Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.5% → 99.9%

Time: 6.4s

Alternatives: 5

Speedup: 0.5×

• 25.4% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+294}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e+294) (* x (fma y y 1.0)) (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e+294) {
		tmp = x * fma(y, y, 1.0);
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e+294)
		tmp = Float64(x * fma(y, y, 1.0));
	else
		tmp = Float64(y * Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e+294], N[(x * N[(y * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.00000000000000007e294

   1. Initial program 99.9%

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

      1. sqr-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. sqr-neg 99.9%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y, y, 1\right)} \]
   4. Add Preprocessing

   if 1.00000000000000007e294 < (*.f64 y y)

   1. Initial program 63.9%

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

      1. *-commutative 63.9%

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

      2. add-sqr-sqrt 63.9%

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

      3. add-sqr-sqrt 31.1%

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

      4. pow2 31.1%

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

      5. pow2 31.1%

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

      6. pow-prod-down 31.1%

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

      7. hypot-1-def 48.3%

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

   4. Applied egg-rr 48.3%

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

   5. Taylor expanded in y around inf 48.3%

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

      1. unpow2 48.3%

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

      2. swap-sqr 31.1%

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

      3. add-sqr-sqrt 63.9%

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

      4. associate-*r* 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+294}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 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 90.9%

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

   1. +-commutative 90.9%

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

   2. distribute-lft-in 90.9%

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

   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-def 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. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+294}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e+294) (* x (+ (* y y) 1.0)) (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e+294) {
		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) <= 1d+294) 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) <= 1e+294) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1e+294:
		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) <= 1e+294)
		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) <= 1e+294)
		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], 1e+294], 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) < 1.00000000000000007e294

   1. Initial program 99.9%

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

   if 1.00000000000000007e294 < (*.f64 y y)

   1. Initial program 63.9%

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

      1. *-commutative 63.9%

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

      2. add-sqr-sqrt 63.9%

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

      3. add-sqr-sqrt 31.1%

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

      4. pow2 31.1%

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

      5. pow2 31.1%

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

      6. pow-prod-down 31.1%

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

      7. hypot-1-def 48.3%

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

   4. Applied egg-rr 48.3%

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

   5. Taylor expanded in y around inf 48.3%

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

      1. unpow2 48.3%

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

      2. swap-sqr 31.1%

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

      3. add-sqr-sqrt 63.9%

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

      4. associate-*r* 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+294}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% 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 92.5%

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

   3. Taylor expanded in y around 0 72.2%

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

   if 1 < y

   1. Initial program 86.7%

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

      1. *-commutative 86.7%

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

      2. add-sqr-sqrt 86.7%

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

      3. add-sqr-sqrt 51.1%

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

      4. pow2 51.1%

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

      5. pow2 51.1%

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

      6. pow-prod-down 51.1%

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

      7. hypot-1-def 58.8%

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

   4. Applied egg-rr 58.8%

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

   5. Taylor expanded in y around inf 57.7%

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

      1. unpow2 57.7%

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

      2. swap-sqr 50.0%

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

      3. add-sqr-sqrt 84.4%

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

      4. associate-*r* 97.3%

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

   7. Applied egg-rr 97.3%

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

4. Final simplification 79.2%

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

Alternative 5: 51.2% 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 90.9%

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

3. Taylor expanded in y around 0 53.6%

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

4. Final simplification 53.6%

   \[\leadsto x \]
5. 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 2024017 
(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))))