Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.2% → 99.9%

Time: 5.7s

Alternatives: 6

Speedup: 1.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(1 + y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (+ 1.0 (* y y))))

C

double code(double x, double y) {
	return x * (1.0 + (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + (y * y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + (y * y));
}

Python

def code(x, y):
	return x * (1.0 + (y * y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + (y * y));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (+ 1.0 (* y y))))

C

double code(double x, double y) {
	return x * (1.0 + (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + (y * y))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + (y * y));
}

Python

def code(x, y):
	return x * (1.0 + (y * y))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + (y * y));
end

Wolfram

code[x_, y_] := N[(x * N[(1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+14) (+ x (* x (* y y))) (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+14) {
		tmp = x + (x * (y * y));
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 5d+14) then
        tmp = x + (x * (y * y))
    else
        tmp = y * (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+14) {
		tmp = x + (x * (y * y));
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e+14:
		tmp = x + (x * (y * y))
	else:
		tmp = y * (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e+14)
		tmp = Float64(x + Float64(x * Float64(y * y)));
	else
		tmp = Float64(y * Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e+14)
		tmp = x + (x * (y * y));
	else
		tmp = y * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e+14], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5e14

   1. Initial program 100.0%

      \[x \cdot \left(1 + y \cdot y\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. pow2 100.0%

         \[\leadsto x \cdot \color{blue}{{y}^{2}} + x \cdot 1 \]

      4. *-rgt-identity 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. unpow2 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 5e14 < (*.f64 y y)

   1. Initial program 86.8%

      \[x \cdot \left(1 + y \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 86.8%

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

      1. add-sqr-sqrt 42.5%

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

      2. pow2 42.5%

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

      3. *-commutative 42.5%

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

      4. sqrt-prod 42.5%

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

      5. sqrt-pow1 49.7%

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

      6. metadata-eval 49.7%

         \[\leadsto {\left({y}^{\color{blue}{1}} \cdot \sqrt{x}\right)}^{2} + x \]

      7. pow1 49.7%

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

   5. Applied egg-rr 49.7%

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

      1. unpow2 49.7%

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

      2. *-commutative 49.7%

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

      3. associate-*r* 49.8%

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

      4. associate-*r* 49.8%

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

      5. add-sqr-sqrt 99.8%

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

   7. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+14) (* x (+ (* y y) 1.0)) (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+14) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 5d+14) then
        tmp = x * ((y * y) + 1.0d0)
    else
        tmp = y * (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+14) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e+14:
		tmp = x * ((y * y) + 1.0)
	else:
		tmp = y * (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e+14)
		tmp = Float64(x * Float64(Float64(y * y) + 1.0));
	else
		tmp = Float64(y * Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e+14)
		tmp = x * ((y * y) + 1.0);
	else
		tmp = y * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e+14], N[(x * N[(N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5e14

   1. Initial program 100.0%

      \[x \cdot \left(1 + y \cdot y\right) \]
   2. Add Preprocessing

   if 5e14 < (*.f64 y y)

   1. Initial program 86.8%

      \[x \cdot \left(1 + y \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 86.8%

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

      1. add-sqr-sqrt 42.5%

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

      2. pow2 42.5%

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

      3. *-commutative 42.5%

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

      4. sqrt-prod 42.5%

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

      5. sqrt-pow1 49.7%

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

      6. metadata-eval 49.7%

         \[\leadsto {\left({y}^{\color{blue}{1}} \cdot \sqrt{x}\right)}^{2} + x \]

      7. pow1 49.7%

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

   5. Applied egg-rr 49.7%

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

      1. unpow2 49.7%

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

      2. *-commutative 49.7%

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

      3. associate-*r* 49.8%

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

      4. associate-*r* 49.8%

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

      5. add-sqr-sqrt 99.8%

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

   7. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 2e-5) x (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-5) {
		tmp = x;
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 2d-5) then
        tmp = x
    else
        tmp = y * (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-5) {
		tmp = x;
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2e-5:
		tmp = x
	else:
		tmp = y * (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e-5)
		tmp = x;
	else
		tmp = Float64(y * Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 2e-5)
		tmp = x;
	else
		tmp = y * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e-5], x, N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.00000000000000016e-5

