sqrt B (should all be same)

Percentage Accurate: 53.8% → 100.0%

Time: 7.6s

Alternatives: 4

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: 53.8% 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 51.4%

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

   1. sqrt-prod 44.3%

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

4. Applied egg-rr 44.3%

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

   1. *-commutative 44.3%

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

6. Simplified 44.3%

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

7. Taylor expanded in x around 0 45.7%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

    \[\leadsto \mathsf{hypot}\left(x, x\right) \]
11. 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 51.4%

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

   1. sqrt-prod 44.3%

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

4. Applied egg-rr 44.3%

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

   1. *-commutative 44.3%

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

6. Simplified 44.3%

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

7. Taylor expanded in x around 0 45.7%

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

9. Simplified 100.0%

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

    1. hypot-udef 51.3%

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

    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. difference-of-squares 0.0%

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

    4. +-inverses 0.0%

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

    5. metadata-eval 0.0%

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

    6. +-inverses 0.0%

       \[\leadsto \sqrt{\frac{\left(x \cdot x + x \cdot x\right) \cdot \log 1}{\color{blue}{0}}} \]

    7. metadata-eval 0.0%

       \[\leadsto \sqrt{\frac{\left(x \cdot x + x \cdot x\right) \cdot \log 1}{\color{blue}{\log 1}}} \]

    8. associate-*r/ 0.0%

       \[\leadsto \sqrt{\color{blue}{\left(x \cdot x + x \cdot x\right) \cdot \frac{\log 1}{\log 1}}} \]

    9. metadata-eval 0.0%

       \[\leadsto \sqrt{\left(x \cdot x + x \cdot x\right) \cdot \frac{\color{blue}{0}}{\log 1}} \]

    10. +-inverses 0.0%

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

    11. metadata-eval 0.0%

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

    12. +-inverses 0.0%

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

    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 6.9%

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

    15. add-sqr-sqrt 6.9%

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

    16. distribute-lft-out 6.9%

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

    17. *-commutative 6.9%

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

    18. *-un-lft-identity 6.9%

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

    19. distribute-rgt-out 6.9%

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

    20. distribute-lft-out 6.9%

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

    21. metadata-eval 6.9%

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

11. Applied egg-rr 6.9%

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

12. Final simplification 6.9%

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

Alternative 3: 11.4% accurate, 35.0× speedup

Math

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

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 51.4%

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

   1. sqrt-prod 44.3%

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

4. Applied egg-rr 44.3%

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

   1. *-commutative 44.3%

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

6. Simplified 44.3%

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

7. Taylor expanded in x around 0 45.7%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around 0 45.7%

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

12. Simplified 12.3%

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

13. Final simplification 12.3%

    \[\leadsto x \cdot -2 \]
14. 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 51.4%

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

   1. sqrt-prod 44.3%

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

4. Applied egg-rr 44.3%

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

   1. *-commutative 44.3%

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

6. Simplified 44.3%

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

7. Taylor expanded in x around 0 45.7%

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

9. Simplified 100.0%

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

    1. hypot-udef 51.3%

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

    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. difference-of-squares 0.0%

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

    4. +-inverses 0.0%

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

    5. metadata-eval 0.0%

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

    6. +-inverses 0.0%

       \[\leadsto \sqrt{\frac{\left(x \cdot x + x \cdot x\right) \cdot \log 1}{\color{blue}{0}}} \]

    7. metadata-eval 0.0%

       \[\leadsto \sqrt{\frac{\left(x \cdot x + x \cdot x\right) \cdot \log 1}{\color{blue}{\log 1}}} \]

    8. associate-*r/ 0.0%

       \[\leadsto \sqrt{\color{blue}{\left(x \cdot x + x \cdot x\right) \cdot \frac{\log 1}{\log 1}}} \]

    9. metadata-eval 0.0%

       \[\leadsto \sqrt{\left(x \cdot x + x \cdot x\right) \cdot \frac{\color{blue}{0}}{\log 1}} \]

    10. +-inverses 0.0%

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

    11. metadata-eval 0.0%

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

    12. +-inverses 0.0%

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

    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 6.9%

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

    15. add-sqr-sqrt 6.9%

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

    16. pow2 6.9%

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

    17. pow2 6.9%

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

11. Applied egg-rr 6.9%

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

13. Simplified 1.7%

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

14. Final simplification 1.7%

    \[\leadsto -2 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))