Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.1% → 99.9%

Time: 6.5s

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: 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, 0.1× 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 95.1%

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

   1. +-commutative 95.1%

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

   2. distribute-lft-in 95.1%

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

   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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

   1. add-sqr-sqrt 45.4%

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

   2. sqrt-unprod 32.3%

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

   3. pow1/2 32.3%

      \[\leadsto \color{blue}{{\left(\left(x \cdot \left(1 + y \cdot y\right)\right) \cdot \left(x \cdot \left(1 + y \cdot y\right)\right)\right)}^{0.5}} \]

   4. pow2 32.3%

      \[\leadsto {\color{blue}{\left({\left(x \cdot \left(1 + y \cdot y\right)\right)}^{2}\right)}}^{0.5} \]

   5. add-sqr-sqrt 31.4%

      \[\leadsto {\left({\color{blue}{\left(\sqrt{x \cdot \left(1 + y \cdot y\right)} \cdot \sqrt{x \cdot \left(1 + y \cdot y\right)}\right)}}^{2}\right)}^{0.5} \]

   6. pow2 31.4%

      \[\leadsto {\left({\color{blue}{\left({\left(\sqrt{x \cdot \left(1 + y \cdot y\right)}\right)}^{2}\right)}}^{2}\right)}^{0.5} \]

   7. pow-pow 31.4%

      \[\leadsto {\color{blue}{\left({\left(\sqrt{x \cdot \left(1 + y \cdot y\right)}\right)}^{\left(2 \cdot 2\right)}\right)}}^{0.5} \]

   8. *-commutative 31.4%

      \[\leadsto {\left({\left(\sqrt{\color{blue}{\left(1 + y \cdot y\right) \cdot x}}\right)}^{\left(2 \cdot 2\right)}\right)}^{0.5} \]

   9. sqrt-prod 31.4%

      \[\leadsto {\left({\color{blue}{\left(\sqrt{1 + y \cdot y} \cdot \sqrt{x}\right)}}^{\left(2 \cdot 2\right)}\right)}^{0.5} \]

   10. hypot-1-def 31.4%

       \[\leadsto {\left({\left(\color{blue}{\mathsf{hypot}\left(1, y\right)} \cdot \sqrt{x}\right)}^{\left(2 \cdot 2\right)}\right)}^{0.5} \]

   11. metadata-eval 31.4%

       \[\leadsto {\left({\left(\mathsf{hypot}\left(1, y\right) \cdot \sqrt{x}\right)}^{\color{blue}{4}}\right)}^{0.5} \]

4. Applied egg-rr 31.4%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{hypot}\left(1, y\right) \cdot \sqrt{x}\right)}^{4}\right)}^{0.5}} \]
5. Step-by-step derivation

   1. unpow1/2 31.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\mathsf{hypot}\left(1, y\right) \cdot \sqrt{x}\right)}^{4}}} \]

6. Simplified 31.4%

   \[\leadsto \color{blue}{\sqrt{{\left(\mathsf{hypot}\left(1, y\right) \cdot \sqrt{x}\right)}^{4}}} \]
7. Step-by-step derivation

   1. sqrt-pow1 46.1%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, y\right) \cdot \sqrt{x}\right)}^{\left(\frac{4}{2}\right)}} \]

   2. metadata-eval 46.1%

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

   3. pow2 46.1%

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

   4. *-commutative 46.1%

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

   5. *-commutative 46.1%

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

   6. swap-sqr 45.5%

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

   7. add-sqr-sqrt 95.1%

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

   8. hypot-undefine 95.1%

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

   9. hypot-undefine 95.1%

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

   10. rem-square-sqrt 95.1%

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

   11. metadata-eval 95.1%

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

   12. add-cube-cbrt 94.8%

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

   13. unpow2 94.8%

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

   14. add-cube-cbrt 94.6%

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

   15. unpow2 94.6%

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

   16. swap-sqr 94.6%

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

   17. unpow2 94.6%

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

8. Applied egg-rr 50.6%

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

   1. fma-undefine 50.6%

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

   2. associate-*l* 50.7%

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

   3. associate-*l* 50.7%

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

   4. associate-*r* 50.7%

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

   5. add-sqr-sqrt 99.9%

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

   6. *-commutative 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 3: 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]

Derivation

1. Initial program 95.1%

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

Alternative 4: 51.9% 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 95.1%

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

3. Taylor expanded in y around 0 50.2%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

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 2024111 
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"
  :precision binary64

  :alt
  (+ x (* (* x y) y))

  (* x (+ 1.0 (* y y))))