Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.3% → 99.9%

Time: 6.4s

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: 94.3% 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 95.9%

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

   1. +-commutative 95.9%

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

   2. distribute-lft-in 95.9%

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

   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%

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

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+195}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{1}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+195) (* x (+ (* y y) 1.0)) (* y (/ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+195) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (x / (1.0 / 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) <= 5d+195) then
        tmp = x * ((y * y) + 1.0d0)
    else
        tmp = y * (x / (1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+195) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (x / (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e+195:
		tmp = x * ((y * y) + 1.0)
	else:
		tmp = y * (x / (1.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e+195)
		tmp = Float64(x * Float64(Float64(y * y) + 1.0));
	else
		tmp = Float64(y * Float64(x / Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e+195)
		tmp = x * ((y * y) + 1.0);
	else
		tmp = y * (x / (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e+195], N[(x * N[(N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.9999999999999998e195

   1. Initial program 99.9%

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

   if 4.9999999999999998e195 < (*.f64 y y)

   1. Initial program 88.9%

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

      1. flip-+ 10.3%

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

      2. associate-*r/ 10.3%

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

      3. metadata-eval 10.3%

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

      4. pow2 10.3%

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

      5. pow2 10.3%

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

      6. pow-prod-up 10.3%

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

      7. metadata-eval 10.3%

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

      8. pow2 10.3%

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

   3. Applied egg-rr 10.3%

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

      1. associate-/l* 10.3%

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

   5. Simplified 10.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{1 - {y}^{2}}{1 - {y}^{4}}}} \]
   6. Step-by-step derivation

      1. clear-num 10.3%

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

      2. add-sqr-sqrt 10.3%

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

      3. sqrt-div 10.3%

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

      4. metadata-eval 10.3%

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

      5. metadata-eval 10.3%

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

      6. metadata-eval 10.3%

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

      7. pow-prod-up 10.3%

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

      8. flip-+ 10.3%

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

      9. metadata-eval 10.3%

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

      10. unpow2 10.3%

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

      11. hypot-udef 10.3%

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

      12. sqrt-div 10.3%

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

      13. metadata-eval 10.3%

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

      14. metadata-eval 10.3%

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

      15. metadata-eval 10.3%

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

      16. pow-prod-up 10.3%

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

   7. Applied egg-rr 89.8%

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

      1. unpow2 89.8%

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

   9. Simplified 89.8%

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

   10. Taylor expanded in y around inf 89.8%

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

       1. *-un-lft-identity 89.8%

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

       2. unpow2 89.8%

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

       3. times-frac 99.8%

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

       4. clear-num 99.8%

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

       5. /-rgt-identity 99.8%

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

   12. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{y \cdot \frac{x}{\frac{1}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+195}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{1}{y}}\\ \end{array} \]

Alternative 3: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(y \cdot y + 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (+ (* y y) 1.0)))

C

double code(double x, double y) {
	return x * ((y * y) + 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * ((y * y) + 1.0d0)
end function

Java

public static double code(double x, double y) {
	return x * ((y * y) + 1.0);
}

Python

def code(x, y):
	return x * ((y * y) + 1.0)

Julia

function code(x, y)
	return Float64(x * Float64(Float64(y * y) + 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = x * ((y * y) + 1.0);
end

Wolfram

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

Derivation

1. Initial program 95.9%

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

2. Final simplification 95.9%

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

Alternative 4: 52.0% 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 95.9%

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

2. Taylor expanded in y around 0 53.0%

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

3. Final simplification 53.0%

   \[\leadsto x \]

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 2023313 
(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))))