sqrt B (should all be same)

Percentage Accurate: 53.7% → 99.4%

Time: 9.4s

Alternatives: 3

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: 53.7% 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {x\_m}^{0.75} \cdot {\left(x\_m \cdot 4\right)}^{0.25} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (pow x_m 0.75) (pow (* x_m 4.0) 0.25)))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(x_m, 0.75) * pow((x_m * 4.0), 0.25);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (x_m ** 0.75d0) * ((x_m * 4.0d0) ** 0.25d0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(x_m, 0.75) * Math.pow((x_m * 4.0), 0.25);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(x_m, 0.75) * math.pow((x_m * 4.0), 0.25)

Julia

x_m = abs(x)
function code(x_m)
	return Float64((x_m ^ 0.75) * (Float64(x_m * 4.0) ^ 0.25))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m ^ 0.75) * ((x_m * 4.0) ^ 0.25);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[x$95$m, 0.75], $MachinePrecision] * N[Power[N[(x$95$m * 4.0), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

   1. add-cbrt-cube 38.1%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\left(2 \cdot x\right) \cdot x} \cdot \sqrt{\left(2 \cdot x\right) \cdot x}\right) \cdot \sqrt{\left(2 \cdot x\right) \cdot x}}} \]

   2. pow3 38.1%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{\left(2 \cdot x\right) \cdot x}\right)}^{3}}} \]

   3. pow1/2 38.1%

      \[\leadsto \sqrt[3]{{\color{blue}{\left({\left(\left(2 \cdot x\right) \cdot x\right)}^{0.5}\right)}}^{3}} \]

   4. associate-*l* 38.1%

      \[\leadsto \sqrt[3]{{\left({\color{blue}{\left(2 \cdot \left(x \cdot x\right)\right)}}^{0.5}\right)}^{3}} \]

   5. *-commutative 38.1%

      \[\leadsto \sqrt[3]{{\left({\color{blue}{\left(\left(x \cdot x\right) \cdot 2\right)}}^{0.5}\right)}^{3}} \]

   6. unpow-prod-down 38.1%

      \[\leadsto \sqrt[3]{{\color{blue}{\left({\left(x \cdot x\right)}^{0.5} \cdot {2}^{0.5}\right)}}^{3}} \]

   7. pow1/2 38.1%

      \[\leadsto \sqrt[3]{{\left(\color{blue}{\sqrt{x \cdot x}} \cdot {2}^{0.5}\right)}^{3}} \]

   8. sqrt-unprod 19.5%

      \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {2}^{0.5}\right)}^{3}} \]

   9. add-sqr-sqrt 20.7%

      \[\leadsto \sqrt[3]{{\left(\color{blue}{x} \cdot {2}^{0.5}\right)}^{3}} \]

   10. pow1/2 20.7%

       \[\leadsto \sqrt[3]{{\left(x \cdot \color{blue}{\sqrt{2}}\right)}^{3}} \]

4. Applied egg-rr 20.7%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(x \cdot \sqrt{2}\right)}^{3}}} \]
5. Step-by-step derivation

   1. rem-cbrt-cube 50.3%

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

   2. add-sqr-sqrt 49.1%

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

   3. associate-*r* 49.1%

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

   4. sqrt-prod 49.3%

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

   5. *-commutative 49.3%

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

   6. add-sqr-sqrt 49.1%

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

   7. associate-*l* 49.1%

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

   8. pow1/2 49.1%

      \[\leadsto \sqrt{\color{blue}{{\left(x \cdot 2\right)}^{0.5}}} \cdot \left(\sqrt{\sqrt{x \cdot 2}} \cdot \sqrt{x}\right) \]

   9. sqrt-pow1 49.2%

      \[\leadsto \color{blue}{{\left(x \cdot 2\right)}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{x \cdot 2}} \cdot \sqrt{x}\right) \]

   10. metadata-eval 49.2%

       \[\leadsto {\left(x \cdot 2\right)}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{x \cdot 2}} \cdot \sqrt{x}\right) \]

