sqrt B (should all be same)

Percentage Accurate: 54.3% → 100.0%

Time: 13.9s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

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(Float64(2.0 * x) * x))
end

MATLAB

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

Wolfram

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

Initial Program: 54.3% accurate, 1.0× speedup

Math

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

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(Float64(2.0 * x) * x))
end

MATLAB

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

Wolfram

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

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

3. Taylor expanded in x around 0 46.8%

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

   1. rem-square-sqrt 45.2%

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

   2. fabs-sqr 45.2%

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

   3. rem-square-sqrt 99.5%

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

   4. rem-sqrt-square 54.6%

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

   5. swap-sqr 54.4%

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

   6. rem-square-sqrt 54.8%

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

   7. unpow2 54.8%

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

   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: 11.3% accurate, 35.0× 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 54.8%

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

   1. associate-*l* 54.8%

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

   2. sqrt-prod 54.6%

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

   3. sqrt-unprod 45.1%

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

   4. add-sqr-sqrt 46.8%

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

4. Applied egg-rr 46.8%

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

5. Taylor expanded in x around 0 46.8%

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

7. Simplified 12.0%

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

   1. add-log-exp 4.5%

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

   2. add-sqr-sqrt 4.5%

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

   3. log-prod 4.5%

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

9. Applied egg-rr 10.7%

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

Alternative 3: 5.4% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.25)

C

double code(double x) {
	return 0.25;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.25d0
end function

Java

public static double code(double x) {
	return 0.25;
}

Python

def code(x):
	return 0.25

Julia

function code(x)
	return 0.25
end

MATLAB

function tmp = code(x)
	tmp = 0.25;
end

Wolfram

code[x_] := 0.25

Derivation

1. Initial program 54.8%

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

   1. associate-*l* 54.8%

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

   2. sqrt-prod 54.6%

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

   3. sqrt-unprod 45.1%

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

   4. add-sqr-sqrt 46.8%

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

4. Applied egg-rr 46.8%

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

5. Taylor expanded in x around 0 46.8%

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

7. Simplified 12.0%

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

   1. add-log-exp 4.5%

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

   2. add-sqr-sqrt 4.5%

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

   3. log-prod 4.5%

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

9. Applied egg-rr 10.7%

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

11. Applied egg-rr 5.4%

    \[\leadsto \color{blue}{0.25} \]
12. Add Preprocessing

Alternative 4: 5.4% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ 0.1111111111111111 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.1111111111111111)

C

double code(double x) {
	return 0.1111111111111111;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.1111111111111111d0
end function

Java

public static double code(double x) {
	return 0.1111111111111111;
}

Python

def code(x):
	return 0.1111111111111111

Julia

function code(x)
	return 0.1111111111111111
end

MATLAB

function tmp = code(x)
	tmp = 0.1111111111111111;
end

Wolfram

code[x_] := 0.1111111111111111

Derivation

1. Initial program 54.8%

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

   1. associate-*l* 54.8%

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

   2. sqrt-prod 54.6%

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

   3. sqrt-unprod 45.1%

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

   4. add-sqr-sqrt 46.8%

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

4. Applied egg-rr 46.8%

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

5. Taylor expanded in x around 0 46.8%

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

7. Simplified 12.0%

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

   1. add-log-exp 4.5%

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

   2. add-sqr-sqrt 4.5%

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

   3. log-prod 4.5%

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

9. Applied egg-rr 10.7%

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

11. Applied egg-rr 5.4%

    \[\leadsto \color{blue}{0.1111111111111111} \]
12. Add Preprocessing

Alternative 5: 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%

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

   1. associate-*l* 54.8%

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

   2. sqrt-prod 54.6%

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

   3. sqrt-unprod 45.1%

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

   4. add-sqr-sqrt 46.8%

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

4. Applied egg-rr 46.8%

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

5. Taylor expanded in x around 0 46.8%

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

7. Simplified 12.0%

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

   1. add-log-exp 4.5%

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

   2. add-sqr-sqrt 4.5%

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

   3. log-prod 4.5%

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

9. Applied egg-rr 10.7%

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

11. Applied egg-rr 3.9%

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

Alternative 6: 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%

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

   1. associate-*l* 54.8%

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

   2. sqrt-prod 54.6%

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

   3. sqrt-unprod 45.1%

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

   4. add-sqr-sqrt 46.8%

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

4. Applied egg-rr 46.8%

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

   1. add-sqr-sqrt 45.1%

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

   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 7.0%

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

6. Applied egg-rr 7.0%

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

8. Simplified 1.7%

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

Reproduce

herbie shell --seed 2024119 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))