Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.1% → 99.9%

Time: 3.6s

Alternatives: 8

Speedup: 0.8×

• 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: 94.1% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1000000000000:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1000000000000.0) (fma x (* y y) x) (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1000000000000.0) {
		tmp = fma(x, (y * y), x);
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1000000000000.0)
		tmp = fma(x, Float64(y * y), x);
	else
		tmp = Float64(y * Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1000000000000.0], N[(x * N[(y * y), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1e12

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

   if 1e12 < (*.f64 y y)

   1. Initial program 87.2%

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

   2. Taylor expanded in y around inf 87.2%

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

      1. unpow2 87.2%

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

      2. associate-*l* 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1000000000000:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{y \cdot x}\\ \mathsf{fma}\left(y \cdot \left(\left(\sqrt[3]{x} \cdot t_0\right) \cdot \sqrt[3]{y}\right), t_0, x\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (cbrt (* y x)))) (fma (* y (* (* (cbrt x) t_0) (cbrt y))) t_0 x)))

C

double code(double x, double y) {
	double t_0 = cbrt((y * x));
	return fma((y * ((cbrt(x) * t_0) * cbrt(y))), t_0, x);
}

Julia

function code(x, y)
	t_0 = cbrt(Float64(y * x))
	return fma(Float64(y * Float64(Float64(cbrt(x) * t_0) * cbrt(y))), t_0, x)
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[(y * x), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(y * N[(N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision] * N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + x), $MachinePrecision]]

Derivation

1. Initial program 93.6%

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

   1. +-commutative 93.6%

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

   2. distribute-lft-in 93.6%

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

   3. *-rgt-identity 93.6%

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

3. Applied egg-rr 93.6%

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

   1. *-commutative 93.6%

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

   2. associate-*r* 99.9%

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

   3. add-cube-cbrt 99.4%

      \[\leadsto y \cdot \color{blue}{\left(\left(\sqrt[3]{y \cdot x} \cdot \sqrt[3]{y \cdot x}\right) \cdot \sqrt[3]{y \cdot x}\right)} + x \]

   4. associate-*r* 99.4%

      \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt[3]{y \cdot x} \cdot \sqrt[3]{y \cdot x}\right)\right) \cdot \sqrt[3]{y \cdot x}} + x \]

   5. fma-def 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(\sqrt[3]{y \cdot x} \cdot \sqrt[3]{y \cdot x}\right), \sqrt[3]{y \cdot x}, x\right)} \]

   6. pow2 99.4%

      \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{{\left(\sqrt[3]{y \cdot x}\right)}^{2}}, \sqrt[3]{y \cdot x}, x\right) \]

   7. *-commutative 99.4%

      \[\leadsto \mathsf{fma}\left(y \cdot {\left(\sqrt[3]{\color{blue}{x \cdot y}}\right)}^{2}, \sqrt[3]{y \cdot x}, x\right) \]

   8. *-commutative 99.4%

      \[\leadsto \mathsf{fma}\left(y \cdot {\left(\sqrt[3]{x \cdot y}\right)}^{2}, \sqrt[3]{\color{blue}{x \cdot y}}, x\right) \]

5. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot {\left(\sqrt[3]{x \cdot y}\right)}^{2}, \sqrt[3]{x \cdot y}, x\right)} \]
6. Step-by-step derivation

   1. unpow2 99.4%

      \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x \cdot y}\right)}, \sqrt[3]{x \cdot y}, x\right) \]

   2. cbrt-prod 99.5%

      \[\leadsto \mathsf{fma}\left(y \cdot \left(\sqrt[3]{x \cdot y} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)}\right), \sqrt[3]{x \cdot y}, x\right) \]

   3. associate-*r* 99.5%

      \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(\left(\sqrt[3]{x \cdot y} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{y}\right)}, \sqrt[3]{x \cdot y}, x\right) \]

   4. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(y \cdot \left(\left(\sqrt[3]{\color{blue}{y \cdot x}} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{y}\right), \sqrt[3]{x \cdot y}, x\right) \]

7. Applied egg-rr 99.5%

   \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\left(\left(\sqrt[3]{y \cdot x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{y}\right)}, \sqrt[3]{x \cdot y}, x\right) \]

8. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(y \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{y \cdot x}\right) \cdot \sqrt[3]{y}\right), \sqrt[3]{y \cdot x}, x\right) \]

Alternative 3: 99.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{y \cdot x}\\ \mathsf{fma}\left(y \cdot {t_0}^{2}, t_0, x\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (cbrt (* y x)))) (fma (* y (pow t_0 2.0)) t_0 x)))

C

double code(double x, double y) {
	double t_0 = cbrt((y * x));
	return fma((y * pow(t_0, 2.0)), t_0, x);
}

Julia

function code(x, y)
	t_0 = cbrt(Float64(y * x))
	return fma(Float64(y * (t_0 ^ 2.0)), t_0, x)
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[(y * x), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(y * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0 + x), $MachinePrecision]]

