Numeric.Log:$clog1p from log-domain-0.10.2.1, A

Percentage Accurate: 99.9% → 100.0%

Time: 5.9s

Alternatives: 6

Speedup: 1.2×

• 43.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))

C

double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) + (x * x)) + (y * y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Python

def code(x, y):
	return ((x * 2.0) + (x * x)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) + Float64(x * x)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) + (x * x)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 2 + x \cdot x\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))

C

double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * 2.0d0) + (x * x)) + (y * y)
end function

Java

public static double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}

Python

def code(x, y):
	return ((x * 2.0) + (x * x)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * 2.0) + Float64(x * x)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * 2.0) + (x * x)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x + 2\right) + y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* x (+ x 2.0)) (* y y)))

C

double code(double x, double y) {
	return (x * (x + 2.0)) + (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * (x + 2.0d0)) + (y * y)
end function

Java

public static double code(double x, double y) {
	return (x * (x + 2.0)) + (y * y);
}

Python

def code(x, y):
	return (x * (x + 2.0)) + (y * y)

Julia

function code(x, y)
	return Float64(Float64(x * Float64(x + 2.0)) + Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * (x + 2.0)) + (y * y);
end

Wolfram

code[x_, y_] := N[(N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot 2 + x \cdot x\right), \color{blue}{\left(y \cdot y\right)}\right) \]

   2. distribute-lft-out N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x + 2\right)\right), \left(y \cdot y\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \left(y \cdot y\right)\right) \]

   6. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(x + 2\right) + y \cdot y} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 96.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-199}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e-199) (* x (+ x 2.0)) (+ (* y y) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-199) {
		tmp = x * (x + 2.0);
	} else {
		tmp = (y * y) + (x * 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-199) then
        tmp = x * (x + 2.0d0)
    else
        tmp = (y * y) + (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-199) {
		tmp = x * (x + 2.0);
	} else {
		tmp = (y * y) + (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2e-199:
		tmp = x * (x + 2.0)
	else:
		tmp = (y * y) + (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e-199)
		tmp = Float64(x * Float64(x + 2.0));
	else
		tmp = Float64(Float64(y * y) + Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 2e-199)
		tmp = x * (x + 2.0);
	else
		tmp = (y * y) + (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e-199], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.99999999999999996e-199

   1. Initial program 100.0%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot 2 + x \cdot x\right), \color{blue}{\left(y \cdot y\right)}\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x + 2\right)\right), \left(y \cdot y\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \left(y \cdot y\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(x + 2\right) + y \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. fma-define N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lft-mult-inverse N/A

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

      9. *-lft-identity N/A

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

      10. fma-define N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(2 + x\right)}\right) \]

      14. +-lowering-+.f64 98.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \color{blue}{x}\right)\right) \]

   7. Simplified 98.1%

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

   if 1.99999999999999996e-199 < (*.f64 y y)

   1. Initial program 100.0%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot 2 + x \cdot x\right), \color{blue}{\left(y \cdot y\right)}\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x + 2\right)\right), \left(y \cdot y\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \left(y \cdot y\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(x + 2\right) + y \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({x}^{2}\right)}, \mathsf{*.f64}\left(y, y\right)\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot x\right), \mathsf{*.f64}\left(\color{blue}{y}, y\right)\right) \]

      2. *-lowering-*.f64 97.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\color{blue}{y}, y\right)\right) \]

   7. Simplified 97.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-199}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+59}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+33}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.4e+59) (* x x) (if (<= x 9.8e+33) (* y y) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+59) {
		tmp = x * x;
	} else if (x <= 9.8e+33) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.4d+59)) then
        tmp = x * x
    else if (x <= 9.8d+33) then
        tmp = y * y
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+59) {
		tmp = x * x;
	} else if (x <= 9.8e+33) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.4e+59:
		tmp = x * x
	elif x <= 9.8e+33:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.4e+59)
		tmp = Float64(x * x);
	elseif (x <= 9.8e+33)
		tmp = Float64(y * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.4e+59)
		tmp = x * x;
	elseif (x <= 9.8e+33)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.4e+59], N[(x * x), $MachinePrecision], If[LessEqual[x, 9.8e+33], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000002e59 or 9.80000000000000027e33 < x

   1. Initial program 100.0%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot 2 + x \cdot x\right), \color{blue}{\left(y \cdot y\right)}\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x + 2\right)\right), \left(y \cdot y\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \left(y \cdot y\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(x + 2\right) + y \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 86.0%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   7. Simplified 86.0%

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

   if -2.4000000000000002e59 < x < 9.80000000000000027e33

   1. Initial program 100.0%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot 2 + x \cdot x\right), \color{blue}{\left(y \cdot y\right)}\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x + 2\right)\right), \left(y \cdot y\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \left(y \cdot y\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(x + 2\right) + y \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

         \[\leadsto y \cdot \color{blue}{y} \]

      2. *-lowering-*.f64 64.1%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{y}\right) \]

