Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 93.7% → 99.9%

Time: 3.7s

Alternatives: 6

Speedup: 0.8×

• 24.7% 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: 93.7% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+70}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e+70) (+ x (* x (* y y))) (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e+70) {
		tmp = x + (x * (y * y));
	} 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+70) then
        tmp = x + (x * (y * y))
    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+70) {
		tmp = x + (x * (y * y));
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2e+70:
		tmp = x + (x * (y * y))
	else:
		tmp = y * (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e+70)
		tmp = Float64(x + Float64(x * Float64(y * y)));
	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+70)
		tmp = x + (x * (y * y));
	else
		tmp = y * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e+70], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.00000000000000015e70

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

   3. Applied egg-rr 100.0%

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

   if 2.00000000000000015e70 < (*.f64 y y)

   1. Initial program 89.0%

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

   2. Taylor expanded in y around inf 89.0%

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

      1. unpow2 89.0%

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

      2. associate-*l* 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+70}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y) {
	return Math.hypot(1.0, y) * (Math.hypot(1.0, y) * x);
}

Python

def code(x, y):
	return math.hypot(1.0, y) * (math.hypot(1.0, y) * x)

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[Sqrt[1.0 ^ 2 + y ^ 2], $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + y ^ 2], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

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

   1. +-commutative 94.4%

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

   2. distribute-lft-in 94.4%

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

   3. *-rgt-identity 94.4%

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

   4. fma-def 94.4%

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

3. Simplified 94.4%

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

   1. fma-udef 94.4%

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

   2. *-rgt-identity 94.4%

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

   3. distribute-lft-in 94.4%

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

   4. +-commutative 94.4%

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

   5. *-commutative 94.4%

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

   6. add-sqr-sqrt 94.4%

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

   7. associate-*l* 94.4%

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

   8. hypot-1-def 94.4%

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

   9. hypot-1-def 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

   \[\leadsto \mathsf{hypot}\left(1, y\right) \cdot \left(\mathsf{hypot}\left(1, y\right) \cdot x\right) \]

Alternative 3: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+70}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e+70) (* x (+ 1.0 (* y y))) (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e+70) {
		tmp = x * (1.0 + (y * y));
	} 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+70) then
        tmp = x * (1.0d0 + (y * y))
    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+70) {
		tmp = x * (1.0 + (y * y));
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2e+70:
		tmp = x * (1.0 + (y * y))
	else:
		tmp = y * (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e+70)
		tmp = Float64(x * Float64(1.0 + Float64(y * y)));
	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+70)
		tmp = x * (1.0 + (y * y));
	else
		tmp = y * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e+70], N[(x * N[(1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.00000000000000015e70

   1. Initial program 99.9%

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

   if 2.00000000000000015e70 < (*.f64 y y)

   1. Initial program 89.0%

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

   2. Taylor expanded in y around inf 89.0%

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

      1. unpow2 89.0%

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

      2. associate-*l* 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+70}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4: 92.5% 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 99.3%

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

   if 1 < (*.f64 y y)

   1. Initial program 90.3%

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

   2. Taylor expanded in y around inf 89.3%

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

      1. unpow2 89.3%

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

      2. *-commutative 89.3%

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

   4. Simplified 89.3%

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

4. Final simplification 93.5%

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

Alternative 5: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 2e-6) x (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-6) {
		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-6) 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-6) {
		tmp = x;
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2e-6:
		tmp = x
	else:
		tmp = y * (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e-6)
		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-6)
		tmp = x;
	else
		tmp = y * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e-6], x, N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.99999999999999991e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.3%

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

   if 1.99999999999999991e-6 < (*.f64 y y)

   1. Initial program 90.3%

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

   2. Taylor expanded in y around inf 89.3%

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

      1. unpow2 89.3%

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

      2. associate-*l* 98.8%

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

   4. Simplified 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 6: 52.2% 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 94.4%

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

2. Taylor expanded in y around 0 44.7%

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

3. Final simplification 44.7%

   \[\leadsto x \]

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 2023229 
(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))))