-+ 6.7%

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

      7. sqrt-unprod 7.1%

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

      8. add-sqr-sqrt 7.1%

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

      9. *-un-lft-identity 7.1%

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

      10. fma-def 7.1%

          \[\leadsto \color{blue}{\mathsf{fma}\left(1, x \cdot x, x \cdot x\right)} \]

   3. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, x \cdot x, x \cdot x\right)} \]

   5. Simplified 7.1%

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

   if -4.999999999999985e-310 < x

   1. Initial program 50.2%

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

   2. Taylor expanded in x around 0 99.2%

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

   4. Simplified 20.3%

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

4. Final simplification 13.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + x\\ \end{array} \]

Alternative 5: 7.0% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return x * x;
}

Python

def code(x):
	return x * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

   \[\sqrt{x \cdot x + x \cdot x} \]
2. Step-by-step derivation

   1. flip-+ 0.0%

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

   2. difference-of-squares 0.0%

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

   3. associate-*r/ 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. flip-+ 6.7%

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

   7. sqrt-unprod 7.0%

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

   8. add-sqr-sqrt 7.0%

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

   9. *-un-lft-identity 7.0%

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

   10. fma-def 7.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(1, x \cdot x, x \cdot x\right)} \]

3. Applied egg-rr 7.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1, x \cdot x, x \cdot x\right)} \]

5. Simplified 7.0%

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

6. Final simplification 7.0%

   \[\leadsto x \cdot x \]

Alternative 6: 1.7% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.0)

C

double code(double x) {
	return -2.0;
}

Fortran

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

Java

public static double code(double x) {
	return -2.0;
}

Python

def code(x):
	return -2.0

Julia

function code(x)
	return -2.0
end

MATLAB

function tmp = code(x)
	tmp = -2.0;
end

Wolfram

code[x_] := -2.0

Derivation

1. Initial program 53.5%

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

2. Taylor expanded in x around 0 48.6%

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

   1. add-sqr-sqrt 47.0%

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

   2. sqrt-prod 53.2%

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

   3. sqrt-prod 53.5%

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

4. Applied egg-rr 53.5%

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

5. Taylor expanded in x around 0 48.6%

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

7. Simplified 1.7%

   \[\leadsto \color{blue}{-2} \]

8. Final simplification 1.7%

   \[\leadsto -2 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "sqrt A (should all be same)"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))