   11. pow1/2 49.2%

       \[\leadsto {\left(x \cdot 2\right)}^{0.25} \cdot \left(\sqrt{\color{blue}{{\left(x \cdot 2\right)}^{0.5}}} \cdot \sqrt{x}\right) \]

   12. sqrt-pow1 49.2%

       \[\leadsto {\left(x \cdot 2\right)}^{0.25} \cdot \left(\color{blue}{{\left(x \cdot 2\right)}^{\left(\frac{0.5}{2}\right)}} \cdot \sqrt{x}\right) \]

   13. metadata-eval 49.2%

       \[\leadsto {\left(x \cdot 2\right)}^{0.25} \cdot \left({\left(x \cdot 2\right)}^{\color{blue}{0.25}} \cdot \sqrt{x}\right) \]

6. Applied egg-rr 49.2%

   \[\leadsto \color{blue}{{\left(x \cdot 2\right)}^{0.25} \cdot \left({\left(x \cdot 2\right)}^{0.25} \cdot \sqrt{x}\right)} \]

7. Taylor expanded in x around 0 45.0%

   \[\leadsto {\left(x \cdot 2\right)}^{0.25} \cdot \color{blue}{e^{0.25 \cdot \left(\log 2 + 3 \cdot \log x\right)}} \]
8. Step-by-step derivation

   1. distribute-rgt-in 45.0%

      \[\leadsto {\left(x \cdot 2\right)}^{0.25} \cdot e^{\color{blue}{\log 2 \cdot 0.25 + \left(3 \cdot \log x\right) \cdot 0.25}} \]

   2. *-commutative 45.0%

      \[\leadsto {\left(x \cdot 2\right)}^{0.25} \cdot e^{\log 2 \cdot 0.25 + \color{blue}{0.25 \cdot \left(3 \cdot \log x\right)}} \]

   3. prod-exp 45.4%

      \[\leadsto {\left(x \cdot 2\right)}^{0.25} \cdot \color{blue}{\left(e^{\log 2 \cdot 0.25} \cdot e^{0.25 \cdot \left(3 \cdot \log x\right)}\right)} \]

   4. exp-to-pow 45.4%

      \[\leadsto {\left(x \cdot 2\right)}^{0.25} \cdot \left(\color{blue}{{2}^{0.25}} \cdot e^{0.25 \cdot \left(3 \cdot \log x\right)}\right) \]

   5. associate-*r* 45.4%

      \[\leadsto {\left(x \cdot 2\right)}^{0.25} \cdot \left({2}^{0.25} \cdot e^{\color{blue}{\left(0.25 \cdot 3\right) \cdot \log x}}\right) \]

   6. metadata-eval 45.4%

      \[\leadsto {\left(x \cdot 2\right)}^{0.25} \cdot \left({2}^{0.25} \cdot e^{\color{blue}{0.75} \cdot \log x}\right) \]

   7. *-commutative 45.4%

      \[\leadsto {\left(x \cdot 2\right)}^{0.25} \cdot \left({2}^{0.25} \cdot e^{\color{blue}{\log x \cdot 0.75}}\right) \]

   8. exp-to-pow 49.3%

      \[\leadsto {\left(x \cdot 2\right)}^{0.25} \cdot \left({2}^{0.25} \cdot \color{blue}{{x}^{0.75}}\right) \]

9. Simplified 49.3%

   \[\leadsto {\left(x \cdot 2\right)}^{0.25} \cdot \color{blue}{\left({2}^{0.25} \cdot {x}^{0.75}\right)} \]
10. Step-by-step derivation

    1. add-log-exp 4.0%

       \[\leadsto \color{blue}{\log \left(e^{{\left(x \cdot 2\right)}^{0.25} \cdot \left({2}^{0.25} \cdot {x}^{0.75}\right)}\right)} \]

    2. *-un-lft-identity 4.0%

       \[\leadsto \log \color{blue}{\left(1 \cdot e^{{\left(x \cdot 2\right)}^{0.25} \cdot \left({2}^{0.25} \cdot {x}^{0.75}\right)}\right)} \]

