Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 93.5% → 99.9%

Time: 5.5s

Alternatives: 4

Speedup: 1.0×

• 25.1% 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: 93.5% 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 90.8%

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

   1. +-commutative 90.8%

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

   2. distribute-lft-in 90.8%

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

   3. pow2 90.8%

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

   4. *-rgt-identity 90.8%

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

4. Applied egg-rr 90.8%

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

   1. add-sqr-sqrt 64.7%

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

   2. pow2 64.7%

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

   3. *-commutative 64.7%

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

   4. sqrt-prod 49.8%

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

   5. sqrt-pow1 54.1%

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

   6. metadata-eval 54.1%

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

   7. pow1 54.1%

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

6. Applied egg-rr 54.1%

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

   1. unpow2 54.1%

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

   2. *-commutative 54.1%

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

   3. associate-*r* 54.1%

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

   4. associate-*r* 54.1%

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

   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 2: 93.5% accurate, 1.0× speedup

Math

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

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(x * Float64(y * y)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. +-commutative 90.8%

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

   2. distribute-lft-in 90.8%

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

   3. pow2 90.8%

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

   4. *-rgt-identity 90.8%

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

4. Applied egg-rr 90.8%

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

   1. unpow2 90.8%

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

6. Applied egg-rr 90.8%

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

7. Final simplification 90.8%

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

Alternative 3: 93.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

3. Final simplification 90.8%

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

Alternative 4: 50.9% 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 90.8%

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

3. Taylor expanded in y around 0 50.5%

   \[\leadsto \color{blue}{x} \]
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 2024135 
(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))))