sqrt A (should all be same)

Percentage Accurate: 54.2% → 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: 54.2% 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 54.5%

   \[\sqrt{x \cdot x + x \cdot x} \]
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: 11.2% 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 54.5%

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

   1. count-2 54.5%

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

   2. sqrt-prod 54.3%

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

   3. sqrt-prod 50.7%

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

   4. add-sqr-sqrt 52.0%

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

3. Applied egg-rr 52.0%

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

   1. add-sqr-sqrt 50.7%

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

   2. associate-*r* 50.7%

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

   3. sqrt-prod 50.9%

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

   4. count-2 50.9%

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

   5. *-commutative 50.9%

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

   6. sqrt-unprod 54.5%

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

   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-+ 13.9%

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

5. Applied egg-rr 8.4%

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

   1. associate-+r+ 8.4%

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

   2. associate--l+ 8.4%

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

7. Applied egg-rr 8.4%

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

   1. +-commutative 8.4%

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

   2. associate-+l- 11.5%

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

   3. +-commutative 11.5%

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

9. Applied egg-rr 11.5%

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

10. Final simplification 11.5%

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

Alternative 3: 6.3% accurate, 35.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (+ x -2.0))

C

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

Fortran

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

Java

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

Python

def code(x):
	return x + -2.0

Julia

function code(x)
	return Float64(x + -2.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

   1. count-2 54.5%

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

   2. sqrt-prod 54.3%

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

   3. sqrt-prod 50.7%

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

   4. add-sqr-sqrt 52.0%

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

3. Applied egg-rr 52.0%

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

   1. add-sqr-sqrt 50.7%

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

   2. associate-*r* 50.7%

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

   3. sqrt-prod 50.9%

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

   4. count-2 50.9%

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

   5. *-commutative 50.9%

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

   6. sqrt-unprod 54.5%

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

   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-+ 13.9%

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

5. Applied egg-rr 8.4%

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

   1. add-exp-log 8.2%

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

   2. log1p-udef 8.2%

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

   3. expm1-udef 11.3%

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

   4. expm1-log1p-u 11.5%

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

   5. *-un-lft-identity 11.5%

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

   6. fma-def 11.5%

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

7. Applied egg-rr 11.5%

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

9. Simplified 6.5%

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

10. Final simplification 6.5%

    \[\leadsto x + -2 \]

Alternative 4: 11.2% 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 54.5%

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

2. Taylor expanded in x around 0 52.0%

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

4. Simplified 11.5%

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

5. Final simplification 11.5%

   \[\leadsto x + x \]

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 54.5%

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

   1. count-2 54.5%

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

   2. sqrt-prod 54.3%

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

   3. sqrt-prod 50.7%

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

   4. add-sqr-sqrt 52.0%

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

3. Applied egg-rr 52.0%

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

   1. add-sqr-sqrt 50.7%

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

   2. associate-*r* 50.7%

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

   3. sqrt-prod 50.9%

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

   4. count-2 50.9%

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

   5. *-commutative 50.9%

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

   6. sqrt-unprod 54.5%

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

   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-+ 13.9%

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

5. Applied egg-rr 11.5%

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

7. Simplified 3.8%

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

8. Final simplification 3.8%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x)
  :name "sqrt A (should all be same)"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))