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

Percentage Accurate: 99.9% → 100.0%

Time: 5.6s

Alternatives: 9

Speedup: 1.5×

• 43.2% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 59.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* y y) (+ (* x 2.0) (* x x))) 1.0) (* x 2.0) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((y * y) + ((x * 2.0) + (x * x))) <= 1.0) {
		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 (((y * y) + ((x * 2.0d0) + (x * x))) <= 1.0d0) 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 (((y * y) + ((x * 2.0) + (x * x))) <= 1.0) {
		tmp = x * 2.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y * y) + ((x * 2.0) + (x * x))) <= 1.0:
		tmp = x * 2.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * y) + Float64(Float64(x * 2.0) + Float64(x * x))) <= 1.0)
		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 (((y * y) + ((x * 2.0) + (x * x))) <= 1.0)
		tmp = x * 2.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) (*.f64 y y)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. lft-mult-inverse N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 78.8

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

   5. Simplified 78.8%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 75.5%

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

   if 1 < (+.f64 (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) (*.f64 y y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 50.8

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

   5. Simplified 50.8%

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

4. Final simplification 56.9%

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

Alternative 3: 99.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x 2.0) (* x x)) 1.0) (fma y y (* x 2.0)) (fma x x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (((x * 2.0) + (x * x)) <= 1.0) {
		tmp = fma(y, y, (x * 2.0));
	} else {
		tmp = fma(x, x, (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * 2.0) + Float64(x * x)) <= 1.0)
		tmp = fma(y, y, Float64(x * 2.0));
	else
		tmp = fma(x, x, Float64(y * y));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 1

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 98.3

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

   7. Simplified 98.3%

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

   if 1 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.3

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

   5. Simplified 98.3%

      \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lower-fma.f64 98.3

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

   7. Applied egg-rr 98.3%

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

4. Final simplification 98.3%

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

Alternative 4: 99.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x 2.0) (* x x)) 1.0) (fma 2.0 x (* y y)) (fma x x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (((x * 2.0) + (x * x)) <= 1.0) {
		tmp = fma(2.0, x, (y * y));
	} else {
		tmp = fma(x, x, (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * 2.0) + Float64(x * x)) <= 1.0)
		tmp = fma(2.0, x, Float64(y * y));
	else
		tmp = fma(x, x, Float64(y * y));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 98.3

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

   5. Simplified 98.3%

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

   if 1 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.3

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

   5. Simplified 98.3%

      \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lower-fma.f64 98.3

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

   7. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, y \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(2, x, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x 2.0) (* x x)) 2e+173) (fma 2.0 x (* y y)) (* x (+ x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (((x * 2.0) + (x * x)) <= 2e+173) {
		tmp = fma(2.0, x, (y * y));
	} else {
		tmp = x * (x + 2.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * 2.0) + Float64(x * x)) <= 2e+173)
		tmp = fma(2.0, x, Float64(y * y));
	else
		tmp = Float64(x * Float64(x + 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], 2e+173], N[(2.0 * x + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 2e173

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 87.7

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

   5. Simplified 87.7%

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

   if 2e173 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. lft-mult-inverse N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 90.6

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

   5. Simplified 90.6%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(2, x, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 10^{+153}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x 2.0) (* x x)) 1e+153) (* y y) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * 2.0) + (x * x)) <= 1e+153) {
		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 * 2.0d0) + (x * x)) <= 1d+153) 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 * 2.0) + (x * x)) <= 1e+153) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * 2.0) + (x * x)) <= 1e+153:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * 2.0) + Float64(x * x)) <= 1e+153)
		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 * 2.0) + (x * x)) <= 1e+153)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], 1e+153], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 1e153

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.9

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

   5. Simplified 60.9%

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

   if 1e153 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.1

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

   5. Simplified 88.1%

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

Alternative 7: 84.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5.5 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5.5e-71) (* x (+ x 2.0)) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5.5e-71) {
		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) <= 5.5d-71) 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) <= 5.5e-71) {
		tmp = x * (x + 2.0);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5.5e-71:
		tmp = x * (x + 2.0)
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5.5e-71)
		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) <= 5.5e-71)
		tmp = x * (x + 2.0);
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5.5e-71], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5.4999999999999997e-71

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l* N/A

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

      5. lft-mult-inverse N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. +-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 94.5

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

   5. Simplified 94.5%

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

   if 5.4999999999999997e-71 < (*.f64 y y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.5

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

   5. Simplified 82.5%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5.5 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 100.0

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-lft-out N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 9: 20.1% 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. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*l* N/A

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

   5. lft-mult-inverse N/A

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

   6. *-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. +-commutative N/A

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

   9. *-lft-identity N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 58.6

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

5. Simplified 58.6%

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

6. Taylor expanded in x around 0

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

8. Simplified 21.1%

   \[\leadsto x \cdot \color{blue}{2} \]
9. 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 2024207 
(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)))