Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.2% → 99.9%

Time: 6.8s

Alternatives: 4

Speedup: 0.5×

• 25.0% 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: 94.2% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.9%

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

   1. +-commutative 92.9%

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

   2. distribute-lft-in 92.9%

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

   3. associate-*r* 99.9%

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

   4. *-rgt-identity 99.9%

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

   5. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 50000000000000:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 50000000000000.0) (* x (+ (* y y) 1.0)) (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 50000000000000.0) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (x * 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) <= 50000000000000.0d0) then
        tmp = x * ((y * y) + 1.0d0)
    else
        tmp = y * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 50000000000000.0) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 50000000000000.0:
		tmp = x * ((y * y) + 1.0)
	else:
		tmp = y * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 50000000000000.0)
		tmp = Float64(x * Float64(Float64(y * y) + 1.0));
	else
		tmp = Float64(y * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 50000000000000.0)
		tmp = x * ((y * y) + 1.0);
	else
		tmp = y * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 50000000000000.0], N[(x * N[(N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5e13

   1. Initial program 100.0%

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

   if 5e13 < (*.f64 y y)

   1. Initial program 86.2%

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

      1. add-cbrt-cube 63.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(x \cdot \left(1 + y \cdot y\right)\right) \cdot \left(x \cdot \left(1 + y \cdot y\right)\right)\right) \cdot \left(x \cdot \left(1 + y \cdot y\right)\right)}} \]

      2. pow3 63.2%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(x \cdot \left(1 + y \cdot y\right)\right)}^{3}}} \]

      3. add-sqr-sqrt 25.9%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{x \cdot \left(1 + y \cdot y\right)} \cdot \sqrt{x \cdot \left(1 + y \cdot y\right)}\right)}}^{3}} \]

      4. pow2 25.9%

         \[\leadsto \sqrt[3]{{\color{blue}{\left({\left(\sqrt{x \cdot \left(1 + y \cdot y\right)}\right)}^{2}\right)}}^{3}} \]

      5. pow-pow 26.0%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{x \cdot \left(1 + y \cdot y\right)}\right)}^{\left(2 \cdot 3\right)}}} \]

      6. *-commutative 26.0%

         \[\leadsto \sqrt[3]{{\left(\sqrt{\color{blue}{\left(1 + y \cdot y\right) \cdot x}}\right)}^{\left(2 \cdot 3\right)}} \]

      7. sqrt-prod 26.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{1 + y \cdot y} \cdot \sqrt{x}\right)}}^{\left(2 \cdot 3\right)}} \]

      8. hypot-1-def 27.4%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\mathsf{hypot}\left(1, y\right)} \cdot \sqrt{x}\right)}^{\left(2 \cdot 3\right)}} \]

      9. metadata-eval 27.4%

         \[\leadsto \sqrt[3]{{\left(\mathsf{hypot}\left(1, y\right) \cdot \sqrt{x}\right)}^{\color{blue}{6}}} \]

   4. Applied egg-rr 27.4%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{hypot}\left(1, y\right) \cdot \sqrt{x}\right)}^{6}}} \]

   5. Taylor expanded in y around inf 27.4%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{x} \cdot y\right)}}^{6}} \]
   6. Step-by-step derivation

      1. pow1/3 27.0%

         \[\leadsto \color{blue}{{\left({\left(\sqrt{x} \cdot y\right)}^{6}\right)}^{0.3333333333333333}} \]

      2. pow-pow 44.5%

         \[\leadsto \color{blue}{{\left(\sqrt{x} \cdot y\right)}^{\left(6 \cdot 0.3333333333333333\right)}} \]

      3. metadata-eval 44.5%

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

      4. pow2 44.5%

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

      5. associate-*r* 44.5%

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

      6. *-commutative 44.5%

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

   7. Applied egg-rr 44.5%

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

   8. Taylor expanded in y around 0 99.7%

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

      1. *-commutative 99.7%

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

   10. Simplified 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 50000000000000:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.0)
		tmp = x;
	else
		tmp = Float64(y * Float64(x * y));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0 63.7%

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

   if 1 < y

   1. Initial program 89.5%

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

      1. add-cbrt-cube 67.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(x \cdot \left(1 + y \cdot y\right)\right) \cdot \left(x \cdot \left(1 + y \cdot y\right)\right)\right) \cdot \left(x \cdot \left(1 + y \cdot y\right)\right)}} \]

      2. pow3 67.7%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(x \cdot \left(1 + y \cdot y\right)\right)}^{3}}} \]

      3. add-sqr-sqrt 30.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{x \cdot \left(1 + y \cdot y\right)} \cdot \sqrt{x \cdot \left(1 + y \cdot y\right)}\right)}}^{3}} \]

      4. pow2 30.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left({\left(\sqrt{x \cdot \left(1 + y \cdot y\right)}\right)}^{2}\right)}}^{3}} \]

      5. pow-pow 30.0%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{x \cdot \left(1 + y \cdot y\right)}\right)}^{\left(2 \cdot 3\right)}}} \]

      6. *-commutative 30.0%

         \[\leadsto \sqrt[3]{{\left(\sqrt{\color{blue}{\left(1 + y \cdot y\right) \cdot x}}\right)}^{\left(2 \cdot 3\right)}} \]

      7. sqrt-prod 30.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{1 + y \cdot y} \cdot \sqrt{x}\right)}}^{\left(2 \cdot 3\right)}} \]

      8. hypot-1-def 31.4%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\mathsf{hypot}\left(1, y\right)} \cdot \sqrt{x}\right)}^{\left(2 \cdot 3\right)}} \]

      9. metadata-eval 31.4%

         \[\leadsto \sqrt[3]{{\left(\mathsf{hypot}\left(1, y\right) \cdot \sqrt{x}\right)}^{\color{blue}{6}}} \]

   4. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{hypot}\left(1, y\right) \cdot \sqrt{x}\right)}^{6}}} \]

   5. Taylor expanded in y around inf 31.4%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{x} \cdot y\right)}}^{6}} \]
   6. Step-by-step derivation

      1. pow1/3 31.0%

         \[\leadsto \color{blue}{{\left({\left(\sqrt{x} \cdot y\right)}^{6}\right)}^{0.3333333333333333}} \]

      2. pow-pow 43.6%

         \[\leadsto \color{blue}{{\left(\sqrt{x} \cdot y\right)}^{\left(6 \cdot 0.3333333333333333\right)}} \]

      3. metadata-eval 43.6%

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

      4. pow2 43.6%

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

      5. associate-*r* 43.6%

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

      6. *-commutative 43.6%

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

   7. Applied egg-rr 43.6%

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

   8. Taylor expanded in y around 0 98.8%

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

      1. *-commutative 98.8%

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

   10. Simplified 98.8%

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

4. Final simplification 72.6%

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

Alternative 4: 52.3% 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 92.9%

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

3. Taylor expanded in y around 0 48.9%

   \[\leadsto \color{blue}{x} \]
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 2024139 
(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))))