Derivation

1. Initial program 93.6%

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

   1. +-commutative 93.6%

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

   2. distribute-lft-in 93.6%

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

   3. *-rgt-identity 93.6%

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

3. Applied egg-rr 93.6%

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

   1. *-commutative 93.6%

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

   2. associate-*r* 99.9%

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

   3. add-cube-cbrt 99.4%

      \[\leadsto y \cdot \color{blue}{\left(\left(\sqrt[3]{y \cdot x} \cdot \sqrt[3]{y \cdot x}\right) \cdot \sqrt[3]{y \cdot x}\right)} + x \]

   4. associate-*r* 99.4%

      \[\leadsto \color{blue}{\left(y \cdot \left(\sqrt[3]{y \cdot x} \cdot \sqrt[3]{y \cdot x}\right)\right) \cdot \sqrt[3]{y \cdot x}} + x \]

   5. fma-def 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(\sqrt[3]{y \cdot x} \cdot \sqrt[3]{y \cdot x}\right), \sqrt[3]{y \cdot x}, x\right)} \]

   6. pow2 99.4%

      \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{{\left(\sqrt[3]{y \cdot x}\right)}^{2}}, \sqrt[3]{y \cdot x}, x\right) \]

   7. *-commutative 99.4%

      \[\leadsto \mathsf{fma}\left(y \cdot {\left(\sqrt[3]{\color{blue}{x \cdot y}}\right)}^{2}, \sqrt[3]{y \cdot x}, x\right) \]

   8. *-commutative 99.4%

      \[\leadsto \mathsf{fma}\left(y \cdot {\left(\sqrt[3]{x \cdot y}\right)}^{2}, \sqrt[3]{\color{blue}{x \cdot y}}, x\right) \]

5. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot {\left(\sqrt[3]{x \cdot y}\right)}^{2}, \sqrt[3]{x \cdot y}, x\right)} \]

6. Final simplification 99.4%

   \[\leadsto \mathsf{fma}\left(y \cdot {\left(\sqrt[3]{y \cdot x}\right)}^{2}, \sqrt[3]{y \cdot x}, x\right) \]

Alternative 4: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1000000000000:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1000000000000.0) (* x (+ (* y y) 1.0)) (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1000000000000.0) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (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) <= 1000000000000.0d0) then
        tmp = x * ((y * y) + 1.0d0)
    else
        tmp = y * (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1000000000000.0) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1000000000000.0:
		tmp = x * ((y * y) + 1.0)
	else:
		tmp = y * (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1000000000000.0)
		tmp = Float64(x * Float64(Float64(y * y) + 1.0));
	else
		tmp = Float64(y * Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 1000000000000.0)
		tmp = x * ((y * y) + 1.0);
	else
		tmp = y * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1000000000000.0], N[(x * N[(N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1e12

   1. Initial program 100.0%

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

   if 1e12 < (*.f64 y y)

   1. Initial program 87.2%

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

   2. Taylor expanded in y around inf 87.2%

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

      1. unpow2 87.2%

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

      2. associate-*l* 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1000000000000:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 5: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1000000000000:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1000000000000.0) (+ x (* x (* y y))) (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1000000000000.0) {
		tmp = x + (x * (y * y));
	} else {
		tmp = y * (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) <= 1000000000000.0d0) then
        tmp = x + (x * (y * y))
    else
        tmp = y * (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1000000000000.0) {
		tmp = x + (x * (y * y));
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1000000000000.0:
		tmp = x + (x * (y * y))
	else:
		tmp = y * (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1000000000000.0)
		tmp = Float64(x + Float64(x * Float64(y * y)));
	else
		tmp = Float64(y * Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 1000000000000.0)
		tmp = x + (x * (y * y));
	else
		tmp = y * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1000000000000.0], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1e12

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

   3. Applied egg-rr 100.0%

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

   if 1e12 < (*.f64 y y)

   1. Initial program 87.2%

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

   2. Taylor expanded in y around inf 87.2%

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

      1. unpow2 87.2%

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

      2. associate-*l* 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1000000000000:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 6: 93.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.0) {
		tmp = x;
	} else {
		tmp = x * (y * 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) <= 1.0d0) then
        tmp = x
    else
        tmp = x * (y * y)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.2%

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

   if 1 < (*.f64 y y)

   1. Initial program 87.5%

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

   2. Taylor expanded in y around inf 86.8%

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

      1. unpow2 86.8%

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

      2. *-commutative 86.8%

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

   4. Simplified 86.8%

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

4. Final simplification 92.4%

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

Alternative 7: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.005:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 0.005) x (* y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.005) {
		tmp = x;
	} else {
		tmp = y * (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) <= 0.005d0) then
        tmp = x
    else
        tmp = y * (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.005) {
		tmp = x;
	} else {
		tmp = y * (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.005:
		tmp = x
	else:
		tmp = y * (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.005)
		tmp = x;
	else
		tmp = Float64(y * Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 0.005)
		tmp = x;
	else
		tmp = y * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.005], x, N[(y * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 0.0050000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.2%

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

   if 0.0050000000000000001 < (*.f64 y y)

   1. Initial program 87.5%

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

   2. Taylor expanded in y around inf 86.8%

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

      1. unpow2 86.8%

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

      2. associate-*l* 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.005:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 8: 52.3% 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 93.6%

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

2. Taylor expanded in y around 0 50.8%

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

3. Final simplification 50.8%

   \[\leadsto x \]

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 2023200 
(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))))