Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.5% → 99.9%

Time: 8.3s

Alternatives: 3

Speedup: 1.2×

• 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: 94.5% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

   1. lift-*.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.9

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

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, y, x\right)} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 2e-9) x (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-9) {
		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-9) 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-9) {
		tmp = x;
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2e-9:
		tmp = x
	else:
		tmp = y * (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e-9)
		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-9)
		tmp = x;
	else
		tmp = y * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e-9], x, N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.00000000000000012e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 99.7%

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

      1. *-rgt-identity 99.7

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

   7. Applied egg-rr 99.7%

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

   if 2.00000000000000012e-9 < (*.f64 y y)

   1. Initial program 86.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 85.1

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

   5. Simplified 85.1%

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

      1. associate-*l* N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 98.4

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

   7. Applied egg-rr 98.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.8% accurate, 14.0× 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 93.6%

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

3. Taylor expanded in y around 0

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

5. Simplified 55.2%

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

   1. *-rgt-identity 55.2

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

7. Applied egg-rr 55.2%

   \[\leadsto \color{blue}{x} \]
8. 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 2024207 
(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))))