Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.1% → 99.9%

Time: 3.6s

Alternatives: 5

Speedup: 1.0×

• 24.8% 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: 94.1% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+258}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 4e+258) (* x (+ (* y y) 1.0)) (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4e+258) {
		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+258) 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+258) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 4e+258:
		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+258)
		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+258)
		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+258], 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) < 4.00000000000000023e258

   1. Initial program 99.9%

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

   if 4.00000000000000023e258 < (*.f64 y y)

   1. Initial program 75.8%

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

      1. add-sqr-sqrt 41.1%

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

      2. pow2 41.1%

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

      3. sqrt-prod 41.1%

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

      4. hypot-1-def 48.7%

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

   3. Applied egg-rr 48.7%

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

   4. Taylor expanded in y around inf 48.7%

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

      1. unpow2 48.7%

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

      2. swap-sqr 41.1%

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

      3. add-sqr-sqrt 75.8%

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

      4. associate-*r* 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+258}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 2: 93.0% accurate, 0.8× 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. Taylor expanded in y around 0 99.7%

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

   if 1 < (*.f64 y y)

   1. Initial program 83.7%

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

   2. Taylor expanded in y around inf 83.1%

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

      1. unpow2 83.1%

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

   4. Simplified 83.1%

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

4. Final simplification 91.6%

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

Alternative 3: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 1e-11) x (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e-11) {
		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) <= 1d-11) 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) <= 1e-11) {
		tmp = x;
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1e-11:
		tmp = x
	else:
		tmp = y * (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e-11)
		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) <= 1e-11)
		tmp = x;
	else
		tmp = y * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e-11], x, N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 9.99999999999999939e-12

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.7%

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

   if 9.99999999999999939e-12 < (*.f64 y y)

   1. Initial program 83.7%

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

      1. add-sqr-sqrt 40.4%

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

      2. pow2 40.4%

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

      3. sqrt-prod 40.4%

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

      4. hypot-1-def 45.5%

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

   3. Applied egg-rr 45.5%

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

   4. Taylor expanded in y around inf 45.0%

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

      1. unpow2 45.0%

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

      2. swap-sqr 39.9%

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

      3. add-sqr-sqrt 83.1%

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

      4. associate-*r* 99.1%

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4: 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 92.0%

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

   1. distribute-rgt-in 92.0%

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

   2. *-un-lft-identity 92.0%

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

   3. +-commutative 92.0%

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

   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 5: 51.8% 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 92.0%

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

2. Taylor expanded in y around 0 53.3%

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

3. Final simplification 53.3%

   \[\leadsto x \]

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 2023290 
(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))))