Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 93.9% → 99.9%

Time: 8.3s

Alternatives: 6

Speedup: 0.5×

• 24.2% 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: 93.9% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+162}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e+162) (* x (fma y y 1.0)) (* y (/ y (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e+162) {
		tmp = x * fma(y, y, 1.0);
	} else {
		tmp = y * (y / (1.0 / x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e+162)
		tmp = Float64(x * fma(y, y, 1.0));
	else
		tmp = Float64(y * Float64(y / Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e+162], N[(x * N[(y * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(y / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.9999999999999999e162

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   if 1.9999999999999999e162 < (*.f64 y y)

   1. Initial program 80.2%

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

      1. +-commutative 80.2%

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

      2. distribute-lft-in 80.2%

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

      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-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. Step-by-step derivation

      1. fma-undefine 99.9%

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

      2. associate-*l* 80.2%

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

      3. unpow2 80.2%

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

      4. *-un-lft-identity 80.2%

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

      5. *-commutative 80.2%

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

      6. distribute-lft-in 80.2%

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

      7. +-commutative 80.2%

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

      8. flip-+ 14.4%

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

      9. metadata-eval 14.4%

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

      10. pow-sqr 14.4%

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

      11. metadata-eval 14.4%

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

      12. associate-/l* 14.4%

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

      13. *-commutative 14.4%

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

      14. clear-num 14.4%

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

      15. clear-num 14.4%

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

      16. *-commutative 14.4%

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

      17. associate-/l* 14.4%

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

      18. metadata-eval 14.4%

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

      19. metadata-eval 14.4%

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

      20. pow-sqr 14.4%

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

   6. Applied egg-rr 80.2%

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

   7. Taylor expanded in y around inf 80.2%

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

      1. associate-/r* 80.2%

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

   9. Simplified 80.2%

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

       1. clear-num 80.2%

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

       2. unpow2 80.2%

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

       3. associate-/l* 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+162}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \]
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 92.6%

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

   1. +-commutative 92.6%

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

   2. distribute-lft-in 92.6%

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

   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. 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+113}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+113) (* x (+ (* y y) 1.0)) (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+113) {
		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) <= 5d+113) 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) <= 5e+113) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e+113:
		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) <= 5e+113)
		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) <= 5e+113)
		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], 5e+113], 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) < 5e113

   1. Initial program 100.0%

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

   if 5e113 < (*.f64 y y)

   1. Initial program 81.9%

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

      1. *-commutative 81.9%

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

      2. add-sqr-sqrt 45.8%

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

      3. add-sqr-sqrt 45.8%

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

      4. pow2 45.8%

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

      5. pow2 45.8%

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

      6. unpow-prod-down 45.7%

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

      7. hypot-1-def 55.5%

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

   4. Applied egg-rr 55.5%

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

   5. Taylor expanded in y around inf 55.5%

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

      1. unpow2 55.5%

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

      2. swap-sqr 45.8%

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

      3. add-sqr-sqrt 81.9%

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

      4. associate-*r* 99.9%

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

   7. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+113}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+162}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e+162) (* x (+ (* y y) 1.0)) (* y (/ y (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e+162) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (y / (1.0 / x));
	}
	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) <= 2d+162) then
        tmp = x * ((y * y) + 1.0d0)
    else
        tmp = y * (y / (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e+162) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (y / (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2e+162:
		tmp = x * ((y * y) + 1.0)
	else:
		tmp = y * (y / (1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e+162)
		tmp = Float64(x * Float64(Float64(y * y) + 1.0));
	else
		tmp = Float64(y * Float64(y / Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 2e+162)
		tmp = x * ((y * y) + 1.0);
	else
		tmp = y * (y / (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e+162], N[(x * N[(N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(y / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.9999999999999999e162

   1. Initial program 100.0%

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

   if 1.9999999999999999e162 < (*.f64 y y)

   1. Initial program 80.2%

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

      1. +-commutative 80.2%

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

      2. distribute-lft-in 80.2%

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

      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-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. Step-by-step derivation

      1. fma-undefine 99.9%

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

      2. associate-*l* 80.2%

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

      3. unpow2 80.2%

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

      4. *-un-lft-identity 80.2%

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

      5. *-commutative 80.2%

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

      6. distribute-lft-in 80.2%

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

      7. +-commutative 80.2%

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

      8. flip-+ 14.4%

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

      9. metadata-eval 14.4%

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

      10. pow-sqr 14.4%

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

      11. metadata-eval 14.4%

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

      12. associate-/l* 14.4%

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

      13. *-commutative 14.4%

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

      14. clear-num 14.4%

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

      15. clear-num 14.4%

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

      16. *-commutative 14.4%

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

      17. associate-/l* 14.4%

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

      18. metadata-eval 14.4%

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

      19. metadata-eval 14.4%

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

      20. pow-sqr 14.4%

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

   6. Applied egg-rr 80.2%

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

   7. Taylor expanded in y around inf 80.2%

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

      1. associate-/r* 80.2%

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

   9. Simplified 80.2%

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

       1. clear-num 80.2%

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

       2. unpow2 80.2%

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

       3. associate-/l* 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+162}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% 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 93.5%

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

   3. Taylor expanded in y around 0 66.8%

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

   if 1 < y

   1. Initial program 89.4%

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

      1. *-commutative 89.4%

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

      2. add-sqr-sqrt 52.1%

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

      3. add-sqr-sqrt 52.1%

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

      4. pow2 52.1%

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

      5. pow2 52.1%

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

      6. unpow-prod-down 52.1%

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

      7. hypot-1-def 59.0%

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

   4. Applied egg-rr 59.0%

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

   5. Taylor expanded in y around inf 59.0%

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

      1. unpow2 59.0%

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

      2. swap-sqr 52.1%

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

      3. add-sqr-sqrt 89.4%

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

      4. associate-*r* 99.9%

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

   7. 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 73.8%

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

Alternative 6: 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 92.6%

   \[x \cdot \left(1 + y \cdot y\right) \]
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 2024047 
(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))))