Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.3% → 99.9%

Time: 8.6s

Alternatives: 7

Speedup: 1.0×

• 24.5% 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: 94.3% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+250}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+250) (+ x (* x (* y y))) (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+250) {
		tmp = x + (x * (y * y));
	} else {
		tmp = y * (x * 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) <= 5d+250) then
        tmp = x + (x * (y * y))
    else
        tmp = y * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+250) {
		tmp = x + (x * (y * y));
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e+250:
		tmp = x + (x * (y * y))
	else:
		tmp = y * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e+250)
		tmp = Float64(x + Float64(x * Float64(y * y)));
	else
		tmp = Float64(y * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e+250)
		tmp = x + (x * (y * y));
	else
		tmp = y * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e+250], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5.0000000000000002e250

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

         \[\leadsto \left(y \cdot y\right) \cdot x + x \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(y \cdot y\right) \cdot x\right), \color{blue}{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(y \cdot y\right)\right), x\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot y\right)\right), x\right) \]

      7. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), x\right) \]

   4. Applied egg-rr 99.9%

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

   if 5.0000000000000002e250 < (*.f64 y y)

   1. Initial program 82.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 82.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   5. Simplified 82.0%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot y\right), \color{blue}{y}\right) \]

      3. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), y\right) \]

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+250}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+250}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+250) (* x (+ (* y y) 1.0)) (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+250) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (x * 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) <= 5d+250) then
        tmp = x * ((y * y) + 1.0d0)
    else
        tmp = y * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+250) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e+250:
		tmp = x * ((y * y) + 1.0)
	else:
		tmp = y * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e+250)
		tmp = Float64(x * Float64(Float64(y * y) + 1.0));
	else
		tmp = Float64(y * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e+250)
		tmp = x * ((y * y) + 1.0);
	else
		tmp = y * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e+250], N[(x * N[(N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5.0000000000000002e250

   1. Initial program 99.9%

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

   if 5.0000000000000002e250 < (*.f64 y y)

   1. Initial program 82.0%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 82.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   5. Simplified 82.0%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot y\right), \color{blue}{y}\right) \]

      3. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), y\right) \]

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+250}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 0.4) x (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.4) {
		tmp = x;
	} else {
		tmp = y * (x * 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) <= 0.4d0) then
        tmp = x
    else
        tmp = y * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.4) {
		tmp = x;
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.4:
		tmp = x
	else:
		tmp = y * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.4)
		tmp = x;
	else
		tmp = Float64(y * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 0.4)
		tmp = x;
	else
		tmp = y * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.4], x, N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 99.1%

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

   if 0.40000000000000002 < (*.f64 y y)

   1. Initial program 90.1%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 88.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   5. Simplified 88.3%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot y\right), \color{blue}{y}\right) \]

      3. *-lowering-*.f64 97.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), y\right) \]

   7. Applied egg-rr 97.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.4:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 0.4) x (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.4) {
		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) <= 0.4d0) 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) <= 0.4) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.4:
		tmp = x
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.4)
		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) <= 0.4)
		tmp = x;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.4], x, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 99.1%

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

   if 0.40000000000000002 < (*.f64 y y)

   1. Initial program 90.1%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 88.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   5. Simplified 88.3%

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

Alternative 5: 58.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x;
	} else {
		tmp = x * 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 <= 1.0d0) then
        tmp = x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 97.7%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 69.6%

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

   if 1 < y

   1. Initial program 87.1%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 84.4%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   5. Simplified 84.4%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot y\right), \color{blue}{y}\right) \]

      3. *-lowering-*.f64 97.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), y\right) \]

   7. Applied egg-rr 97.2%

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

   9. Applied egg-rr 31.8%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.5%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

      \[\leadsto \left(y \cdot y\right) \cdot x + x \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(y \cdot y\right) \cdot x\right), \color{blue}{x}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(y \cdot y\right)\right), x\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot y\right)\right), x\right) \]

   7. *-lowering-*.f64 95.5%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, y\right)\right), x\right) \]

4. Applied egg-rr 95.5%

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

   1. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(x \cdot y\right) \cdot y\right), x\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x \cdot y\right), y\right), x\right) \]

   3. *-lowering-*.f64 99.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), y\right), x\right) \]

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 7: 52.4% 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 95.5%

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

3. Taylor expanded in y around 0

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

5. Simplified 56.5%

   \[\leadsto \color{blue}{x} \]
6. 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 2024161 
(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))))