Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 93.8% → 99.9%

Time: 6.5s

Alternatives: 6

Speedup: 0.6×

• 25.6% 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: 93.8% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 3 \cdot 10^{+293}:\\ \;\;\;\;x + x \cdot {y}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{1}{y}}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 3e+293) (+ x (* x (pow y 2.0))) (/ y (/ (/ 1.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 3e+293) {
		tmp = x + (x * pow(y, 2.0));
	} else {
		tmp = y / ((1.0 / y) / 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) <= 3d+293) then
        tmp = x + (x * (y ** 2.0d0))
    else
        tmp = y / ((1.0d0 / y) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 3e+293) {
		tmp = x + (x * Math.pow(y, 2.0));
	} else {
		tmp = y / ((1.0 / y) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 3e+293:
		tmp = x + (x * math.pow(y, 2.0))
	else:
		tmp = y / ((1.0 / y) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 3e+293)
		tmp = Float64(x + Float64(x * (y ^ 2.0)));
	else
		tmp = Float64(y / Float64(Float64(1.0 / y) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 3e+293)
		tmp = x + (x * (y ^ 2.0));
	else
		tmp = y / ((1.0 / y) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 3e+293], N[(x + N[(x * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 3.00000000000000013e293

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft1-in 99.9%

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

      4. *-commutative 99.9%

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

      5. pow2 99.9%

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

   3. Applied egg-rr 99.9%

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

   if 3.00000000000000013e293 < (*.f64 y y)

   1. Initial program 77.5%

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

      1. flip-+ 2.7%

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

      2. associate-*r/ 2.7%

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

      3. associate-/l* 2.7%

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

      4. clear-num 2.7%

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

      5. flip-+ 77.5%

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

      6. +-commutative 77.5%

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

      7. fma-def 77.5%

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

   3. Applied egg-rr 77.5%

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

   4. Taylor expanded in y around inf 77.5%

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

      1. associate-/r/ 77.5%

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

      2. unpow2 77.5%

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

      3. associate-*r* 99.8%

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

      4. /-rgt-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. remove-double-div 99.8%

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

      3. div-inv 99.8%

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

      4. clear-num 99.8%

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

      5. un-div-inv 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 3 \cdot 10^{+293}:\\ \;\;\;\;x + x \cdot {y}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{1}{y}}{x}}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y) {
	return Math.hypot(1.0, y) * (x * Math.hypot(1.0, y));
}

Python

def code(x, y):
	return math.hypot(1.0, y) * (x * math.hypot(1.0, y))

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[Sqrt[1.0 ^ 2 + y ^ 2], $MachinePrecision] * N[(x * N[Sqrt[1.0 ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.0%

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

   1. add-sqr-sqrt 52.0%

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

   2. pow2 52.0%

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

   3. sqrt-prod 52.0%

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

   4. hypot-1-def 56.7%

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

3. Applied egg-rr 56.7%

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

   1. *-commutative 56.7%

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

5. Simplified 56.7%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, y\right) \cdot \sqrt{x}\right)}^{2}} \]
6. Step-by-step derivation

   1. unpow2 56.7%

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

   2. swap-sqr 52.0%

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

   3. unpow2 52.0%

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

   4. hypot-1-def 52.0%

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

   5. unpow2 52.0%

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

   6. +-commutative 52.0%

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

   7. unpow2 52.0%

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

   8. fma-udef 52.0%

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

   9. pow2 52.0%

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

   10. add-sqr-sqrt 52.0%

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

   11. add-sqr-sqrt 93.0%

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

   12. /-rgt-identity 93.0%

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

   13. *-commutative 93.0%

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

   14. add-sqr-sqrt 93.0%

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

   15. associate-*r* 92.9%

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

   16. /-rgt-identity 92.9%

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

   17. fma-udef 92.9%

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

   18. unpow2 92.9%

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

   19. +-commutative 92.9%

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

   20. unpow2 92.9%

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

   21. hypot-1-def 92.9%

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

   22. fma-udef 92.9%

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

   23. unpow2 92.9%

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

   24. +-commutative 92.9%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 3 \cdot 10^{+195}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 3e+195) (* x (+ 1.0 (* y y))) (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 3e+195) {
		tmp = x * (1.0 + (y * y));
	} 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) <= 3d+195) then
        tmp = x * (1.0d0 + (y * y))
    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) <= 3e+195) {
		tmp = x * (1.0 + (y * y));
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 3e+195:
		tmp = x * (1.0 + (y * y))
	else:
		tmp = y * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 3e+195)
		tmp = Float64(x * Float64(1.0 + Float64(y * y)));
	else
		tmp = Float64(y * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 3e+195)
		tmp = x * (1.0 + (y * y));
	else
		tmp = y * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 3e+195], N[(x * N[(1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 3.0000000000000001e195

