sqrt A (should all be same)

Percentage Accurate: 53.9% → 100.0%

Time: 3.9s

Alternatives: 4

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: 53.9% 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 50.7%

   \[\sqrt{x \cdot x + x \cdot x} \]
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: 11.1% 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 50.7%

   \[\sqrt{x \cdot x + x \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 50.7%

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

5. Simplified 11.2%

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

   1. add-sqr-sqrt 10.2%

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

   2. sqrt-unprod 13.5%

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

   3. swap-sqr 13.5%

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

   4. metadata-eval 13.5%

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

   5. metadata-eval 13.5%

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

   6. swap-sqr 13.5%

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

   7. sqrt-unprod 10.2%

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

   8. add-sqr-sqrt 11.2%

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

   9. add-log-exp 4.4%

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

   10. exp-lft-sqr 4.4%

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

   11. log-prod 4.4%

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

   12. add-log-exp 7.0%

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

   13. add-log-exp 11.2%

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

7. Applied egg-rr 11.2%

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

Alternative 3: 11.1% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 50.7%

   \[\sqrt{x \cdot x + x \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 50.7%

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

5. Simplified 11.2%

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

   1. add-sqr-sqrt 10.2%

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

   2. sqrt-unprod 13.5%

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

   3. swap-sqr 13.5%

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

   4. metadata-eval 13.5%

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

   5. metadata-eval 13.5%

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

   6. swap-sqr 13.5%

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

   7. sqrt-unprod 10.2%

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

   8. add-sqr-sqrt 11.2%

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

   9. add-log-exp 4.4%

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

   10. exp-lft-sqr 4.4%

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

   11. log-prod 4.4%

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

   12. add-log-exp 7.0%

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

   13. add-log-exp 11.2%

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

7. Applied egg-rr 11.2%

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

   1. *-un-lft-identity 11.2%

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

   2. *-commutative 11.2%

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

   3. fma-define 11.2%

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

9. Applied egg-rr 11.2%

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

11. Simplified 6.9%

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

12. Taylor expanded in x around inf 11.2%

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

Alternative 4: 1.7% 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 50.7%

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

   1. flip-+ 0.0%

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

   2. difference-of-squares 0.0%

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

   3. associate-*r/ 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. flip-+ 6.8%

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

   7. sqrt-unprod 7.1%

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

   8. add-sqr-sqrt 7.1%

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

   9. add-log-exp 5.7%

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

   10. *-un-lft-identity 5.7%

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

   11. log-prod 5.7%

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

   12. metadata-eval 5.7%

       \[\leadsto \color{blue}{0} + \log \left(e^{x \cdot x + x \cdot x}\right) \]

   13. flip-+ 0.0%

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

   14. add-log-exp 0.0%

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

4. Applied egg-rr 0.0%

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

6. Simplified 1.6%

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

Reproduce

herbie shell --seed 2024144 
(FPCore (x)
  :name "sqrt A (should all be same)"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))