Numeric.Integration.TanhSinh:everywhere from integration-0.2.1

Percentage Accurate: 94.4% → 99.9%

Time: 11.0s

Alternatives: 7

Speedup: 1.0×

• 24.7% 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: 94.4% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{1}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+307) (* x (+ (* y y) 1.0)) (* y (/ x (/ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+307) {
		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+307) 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+307) {
		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+307:
		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+307)
		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+307)
		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+307], 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) < 5e307

   1. Initial program 99.9%

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

   if 5e307 < (*.f64 y y)

   1. Initial program 77.2%

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

      1. distribute-rgt-in 77.2%

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

      2. *-un-lft-identity 77.2%

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

      3. +-commutative 77.2%

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

      4. associate-*l* 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. flip3-+ 3.6%

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

      3. clear-num 3.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(y \cdot \left(y \cdot x\right)\right) \cdot \left(y \cdot \left(y \cdot x\right)\right) - x \cdot \left(y \cdot \left(y \cdot x\right)\right)\right)}{{x}^{3} + {\left(y \cdot \left(y \cdot x\right)\right)}^{3}}}} \]

      4. clear-num 3.6%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(y \cdot \left(y \cdot x\right)\right)}^{3}}{x \cdot x + \left(\left(y \cdot \left(y \cdot x\right)\right) \cdot \left(y \cdot \left(y \cdot x\right)\right) - x \cdot \left(y \cdot \left(y \cdot x\right)\right)\right)}}}} \]

      5. flip3-+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-*r* 77.2%

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

      8. *-commutative 77.2%

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

      9. fma-def 77.2%

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

   5. Applied egg-rr 77.2%

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

   6. Taylor expanded in y around inf 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-/r* 77.2%

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

      3. unpow2 77.2%

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

      4. associate-/r* 81.2%

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

   8. Simplified 81.2%

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

      1. clear-num 81.2%

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

      2. *-un-lft-identity 81.2%

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

      3. div-inv 81.2%

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

      4. times-frac 99.9%

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

      5. clear-num 99.9%

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

      6. /-rgt-identity 99.9%

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{1}{y}}\\ \end{array} \]

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 94.8%

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

   1. distribute-rgt-in 94.8%

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

   2. *-un-lft-identity 94.8%

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

   3. +-commutative 94.8%

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

   4. associate-*l* 99.9%

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

3. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. fma-def 99.9%

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

   3. *-commutative 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+124}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e+124) (* x (+ (* y y) 1.0)) (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e+124) {
		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) <= 1d+124) 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) <= 1e+124) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1e+124:
		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) <= 1e+124)
		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) <= 1e+124)
		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], 1e+124], 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) < 9.99999999999999948e123

   1. Initial program 99.9%

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

   if 9.99999999999999948e123 < (*.f64 y y)

   1. Initial program 87.3%

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

      1. distribute-rgt-in 87.3%

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

      2. *-un-lft-identity 87.3%

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

      3. +-commutative 87.3%

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

      4. associate-*l* 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. flip3-+ 15.5%

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

      3. clear-num 15.4%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(y \cdot \left(y \cdot x\right)\right) \cdot \left(y \cdot \left(y \cdot x\right)\right) - x \cdot \left(y \cdot \left(y \cdot x\right)\right)\right)}{{x}^{3} + {\left(y \cdot \left(y \cdot x\right)\right)}^{3}}}} \]

      4. clear-num 15.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(y \cdot \left(y \cdot x\right)\right)}^{3}}{x \cdot x + \left(\left(y \cdot \left(y \cdot x\right)\right) \cdot \left(y \cdot \left(y \cdot x\right)\right) - x \cdot \left(y \cdot \left(y \cdot x\right)\right)\right)}}}} \]

      5. flip3-+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-*r* 87.2%

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

      8. *-commutative 87.2%

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

      9. fma-def 87.2%

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

   5. Applied egg-rr 87.2%

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

   6. Taylor expanded in y around inf 87.2%

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

      1. *-commutative 87.2%

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

      2. associate-/r* 87.1%

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

      3. unpow2 87.1%

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

      4. associate-/r* 89.4%

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

   8. Simplified 89.4%

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

      1. clear-num 89.5%

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

      2. div-inv 87.2%

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

      3. associate-/l/ 87.2%

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

      4. remove-double-div 87.3%

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

      5. associate-*l* 99.8%

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

   10. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+124}:\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 4: 93.4% 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 99.0%

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

   if 1 < (*.f64 y y)

   1. Initial program 89.8%

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

   2. Taylor expanded in y around inf 88.8%

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

      1. unpow2 88.8%

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

   4. Simplified 88.8%

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

4. Final simplification 93.7%

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

Alternative 5: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 0.002) x (* y (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.002) {
		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 * y) <= 0.002d0) 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 * y) <= 0.002) {
		tmp = x;
	} else {
		tmp = y * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.002:
		tmp = x
	else:
		tmp = y * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.002)
		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 * y) <= 0.002)
		tmp = x;
	else
		tmp = y * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.002], x, N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2e-3

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.0%

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

   if 2e-3 < (*.f64 y y)

   1. Initial program 89.8%

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

      1. distribute-rgt-in 89.8%

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

      2. *-un-lft-identity 89.8%

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

      3. +-commutative 89.8%

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

      4. associate-*l* 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

      2. flip3-+ 22.9%

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

      3. clear-num 22.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(y \cdot \left(y \cdot x\right)\right) \cdot \left(y \cdot \left(y \cdot x\right)\right) - x \cdot \left(y \cdot \left(y \cdot x\right)\right)\right)}{{x}^{3} + {\left(y \cdot \left(y \cdot x\right)\right)}^{3}}}} \]

      4. clear-num 22.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(y \cdot \left(y \cdot x\right)\right)}^{3}}{x \cdot x + \left(\left(y \cdot \left(y \cdot x\right)\right) \cdot \left(y \cdot \left(y \cdot x\right)\right) - x \cdot \left(y \cdot \left(y \cdot x\right)\right)\right)}}}} \]

      5. flip3-+ 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-*r* 89.7%

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

      8. *-commutative 89.7%

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

      9. fma-def 89.7%

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

   5. Applied egg-rr 89.7%

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

   6. Taylor expanded in y around inf 88.7%

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

      1. *-commutative 88.7%

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

      2. associate-/r* 88.5%

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

      3. unpow2 88.5%

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

      4. associate-/r* 90.3%

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

   8. Simplified 90.3%

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

      1. clear-num 90.5%

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

      2. div-inv 88.6%

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

      3. associate-/l/ 88.7%

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

      4. remove-double-div 88.8%

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

      5. associate-*l* 98.7%

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

   10. Applied egg-rr 98.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

   1. distribute-rgt-in 94.8%

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

   2. *-un-lft-identity 94.8%

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

   3. +-commutative 94.8%

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

   4. associate-*l* 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 7: 52.5% 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 94.8%

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

2. Taylor expanded in y around 0 50.6%

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

3. Final simplification 50.6%

   \[\leadsto x \]

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 2023285 
(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))))