    3. log-prod 4.0%

       \[\leadsto \color{blue}{\log 1 + \log \left(e^{{\left(x \cdot 2\right)}^{0.25} \cdot \left({2}^{0.25} \cdot {x}^{0.75}\right)}\right)} \]

    4. metadata-eval 4.0%

       \[\leadsto \color{blue}{0} + \log \left(e^{{\left(x \cdot 2\right)}^{0.25} \cdot \left({2}^{0.25} \cdot {x}^{0.75}\right)}\right) \]

    5. add-log-exp 49.3%

       \[\leadsto 0 + \color{blue}{{\left(x \cdot 2\right)}^{0.25} \cdot \left({2}^{0.25} \cdot {x}^{0.75}\right)} \]

    6. associate-*r* 49.2%

       \[\leadsto 0 + \color{blue}{\left({\left(x \cdot 2\right)}^{0.25} \cdot {2}^{0.25}\right) \cdot {x}^{0.75}} \]

    7. *-commutative 49.2%

       \[\leadsto 0 + \color{blue}{{x}^{0.75} \cdot \left({\left(x \cdot 2\right)}^{0.25} \cdot {2}^{0.25}\right)} \]

    8. unpow-prod-down 49.1%

       \[\leadsto 0 + {x}^{0.75} \cdot \left(\color{blue}{\left({x}^{0.25} \cdot {2}^{0.25}\right)} \cdot {2}^{0.25}\right) \]

    9. associate-*l* 49.1%

       \[\leadsto 0 + {x}^{0.75} \cdot \color{blue}{\left({x}^{0.25} \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right)} \]

    10. pow-prod-down 49.3%

        \[\leadsto 0 + {x}^{0.75} \cdot \left({x}^{0.25} \cdot \color{blue}{{\left(2 \cdot 2\right)}^{0.25}}\right) \]

    11. pow-prod-down 49.4%

        \[\leadsto 0 + {x}^{0.75} \cdot \color{blue}{{\left(x \cdot \left(2 \cdot 2\right)\right)}^{0.25}} \]

    12. metadata-eval 49.4%

        \[\leadsto 0 + {x}^{0.75} \cdot {\left(x \cdot \color{blue}{4}\right)}^{0.25} \]

11. Applied egg-rr 49.4%

    \[\leadsto \color{blue}{0 + {x}^{0.75} \cdot {\left(x \cdot 4\right)}^{0.25}} \]
12. Step-by-step derivation

    1. +-lft-identity 49.4%

       \[\leadsto \color{blue}{{x}^{0.75} \cdot {\left(x \cdot 4\right)}^{0.25}} \]

13. Simplified 49.4%

    \[\leadsto \color{blue}{{x}^{0.75} \cdot {\left(x \cdot 4\right)}^{0.25}} \]
14. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \sqrt{x\_m \cdot 2} \cdot \sqrt{x\_m} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (sqrt (* x_m 2.0)) (sqrt x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return sqrt((x_m * 2.0)) * sqrt(x_m);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = sqrt((x_m * 2.0d0)) * sqrt(x_m)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.sqrt((x_m * 2.0)) * Math.sqrt(x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.sqrt((x_m * 2.0)) * math.sqrt(x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(sqrt(Float64(x_m * 2.0)) * sqrt(x_m))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Sqrt[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

   1. sqrt-prod 49.3%

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

4. Applied egg-rr 49.3%

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

   1. *-commutative 49.3%

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

6. Simplified 49.3%

   \[\leadsto \color{blue}{\sqrt{x \cdot 2} \cdot \sqrt{x}} \]
7. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot \sqrt{2} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* x_m (sqrt 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * sqrt(2.0);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = x_m * sqrt(2.0d0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * Math.sqrt(2.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * math.sqrt(2.0)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * sqrt(2.0))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

   1. associate-*l* 53.3%

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

   2. sqrt-prod 53.1%

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

   3. sqrt-unprod 49.1%

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

   4. add-sqr-sqrt 50.3%

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

4. Applied egg-rr 50.3%

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

5. Final simplification 50.3%

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

Reproduce

herbie shell --seed 2024103 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))