   7. Simplified 64.1%

      \[\leadsto \color{blue}{y \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 59.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-9}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e-9) (* x x) (if (<= x 2.0) (* x 2.0) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-9) {
		tmp = x * x;
	} else if (x <= 2.0) {
		tmp = x * 2.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.8d-9)) then
        tmp = x * x
    else if (x <= 2.0d0) then
        tmp = x * 2.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-9) {
		tmp = x * x;
	} else if (x <= 2.0) {
		tmp = x * 2.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.8e-9:
		tmp = x * x
	elif x <= 2.0:
		tmp = x * 2.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.8e-9)
		tmp = Float64(x * x);
	elseif (x <= 2.0)
		tmp = Float64(x * 2.0);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.8e-9)
		tmp = x * x;
	elseif (x <= 2.0)
		tmp = x * 2.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.8e-9], N[(x * x), $MachinePrecision], If[LessEqual[x, 2.0], N[(x * 2.0), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.79999999999999984e-9 or 2 < x

   1. Initial program 100.0%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot 2 + x \cdot x\right), \color{blue}{\left(y \cdot y\right)}\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x + 2\right)\right), \left(y \cdot y\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \left(y \cdot y\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(x + 2\right) + y \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 75.9%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   7. Simplified 75.9%

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

   if -2.79999999999999984e-9 < x < 2

   1. Initial program 100.0%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot 2 + x \cdot x\right), \color{blue}{\left(y \cdot y\right)}\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x + 2\right)\right), \left(y \cdot y\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \left(y \cdot y\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(x + 2\right) + y \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. fma-define N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lft-mult-inverse N/A

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

      9. *-lft-identity N/A

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

      10. fma-define N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(2 + x\right)}\right) \]

      14. +-lowering-+.f64 43.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \color{blue}{x}\right)\right) \]

   7. Simplified 43.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{2}\right) \]
   9. Step-by-step derivation

   10. Simplified 42.3%

       \[\leadsto x \cdot \color{blue}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 83.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 7.5 \cdot 10^{-50}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 7.5e-50) (* x (+ x 2.0)) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 7.5e-50) {
		tmp = x * (x + 2.0);
	} else {
		tmp = 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) <= 7.5d-50) then
        tmp = x * (x + 2.0d0)
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 7.5e-50) {
		tmp = x * (x + 2.0);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 7.5e-50:
		tmp = x * (x + 2.0)
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 7.5e-50)
		tmp = Float64(x * Float64(x + 2.0));
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 7.5e-50)
		tmp = x * (x + 2.0);
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 7.5e-50], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 7.5e-50

   1. Initial program 100.0%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot 2 + x \cdot x\right), \color{blue}{\left(y \cdot y\right)}\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x + 2\right)\right), \left(y \cdot y\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \left(y \cdot y\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(x + 2\right) + y \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. fma-define N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lft-mult-inverse N/A

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

      9. *-lft-identity N/A

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

      10. fma-define N/A

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

      11. unpow2 N/A

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

      12. distribute-rgt-in N/A

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

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(2 + x\right)}\right) \]

      14. +-lowering-+.f64 88.5%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \color{blue}{x}\right)\right) \]

   7. Simplified 88.5%

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

   if 7.5e-50 < (*.f64 y y)

   1. Initial program 100.0%

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot 2 + x \cdot x\right), \color{blue}{\left(y \cdot y\right)}\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x + 2\right)\right), \left(y \cdot y\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \left(y \cdot y\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(x + 2\right) + y \cdot y} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

         \[\leadsto y \cdot \color{blue}{y} \]

      2. *-lowering-*.f64 81.5%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{y}\right) \]

   7. Simplified 81.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 7.5 \cdot 10^{-50}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 21.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 2 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x 2.0))

C

double code(double x, double y) {
	return x * 2.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 2.0d0
end function

Java

public static double code(double x, double y) {
	return x * 2.0;
}

Python

def code(x, y):
	return x * 2.0

Julia

function code(x, y)
	return Float64(x * 2.0)
end

MATLAB

function tmp = code(x, y)
	tmp = x * 2.0;
end

Wolfram

code[x_, y_] := N[(x * 2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot 2 + x \cdot x\right), \color{blue}{\left(y \cdot y\right)}\right) \]

   2. distribute-lft-out N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(2 + x\right)\right), \left(\color{blue}{y} \cdot y\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(x + 2\right)\right), \left(y \cdot y\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \left(y \cdot y\right)\right) \]

   6. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, 2\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(x + 2\right) + y \cdot y} \]
4. Add Preprocessing

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. associate-*l* N/A

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

   5. fma-define N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lft-mult-inverse N/A

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

   9. *-lft-identity N/A

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

   10. fma-define N/A

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

   11. unpow2 N/A

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

   12. distribute-rgt-in N/A

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

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(2 + x\right)}\right) \]

   14. +-lowering-+.f64 61.6%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \color{blue}{x}\right)\right) \]

7. Simplified 61.6%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{2}\right) \]
9. Step-by-step derivation

10. Simplified 20.7%

    \[\leadsto x \cdot \color{blue}{2} \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot y + \left(2 \cdot x + x \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* y y) (+ (* 2.0 x) (* x x))))

C

double code(double x, double y) {
	return (y * y) + ((2.0 * x) + (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y * y) + ((2.0d0 * x) + (x * x))
end function

Java

public static double code(double x, double y) {
	return (y * y) + ((2.0 * x) + (x * x));
}

Python

def code(x, y):
	return (y * y) + ((2.0 * x) + (x * x))

Julia

function code(x, y)
	return Float64(Float64(y * y) + Float64(Float64(2.0 * x) + Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (y * y) + ((2.0 * x) + (x * x));
end

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(N[(2.0 * x), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024145 
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* y y) (+ (* 2 x) (* x x))))

  (+ (+ (* x 2.0) (* x x)) (* y y)))