   1. Initial program 99.9%

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

   if 3.0000000000000001e195 < (*.f64 y y)

   1. Initial program 82.5%

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

      1. flip-+ 11.6%

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

      2. associate-*r/ 11.6%

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

      3. associate-/l* 11.6%

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

      4. clear-num 11.6%

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

      5. flip-+ 82.5%

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

      6. +-commutative 82.5%

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

      7. fma-def 82.5%

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

   3. Applied egg-rr 82.5%

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

   4. Taylor expanded in y around inf 82.5%

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

      1. associate-/r/ 82.5%

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

      2. unpow2 82.5%

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

      3. associate-*r* 99.8%

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

      4. /-rgt-identity 99.8%

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

   6. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 3 \cdot 10^{+195}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 3 \cdot 10^{+293}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{1}{y}}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 3e+293) (* x (+ 1.0 (* y y))) (/ y (/ (/ 1.0 y) x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 3e+293) {
		tmp = x * (1.0 + (y * y));
	} else {
		tmp = y / ((1.0 / y) / 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) <= 3d+293) then
        tmp = x * (1.0d0 + (y * y))
    else
        tmp = y / ((1.0d0 / y) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 3e+293) {
		tmp = x * (1.0 + (y * y));
	} else {
		tmp = y / ((1.0 / y) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 3e+293:
		tmp = x * (1.0 + (y * y))
	else:
		tmp = y / ((1.0 / y) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 3e+293)
		tmp = Float64(x * Float64(1.0 + Float64(y * y)));
	else
		tmp = Float64(y / Float64(Float64(1.0 / y) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 3e+293)
		tmp = x * (1.0 + (y * y));
	else
		tmp = y / ((1.0 / y) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 3e+293], N[(x * N[(1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 3.00000000000000013e293

   1. Initial program 99.9%

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

   if 3.00000000000000013e293 < (*.f64 y y)

   1. Initial program 77.5%

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

      1. flip-+ 2.7%

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

      2. associate-*r/ 2.7%

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

      3. associate-/l* 2.7%

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

      4. clear-num 2.7%

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

      5. flip-+ 77.5%

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

      6. +-commutative 77.5%

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

      7. fma-def 77.5%

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

   3. Applied egg-rr 77.5%

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

   4. Taylor expanded in y around inf 77.5%

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

      1. associate-/r/ 77.5%

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

      2. unpow2 77.5%

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

      3. associate-*r* 99.8%

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

      4. /-rgt-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. remove-double-div 99.8%

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

      3. div-inv 99.8%

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

      4. clear-num 99.8%

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

      5. un-div-inv 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 3 \cdot 10^{+293}:\\ \;\;\;\;x \cdot \left(1 + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\frac{1}{y}}{x}}\\ \end{array} \]

Alternative 5: 75.0% 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 94.5%

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

   2. Taylor expanded in y around 0 62.9%

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

   if 1 < y

   1. Initial program 88.7%

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

      1. flip-+ 32.2%

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

      2. associate-*r/ 29.5%

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

      3. associate-/l* 32.2%

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

      4. clear-num 32.2%

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

      5. flip-+ 88.7%

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

      6. +-commutative 88.7%

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

      7. fma-def 88.7%

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

   3. Applied egg-rr 88.7%

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

   4. Taylor expanded in y around inf 87.0%

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

      1. associate-/r/ 87.0%

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

      2. unpow2 87.0%

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

      3. associate-*r* 98.0%

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

      4. /-rgt-identity 98.0%

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

   6. Applied egg-rr 98.0%

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

4. Final simplification 72.1%

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

Alternative 6: 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 93.0%

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

2. Taylor expanded in y around 0 47.6%

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

3. Final simplification 47.6%

   \[\leadsto x \]

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 2023326 
(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))))