Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.1% → 99.9%

Time: 4.6s

Alternatives: 4

Speedup: 0.5×

• 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: 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 93.3%

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

   1. +-commutative 93.3%

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

   2. distribute-lft-in 93.3%

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

   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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+232}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 4e+232) (* x (+ (* y y) 1.0)) (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4e+232) {
		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) <= 4d+232) 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) <= 4e+232) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 4e+232:
		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) <= 4e+232)
		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) <= 4e+232)
		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], 4e+232], 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) < 4.00000000000000023e232

   1. Initial program 99.9%

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

   if 4.00000000000000023e232 < (*.f64 y y)

   1. Initial program 79.5%

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

      1. add-sqr-sqrt 46.5%

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

      2. sqrt-unprod 44.5%

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

      3. pow1/2 44.5%

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

      4. pow2 44.5%

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

      5. add-sqr-sqrt 44.4%

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

      6. pow2 44.4%

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

      7. pow-pow 44.4%

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

      8. *-commutative 44.4%

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

      9. sqrt-prod 44.4%

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

      10. hypot-1-def 49.0%

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

      11. metadata-eval 49.0%

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

   4. Applied egg-rr 49.0%

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

      1. unpow1/2 49.0%

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

   6. Simplified 49.0%

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

   7. Taylor expanded in y around inf 49.0%

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

      1. sqrt-pow1 61.2%

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

      2. metadata-eval 61.2%

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

      3. unpow2 61.2%

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

      4. swap-sqr 46.6%

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

      5. add-sqr-sqrt 79.5%

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

      6. associate-*l* 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+232}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% 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 96.5%

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

   3. Taylor expanded in y around 0 69.7%

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

   if 1 < y

   1. Initial program 83.9%

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

      1. add-sqr-sqrt 46.6%

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

      2. sqrt-unprod 36.9%

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

      3. pow1/2 36.9%

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

      4. pow2 36.9%

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

      5. add-sqr-sqrt 36.5%

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

      6. pow2 36.5%

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

      7. pow-pow 36.5%

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

      8. *-commutative 36.5%

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

      9. sqrt-prod 36.5%

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

      10. hypot-1-def 40.9%

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

      11. metadata-eval 40.9%

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

   4. Applied egg-rr 40.9%

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

      1. unpow1/2 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in y around inf 40.9%

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

      1. sqrt-pow1 57.9%

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

      2. metadata-eval 57.9%

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

      3. unpow2 57.9%

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

      4. swap-sqr 46.5%

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

      5. add-sqr-sqrt 82.8%

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

      6. associate-*l* 98.7%

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

   9. Applied egg-rr 98.7%

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

4. Final simplification 77.0%

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

Alternative 4: 51.8% 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 93.3%

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

3. Taylor expanded in y around 0 53.3%

   \[\leadsto \color{blue}{x} \]
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 2024130 
(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))))