sqrt A (should all be same)

Percentage Accurate: 54.7% → 100.0%

Time: 21.5s

Alternatives: 6

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: 54.7% 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 53.1%

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

   1. hypot-define 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. Add Preprocessing
5. Add Preprocessing

Alternative 2: 7.0% accurate, 21.4× 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 53.1%

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

   1. hypot-define 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. Add Preprocessing
5. Step-by-step derivation

   1. hypot-undefine 53.1%

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

   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%

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

6. Applied egg-rr 6.9%

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

7. Final simplification 6.9%

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

Alternative 3: 5.4% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 64 \end{array} \]

FPCore

(FPCore (x) :precision binary64 64.0)

C

double code(double x) {
	return 64.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 64.0d0
end function

Java

public static double code(double x) {
	return 64.0;
}

Python

def code(x):
	return 64.0

Julia

function code(x)
	return 64.0
end

MATLAB

function tmp = code(x)
	tmp = 64.0;
end

Wolfram

code[x_] := 64.0

Derivation

1. Initial program 53.1%

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

   1. hypot-define 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. Add Preprocessing

6. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{{\left(\frac{0}{0}\right)}^{6}} \]

8. Simplified 5.3%

   \[\leadsto \color{blue}{64} \]
9. Add Preprocessing

Alternative 4: 5.4% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 32 \end{array} \]

FPCore

(FPCore (x) :precision binary64 32.0)

C

double code(double x) {
	return 32.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 32.0d0
end function

Java

public static double code(double x) {
	return 32.0;
}

Python

def code(x):
	return 32.0

Julia

function code(x)
	return 32.0
end

MATLAB

function tmp = code(x)
	tmp = 32.0;
end

Wolfram

code[x_] := 32.0

Derivation

1. Initial program 53.1%

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

   1. hypot-define 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. Add Preprocessing

6. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{{\left(\frac{0}{0}\right)}^{5}} \]

8. Simplified 5.3%

   \[\leadsto \color{blue}{32} \]
9. Add Preprocessing

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 53.1%

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

   1. hypot-define 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. Add Preprocessing

5. Taylor expanded in x around 0 54.1%

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

7. Simplified 10.8%

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

   1. add-sqr-sqrt 9.5%

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

   2. sqrt-unprod 13.7%

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

   3. swap-sqr 13.7%

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

   4. metadata-eval 13.7%

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

   5. metadata-eval 13.7%

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

   6. swap-sqr 13.7%

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

   7. sqrt-unprod 10.9%

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

   8. add-log-exp 3.2%

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

   9. add-sqr-sqrt 4.5%

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

   10. add-sqr-sqrt 4.5%

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

   11. sqrt-unprod 4.5%

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

   12. add-sqr-sqrt 3.2%

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

   13. sqrt-unprod 4.4%

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

   14. swap-sqr 4.4%

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

   15. metadata-eval 4.4%

       \[\leadsto \log \left(\sqrt{e^{\sqrt{\left(x \cdot x\right) \cdot \color{blue}{4}}} \cdot e^{x \cdot 2}}\right) \]

   16. metadata-eval 4.4%

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

   17. swap-sqr 4.4%

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

   18. sqrt-unprod 1.2%

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

   19. add-sqr-sqrt 2.7%

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

   20. exp-prod 2.8%

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

   21. exp-prod 2.8%

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

9. Applied egg-rr 3.9%

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

Alternative 6: 1.6% accurate, 107.0× 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 53.1%

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

   1. hypot-define 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. Add Preprocessing
5. Step-by-step derivation

   1. hypot-undefine 53.1%

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

   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%

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

6. Applied egg-rr 6.9%

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

8. Simplified 1.7%

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

Reproduce

herbie shell --seed 2024123 
(FPCore (x)
  :name "sqrt A (should all be same)"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))