Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.5% → 99.9%

Time: 5.6s

Alternatives: 5

Speedup: 1.0×

• 26.6% 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: 94.5% 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, 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 95.9%

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

   1. distribute-rgt-in 95.9%

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

   2. *-un-lft-identity 95.9%

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

   3. +-commutative 95.9%

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

   4. associate-*l* 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 4e+16) (* x (+ (* y y) 1.0)) (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4e+16) {
		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) <= 4d+16) 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) <= 4e+16) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 4e+16:
		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) <= 4e+16)
		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) <= 4e+16)
		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], 4e+16], 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) < 4e16

   1. Initial program 100.0%

      \[x \cdot \left(1 + y \cdot y\right) \]

   if 4e16 < (*.f64 y y)

   1. Initial program 91.5%

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

      1. distribute-rgt-in 91.5%

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

      2. *-un-lft-identity 91.5%

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

      3. flip-+ 18.3%

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

      4. associate-*l* 18.3%

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

      5. associate-*l* 18.3%

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

      6. associate-*l* 21.9%

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

   3. Applied egg-rr 21.9%

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

      1. difference-of-squares 22.0%

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

      2. associate-*r* 18.8%

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

      3. distribute-rgt1-in 18.8%

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

      4. fma-udef 18.8%

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

      5. *-commutative 18.8%

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

      6. associate-/l* 33.4%

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

      7. *-inverses 91.5%

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

      8. associate-/l* 91.4%

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

   5. Simplified 91.4%

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

   6. Taylor expanded in y around inf 91.4%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{{y}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 91.4%

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{y \cdot y}}} \]

   8. Simplified 91.4%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{y \cdot y}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 91.5%

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

      2. /-rgt-identity 91.5%

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

      3. associate-*r* 99.9%

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

   10. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 3: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 5e-17) x (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-17) {
		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) <= 5d-17) 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) <= 5e-17) {
		tmp = x;
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e-17:
		tmp = x
	else:
		tmp = y * (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-17)
		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) <= 5e-17)
		tmp = x;
	else
		tmp = y * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-17], x, N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.9999999999999999e-17

   1. Initial program 100.0%

      \[x \cdot \left(1 + y \cdot y\right) \]

   2. Taylor expanded in y around 0 100.0%

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

   if 4.9999999999999999e-17 < (*.f64 y y)

   1. Initial program 91.8%

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

      1. distribute-rgt-in 91.8%

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

      2. *-un-lft-identity 91.8%

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

      3. flip-+ 19.1%

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

      4. associate-*l* 19.1%

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

      5. associate-*l* 19.2%

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

      6. associate-*l* 22.6%

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

   3. Applied egg-rr 22.6%

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

      1. difference-of-squares 22.8%

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

      2. associate-*r* 19.8%

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

      3. distribute-rgt1-in 19.8%

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

      4. fma-udef 19.8%

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

      5. *-commutative 19.8%

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

      6. associate-/l* 35.2%

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

      7. *-inverses 91.8%

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

      8. associate-/l* 91.7%

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

   5. Simplified 91.7%

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

   6. Taylor expanded in y around inf 90.6%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{{y}^{2}}}} \]
   7. Step-by-step derivation

      1. unpow2 90.6%

         \[\leadsto \frac{x}{\frac{1}{\color{blue}{y \cdot y}}} \]

   8. Simplified 90.6%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{y \cdot y}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 90.6%

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

      2. /-rgt-identity 90.6%

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

      3. associate-*r* 98.7%

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

   10. Applied egg-rr 98.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4: 72.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 97.5%

      \[x \cdot \left(1 + y \cdot y\right) \]

   2. Taylor expanded in y around 0 66.2%

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

   if 1 < y

   1. Initial program 90.3%

      \[x \cdot \left(1 + y \cdot y\right) \]

   2. Taylor expanded in y around inf 89.0%

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

      1. unpow2 89.0%

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

   4. Simplified 89.0%

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

4. Final simplification 71.5%

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

Alternative 5: 50.7% 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 95.9%

   \[x \cdot \left(1 + y \cdot y\right) \]

2. Taylor expanded in y around 0 52.2%

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

3. Final simplification 52.2%

   \[\leadsto x \]

Developer target: 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 2023282 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"
  :precision binary64

  :herbie-target
  (+ x (* (* x y) y))

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