sqrt A (should all be same)

Percentage Accurate: 54.9% → 100.0%

Time: 2.3s

Alternatives: 5

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: 54.9% 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 58.6%

   \[\sqrt{x \cdot x + x \cdot x} \]
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: 8.7% accurate, 15.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2.0) {
		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 * x) <= 2.0d0) 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 * x) <= 2.0) {
		tmp = x * x;
	} else {
		tmp = x + -2.0;
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2.0)
		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 * x) <= 2.0)
		tmp = x * x;
	else
		tmp = x + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2

   1. Initial program 60.4%

      \[\sqrt{x \cdot x + x \cdot x} \]
   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-+ 6.9%

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

      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 2 < (*.f64 x x)

   1. Initial program 56.6%

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

   2. Taylor expanded in x around 0 49.2%

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

      1. add-sqr-sqrt 48.6%

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

      2. associate-*r* 48.7%

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

      3. sqrt-prod 48.9%

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

      4. count-2 48.9%

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

      5. *-commutative 48.9%

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

      6. sqrt-unprod 56.6%

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

      7. flip-+ 0.0%

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

      8. +-inverses 0.0%

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

      9. +-inverses 0.0%

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

      10. associate-*r/ 0.0%

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

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

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

      13. distribute-lft-out-- 0.0%

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

      14. difference-of-squares 0.0%

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

      15. +-inverses 0.0%

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

      16. +-inverses 0.0%

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

      17. associate-*r/ 0.0%

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

      18. +-inverses 0.0%

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

      19. +-inverses 0.0%

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

      20. flip-+ 14.0%

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

   4. Applied egg-rr 10.4%

      \[\leadsto \color{blue}{\left(1 + \left(x + x\right)\right) - 1} \]
   5. Step-by-step derivation

      1. add-exp-log 10.0%

         \[\leadsto \color{blue}{e^{\log \left(1 + \left(x + x\right)\right)}} - 1 \]

      2. log1p-udef 10.0%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(x + x\right)}} - 1 \]

      3. expm1-udef 10.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + x\right)\right)} \]

      4. expm1-log1p-u 10.4%

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

      5. *-un-lft-identity 10.4%

         \[\leadsto \color{blue}{1 \cdot x} + x \]

      6. fma-def 10.4%

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

   6. Applied egg-rr 10.4%

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

   8. Simplified 10.4%

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

4. Final simplification 8.7%

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

Alternative 3: 6.9% 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 58.6%

   \[\sqrt{x \cdot x + x \cdot x} \]
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 7.0%

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

   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 7.1%

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

6. Final simplification 7.1%

   \[\leadsto x \cdot x \]

Alternative 4: 11.1% accurate, 35.7× speedup

Math

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

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 58.6%

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

2. Taylor expanded in x around 0 49.2%

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

4. Simplified 11.0%

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

5. Final simplification 11.0%

   \[\leadsto x + x \]

Alternative 5: 3.8% accurate, 107.0× 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 58.6%

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

2. Taylor expanded in x around 0 49.2%

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

   1. add-sqr-sqrt 47.9%

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

   2. associate-*r* 47.9%

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

   3. sqrt-prod 48.1%

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

   4. count-2 48.1%

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

   5. *-commutative 48.1%

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

   6. sqrt-unprod 58.6%

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

   7. flip-+ 0.0%

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

   8. +-inverses 0.0%

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

   9. +-inverses 0.0%

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

   10. associate-*r/ 0.0%

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

   11. +-inverses 0.0%

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

   12. +-inverses 0.0%

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

   13. distribute-lft-out-- 0.0%

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

   14. difference-of-squares 0.0%

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

   15. +-inverses 0.0%

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

   16. +-inverses 0.0%

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

   17. associate-*r/ 0.0%

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

   18. +-inverses 0.0%

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

   19. +-inverses 0.0%

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

   20. flip-+ 14.4%

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

4. Applied egg-rr 11.0%

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

6. Simplified 3.8%

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

7. Final simplification 3.8%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023222 
(FPCore (x)
  :name "sqrt A (should all be same)"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))