sqrt C (should all be same)

Percentage Accurate: 54.3% → 100.0%

Time: 11.8s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* 2.0 (* x x))))

C

double code(double x) {
	return sqrt((2.0 * (x * x)));
}

Fortran

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

Java

public static double code(double x) {
	return Math.sqrt((2.0 * (x * x)));
}

Python

def code(x):
	return math.sqrt((2.0 * (x * x)))

Julia

function code(x)
	return sqrt(Float64(2.0 * Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((2.0 * (x * x)));
end

Wolfram

code[x_] := N[Sqrt[N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 54.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* 2.0 (* x x))))

C

double code(double x) {
	return sqrt((2.0 * (x * x)));
}

Fortran

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

Java

public static double code(double x) {
	return Math.sqrt((2.0 * (x * x)));
}

Python

def code(x):
	return math.sqrt((2.0 * (x * x)))

Julia

function code(x)
	return sqrt(Float64(2.0 * Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((2.0 * (x * x)));
end

Wolfram

code[x_] := N[Sqrt[N[(2.0 * 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 54.8%

   \[\sqrt{2 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 52.1%

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

   1. rem-square-sqrt 51.0%

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

   2. fabs-sqr 51.0%

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

   3. rem-square-sqrt 99.4%

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

   4. rem-sqrt-square 54.7%

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

   5. swap-sqr 54.4%

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

   6. unpow2 54.4%

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

   7. rem-square-sqrt 54.8%

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

   8. *-commutative 54.8%

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

   9. count-2 54.8%

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

   10. unpow2 54.8%

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

   11. unpow2 54.8%

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

   12. hypot-define 100.0%

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

5. Simplified 100.0%

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

Alternative 2: 7.0% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x 2.0)))

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * (x * 2.0)

Julia

function code(x)
	return Float64(x * Float64(x * 2.0))
end

MATLAB

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

Wolfram

code[x_] := N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

   \[\sqrt{2 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 52.1%

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

   1. rem-square-sqrt 51.0%

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

   2. fabs-sqr 51.0%

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

   3. rem-square-sqrt 99.4%

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

   4. rem-sqrt-square 54.7%

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

   5. swap-sqr 54.4%

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

   6. unpow2 54.4%

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

   7. rem-square-sqrt 54.8%

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

   8. *-commutative 54.8%

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

   9. count-2 54.8%

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

   10. unpow2 54.8%

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

   11. unpow2 54.8%

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

   12. hypot-define 100.0%

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

5. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]
6. Step-by-step derivation

   1. hypot-undefine 54.8%

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

   2. flip-+ 0.0%

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

   3. +-inverses 0.0%

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

   4. metadata-eval 0.0%

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

   5. +-inverses 0.0%

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

   6. +-inverses 0.0%

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

   7. +-inverses 0.0%

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

   8. metadata-eval 0.0%

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

   9. +-inverses 0.0%

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

   10. +-inverses 0.0%

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

   11. frac-times 0.0%

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

   12. flip-+ 0.0%

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

   13. flip-+ 6.7%

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

   14. sqrt-unprod 7.1%

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

   15. add-sqr-sqrt 7.1%

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

   16. count-2 7.1%

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

   17. associate-*r* 7.1%

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

7. Applied egg-rr 7.1%

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

8. Final simplification 7.1%

   \[\leadsto x \cdot \left(x \cdot 2\right) \]
9. Add Preprocessing

Alternative 3: 3.8% accurate, 105.0× 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 54.8%

   \[\sqrt{2 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sqrt-prod 54.6%

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

   2. sqrt-prod 51.1%

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

   3. add-sqr-sqrt 52.1%

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

4. Applied egg-rr 52.1%

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

5. Taylor expanded in x around 0 52.1%

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

7. Simplified 11.2%

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

   1. add-sqr-sqrt 9.9%

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

   2. sqrt-unprod 14.0%

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

   3. swap-sqr 14.0%

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

   4. sqrt-prod 14.0%

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

   5. sqrt-unprod 10.5%

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

   6. metadata-eval 10.5%

      \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \sqrt{\color{blue}{4}} \]

   7. metadata-eval 10.5%

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

   8. add-sqr-sqrt 11.3%

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

   9. add-log-exp 4.1%

      \[\leadsto \color{blue}{\log \left(e^{x \cdot 2}\right)} \]

   10. *-commutative 4.1%

       \[\leadsto \log \left(e^{\color{blue}{2 \cdot x}}\right) \]

   11. add-sqr-sqrt 4.1%

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

   12. sqrt-unprod 4.1%

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

9. Applied egg-rr 3.7%

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

Alternative 4: 1.7% accurate, 105.0× 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 54.8%

   \[\sqrt{2 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sqrt-prod 54.6%

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

   2. sqrt-prod 51.1%

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

   3. add-sqr-sqrt 52.1%

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

4. Applied egg-rr 52.1%

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

   1. add-sqr-sqrt 51.1%

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

   2. sqrt-unprod 54.6%

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

   3. sqrt-prod 54.8%

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

   4. count-2 54.8%

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

   5. flip-+ 0.0%

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

   6. +-inverses 0.0%

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

   7. metadata-eval 0.0%

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

   8. +-inverses 0.0%

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

   9. +-inverses 0.0%

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

   10. +-inverses 0.0%

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

   11. metadata-eval 0.0%

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

   12. +-inverses 0.0%

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

   13. +-inverses 0.0%

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

   14. frac-times 0.0%

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

   15. flip-+ 0.0%

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

   16. flip-+ 6.7%

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

   17. sqrt-unprod 7.1%

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

6. Applied egg-rr 7.1%

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

8. Simplified 1.6%

   \[\leadsto \color{blue}{-2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024118 
(FPCore (x)
  :name "sqrt C (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (* x x))))