   1. Initial program 100.0%

      \[x \cdot \left(1 + y \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.2%

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

   if 2.00000000000000016e-5 < (*.f64 y y)

   1. Initial program 86.9%

      \[x \cdot \left(1 + y \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 86.8%

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

      1. add-sqr-sqrt 41.8%

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

      2. pow2 41.8%

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

      3. *-commutative 41.8%

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

      4. sqrt-prod 41.8%

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

      5. sqrt-pow1 49.0%

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

      6. metadata-eval 49.0%

         \[\leadsto {\left({y}^{\color{blue}{1}} \cdot \sqrt{x}\right)}^{2} + x \]

      7. pow1 49.0%

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

   5. Applied egg-rr 49.0%

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

      1. unpow2 49.0%

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

      2. *-commutative 49.0%

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

      3. associate-*r* 49.0%

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

      4. associate-*r* 49.0%

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

      5. add-sqr-sqrt 99.8%

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

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(y \cdot x\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 1.0) x (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.0) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 1.0d0) then
        tmp = x
    else
        tmp = x * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.0) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1.0:
		tmp = x
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1.0)
		tmp = x;
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 1.0)
		tmp = x;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1.0], x, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1

   1. Initial program 100.0%

      \[x \cdot \left(1 + y \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.2%

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

   if 1 < (*.f64 y y)

   1. Initial program 86.9%

      \[x \cdot \left(1 + y \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 86.8%

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

      1. unpow2 87.0%

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

   5. Applied egg-rr 86.7%

      \[\leadsto x \cdot \color{blue}{\left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + y \cdot \left(y \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (* y (* y x))))

C

double code(double x, double y) {
	return x + (y * (y * x));
}

Fortran

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

Java

public static double code(double x, double y) {
	return x + (y * (y * x));
}

Python

def code(x, y):
	return x + (y * (y * x))

Julia

function code(x, y)
	return Float64(x + Float64(y * Float64(y * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (y * (y * x));
end

Wolfram

code[x_, y_] := N[(x + N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.4%

   \[x \cdot \left(1 + y \cdot y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative 93.4%

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

   2. distribute-lft-in 93.4%

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

   3. pow2 93.4%

      \[\leadsto x \cdot \color{blue}{{y}^{2}} + x \cdot 1 \]

   4. *-rgt-identity 93.4%

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

4. Applied egg-rr 93.4%

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

   1. add-sqr-sqrt 61.5%

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

   2. pow2 61.5%

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

   3. *-commutative 61.5%

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

   4. sqrt-prod 45.5%

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

   5. sqrt-pow1 49.1%

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

   6. metadata-eval 49.1%

      \[\leadsto {\left({y}^{\color{blue}{1}} \cdot \sqrt{x}\right)}^{2} + x \]

   7. pow1 49.1%

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

6. Applied egg-rr 49.1%

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

   1. unpow2 49.1%

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

   2. *-commutative 49.1%

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

   3. associate-*r* 49.1%

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

   4. associate-*r* 49.1%

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

   5. add-sqr-sqrt 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

   \[\leadsto x + y \cdot \left(y \cdot x\right) \]
10. Add Preprocessing

Alternative 6: 52.3% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 93.4%

   \[x \cdot \left(1 + y \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0 51.3%

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

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(x \cdot y\right) \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (* (* x y) y)))

C

double code(double x, double y) {
	return x + ((x * y) * y);
}

Fortran

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

Java

public static double code(double x, double y) {
	return x + ((x * y) * y);
}

Python

def code(x, y):
	return x + ((x * y) * y)

Julia

function code(x, y)
	return Float64(x + Float64(Float64(x * y) * y))
end

MATLAB

function tmp = code(x, y)
	tmp = x + ((x * y) * y);
end

Wolfram

code[x_, y_] := N[(x + N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024132 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (* (* x y) y)))

  (* x (+ 1.0 (* y y))))