sqrt A (should all be same)

Percentage Accurate: 54.1% → 100.0%

Time: 3.4s

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: 54.1% 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 52.7%

   \[\sqrt{x \cdot x + x \cdot x} \]
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: 11.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ x + \left(\left(x + 1\right) + -1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ x (+ (+ x 1.0) -1.0)))

C

double code(double x) {
	return x + ((x + 1.0) + -1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + ((x + 1.0d0) + (-1.0d0))
end function

Java

public static double code(double x) {
	return x + ((x + 1.0) + -1.0);
}

Python

def code(x):
	return x + ((x + 1.0) + -1.0)

Julia

function code(x)
	return Float64(x + Float64(Float64(x + 1.0) + -1.0))
end

MATLAB

function tmp = code(x)
	tmp = x + ((x + 1.0) + -1.0);
end

Wolfram

code[x_] := N[(x + N[(N[(x + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7%

   \[\sqrt{x \cdot x + x \cdot x} \]
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. Taylor expanded in x around 0 53.0%

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

   1. *-commutative 53.0%

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

   2. add-sqr-sqrt 51.8%

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

   3. associate-*r* 51.8%

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

   4. sqrt-prod 52.1%

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

   5. count-2 52.1%

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

   6. *-commutative 52.1%

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

   7. sqrt-unprod 52.7%

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

   8. flip-+ 0.0%

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

   9. +-inverses 0.0%

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

   10. +-inverses 0.0%

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

   11. associate-*r/ 0.0%

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

   12. +-inverses 0.0%

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

   13. +-inverses 0.0%

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

   14. distribute-lft-out-- 0.0%

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

   15. difference-of-squares 0.0%

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

   16. +-inverses 0.0%

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

   17. +-inverses 0.0%

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

   18. associate-*r/ 0.0%

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

   19. +-inverses 0.0%

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

   20. +-inverses 0.0%

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

   21. flip-+ 13.6%

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

6. Applied egg-rr 8.3%

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

   1. associate-+r+ 8.3%

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

   2. associate--l+ 8.3%

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

8. Applied egg-rr 8.3%

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

   1. +-commutative 8.3%

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

   2. associate-+l- 11.6%

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

10. Applied egg-rr 11.6%

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

11. Final simplification 11.6%

    \[\leadsto x + \left(\left(x + 1\right) + -1\right) \]

Alternative 3: 8.5% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.0) 0.0 (+ x -2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 0.0;
	} else {
		tmp = x + -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = 0.0d0
    else
        tmp = x + (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 0.0;
	} else {
		tmp = x + -2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = 0.0
	else:
		tmp = x + -2.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 2.0], 0.0, N[(x + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 53.4%

      \[\sqrt{x \cdot x + x \cdot x} \]
   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. Taylor expanded in x around 0 37.5%

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

      1. *-commutative 37.5%

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

      2. add-sqr-sqrt 36.0%

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

      3. associate-*r* 36.0%

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

      4. sqrt-prod 36.2%

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

      5. count-2 36.2%

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

      6. *-commutative 36.2%

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

      7. sqrt-unprod 53.4%

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

      8. flip-+ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. associate-*r/ 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. distribute-lft-out-- 0.0%

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

      15. difference-of-squares 0.0%

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

      16. +-inverses 0.0%

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

      17. +-inverses 0.0%

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

      18. associate-*r/ 0.0%

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

      19. +-inverses 0.0%

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

      20. +-inverses 0.0%

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

      21. flip-+ 13.7%

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

   6. Applied egg-rr 8.7%

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

   8. Simplified 4.4%

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

   if 2 < x

   1. Initial program 50.5%

      \[\sqrt{x \cdot x + x \cdot x} \]
   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. Taylor expanded in x around 0 99.4%

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

      1. *-commutative 99.4%

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

      2. add-sqr-sqrt 99.1%

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

      3. associate-*r* 99.2%

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

      4. sqrt-prod 99.6%

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

      5. count-2 99.6%

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

      6. *-commutative 99.6%

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

      7. sqrt-unprod 50.5%

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

      8. flip-+ 0.0%

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

      9. +-inverses 0.0%

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

      10. +-inverses 0.0%

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

      11. associate-*r/ 0.0%

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

      12. +-inverses 0.0%

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

      13. +-inverses 0.0%

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

      14. distribute-lft-out-- 0.0%

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

      15. difference-of-squares 0.0%

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

      16. +-inverses 0.0%

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

      17. +-inverses 0.0%

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

      18. associate-*r/ 0.0%

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

      19. +-inverses 0.0%

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

      20. +-inverses 0.0%

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

      21. flip-+ 13.2%

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

   6. Applied egg-rr 20.3%

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

      1. add-exp-log 20.3%

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

      2. log1p-udef 20.3%

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

      3. expm1-udef 20.3%

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

      4. expm1-log1p-u 20.3%

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

      5. *-un-lft-identity 20.3%

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

      6. fma-def 20.3%

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

   8. Applied egg-rr 20.3%

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

   10. Simplified 20.3%

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

4. Final simplification 8.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x + -2\\ \end{array} \]

Alternative 4: 11.3% 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 52.7%

   \[\sqrt{x \cdot x + x \cdot x} \]
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. Taylor expanded in x around 0 53.0%

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

   1. *-commutative 53.0%

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

   2. add-sqr-sqrt 51.8%

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

   3. associate-*r* 51.8%

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

   4. sqrt-prod 52.1%

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

   5. count-2 52.1%

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

   6. *-commutative 52.1%

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

   7. sqrt-unprod 52.7%

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

   8. flip-+ 0.0%

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

   9. +-inverses 0.0%

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

   10. +-inverses 0.0%

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

   11. associate-*r/ 0.0%

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

   12. +-inverses 0.0%

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

   13. +-inverses 0.0%

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

   14. distribute-lft-out-- 0.0%

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

   15. difference-of-squares 0.0%

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

   16. +-inverses 0.0%

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

   17. +-inverses 0.0%

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

   18. associate-*r/ 0.0%

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

   19. +-inverses 0.0%

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

   20. +-inverses 0.0%

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

   21. flip-+ 13.6%

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

6. Applied egg-rr 11.6%

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

7. Final simplification 11.6%

   \[\leadsto x + x \]

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 52.7%

   \[\sqrt{x \cdot x + x \cdot x} \]
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. Taylor expanded in x around 0 53.0%

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

   1. *-commutative 53.0%

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

   2. add-sqr-sqrt 51.8%

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

   3. associate-*r* 51.8%

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

   4. sqrt-prod 52.1%

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

   5. count-2 52.1%

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

   6. *-commutative 52.1%

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

   7. sqrt-unprod 52.7%

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

   8. flip-+ 0.0%

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

   9. +-inverses 0.0%

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

   10. +-inverses 0.0%

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

   11. associate-*r/ 0.0%

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

   12. +-inverses 0.0%

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

   13. +-inverses 0.0%

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

   14. distribute-lft-out-- 0.0%

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

   15. difference-of-squares 0.0%

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

   16. +-inverses 0.0%

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

   17. +-inverses 0.0%

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

   18. associate-*r/ 0.0%

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

   19. +-inverses 0.0%

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

   20. +-inverses 0.0%

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

   21. flip-+ 13.6%

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

6. Applied egg-rr 11.6%

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

8. Simplified 3.8%

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

9. Final simplification 3.8%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023278 
(FPCore (x)
  :name "sqrt A (should all be same)"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))