Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.2% → 99.9%

Time: 5.8s

Alternatives: 5

Speedup: 0.6×

• 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 96.2%

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

   1. +-commutative 96.2%

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

   2. distribute-lft-in 96.2%

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

   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%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+249}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 4e+249) (* x (+ (* y y) 1.0)) (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4e+249) {
		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+249) 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+249) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 4e+249:
		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+249)
		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+249)
		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+249], 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) < 3.9999999999999997e249

   1. Initial program 99.9%

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

   if 3.9999999999999997e249 < (*.f64 y y)

   1. Initial program 87.3%

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

      1. flip-+ 11.4%

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

      2. associate-*r/ 11.4%

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

      3. metadata-eval 11.4%

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

      4. pow2 11.4%

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

      5. pow2 11.4%

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

      6. pow-prod-up 11.4%

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

      7. metadata-eval 11.4%

         \[\leadsto \frac{x \cdot \left(1 - {y}^{\color{blue}{4}}\right)}{1 - y \cdot y} \]

      8. pow2 11.4%

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

   3. Applied egg-rr 11.4%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 - {y}^{4}\right)}{1 - {y}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* 11.4%

         \[\leadsto \color{blue}{\frac{x}{\frac{1 - {y}^{2}}{1 - {y}^{4}}}} \]

   5. Simplified 11.4%

      \[\leadsto \color{blue}{\frac{x}{\frac{1 - {y}^{2}}{1 - {y}^{4}}}} \]

   6. Taylor expanded in y around inf 87.3%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{{y}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 87.3%

         \[\leadsto \color{blue}{\frac{x}{1} \cdot {y}^{2}} \]

      2. /-rgt-identity 87.3%

         \[\leadsto \color{blue}{x} \cdot {y}^{2} \]

      3. unpow2 87.3%

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

      4. associate-*r* 99.9%

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

   8. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+249}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 3: 75.7% accurate, 1.0× 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 98.4%

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

   2. Taylor expanded in y around 0 65.5%

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

   if 1 < y

   1. Initial program 90.0%

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

      1. flip-+ 42.4%

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

      2. associate-*r/ 42.4%

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

      3. metadata-eval 42.4%

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

      4. pow2 42.4%

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

      5. pow2 42.4%

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

      6. pow-prod-up 42.4%

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

      7. metadata-eval 42.4%

         \[\leadsto \frac{x \cdot \left(1 - {y}^{\color{blue}{4}}\right)}{1 - y \cdot y} \]

      8. pow2 42.4%

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

   3. Applied egg-rr 42.4%

      \[\leadsto \color{blue}{\frac{x \cdot \left(1 - {y}^{4}\right)}{1 - {y}^{2}}} \]
   4. Step-by-step derivation

      1. associate-/l* 42.3%

         \[\leadsto \color{blue}{\frac{x}{\frac{1 - {y}^{2}}{1 - {y}^{4}}}} \]

   5. Simplified 42.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{1 - {y}^{2}}{1 - {y}^{4}}}} \]

   6. Taylor expanded in y around inf 87.5%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{{y}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 87.6%

         \[\leadsto \color{blue}{\frac{x}{1} \cdot {y}^{2}} \]

      2. /-rgt-identity 87.6%

         \[\leadsto \color{blue}{x} \cdot {y}^{2} \]

      3. unpow2 87.6%

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

      4. associate-*r* 97.3%

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

   8. 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 73.8%

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

Alternative 4: 57.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 98.4%

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

   2. Taylor expanded in y around 0 65.5%

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

   if 1 < y

   1. Initial program 90.0%

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

      1. +-commutative 90.0%

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

      2. distribute-lft-in 90.0%

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

      3. associate-*r* 99.8%

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

      4. *-rgt-identity 99.8%

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

      5. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. associate-*l* 90.0%

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

      3. unpow2 90.0%

         \[\leadsto x \cdot \color{blue}{{y}^{2}} + x \]

      4. *-un-lft-identity 90.0%

         \[\leadsto x \cdot {y}^{2} + \color{blue}{1 \cdot x} \]

      5. *-commutative 90.0%

         \[\leadsto x \cdot {y}^{2} + \color{blue}{x \cdot 1} \]

      6. distribute-lft-in 90.0%

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

      7. +-commutative 90.0%

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

      8. flip-+ 42.4%

         \[\leadsto x \cdot \color{blue}{\frac{1 \cdot 1 - {y}^{2} \cdot {y}^{2}}{1 - {y}^{2}}} \]

      9. metadata-eval 42.4%

         \[\leadsto x \cdot \frac{\color{blue}{1} - {y}^{2} \cdot {y}^{2}}{1 - {y}^{2}} \]

      10. pow-prod-up 42.4%

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

      11. metadata-eval 42.4%

          \[\leadsto x \cdot \frac{1 - {y}^{\color{blue}{4}}}{1 - {y}^{2}} \]

      12. clear-num 42.3%

          \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{1 - {y}^{2}}{1 - {y}^{4}}}} \]

      13. div-inv 42.3%

          \[\leadsto \color{blue}{\frac{x}{\frac{1 - {y}^{2}}{1 - {y}^{4}}}} \]

      14. add-sqr-sqrt 42.2%

          \[\leadsto \frac{x}{\color{blue}{\sqrt{\frac{1 - {y}^{2}}{1 - {y}^{4}}} \cdot \sqrt{\frac{1 - {y}^{2}}{1 - {y}^{4}}}}} \]

      15. associate-/r* 42.1%

          \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{\frac{1 - {y}^{2}}{1 - {y}^{4}}}}}{\sqrt{\frac{1 - {y}^{2}}{1 - {y}^{4}}}}} \]

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in y around inf 97.3%

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

   7. Taylor expanded in y around 0 23.5%

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

4. Final simplification 54.5%

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

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 96.2%

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

2. Taylor expanded in y around 0 49.7%

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

3. Final simplification 49.7%

   \[\leadsto x \]

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 2023322 
(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))))