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

Percentage Accurate: 99.9% → 100.0%

Time: 5.3s

Alternatives: 7

Speedup: 1.5×

• 43.6% 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, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(2 + x\right) \cdot x} + y \cdot y \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(2, x, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x 2.0) (* x x)) 5e+111) (fma 2.0 x (* y y)) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * 2.0) + (x * x)) <= 5e+111) {
		tmp = fma(2.0, x, (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)) <= 5e+111)
		tmp = fma(2.0, x, Float64(y * y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], 5e+111], N[(2.0 * x + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 4.9999999999999997e111

   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 93.7

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

   5. Applied rewrites 93.7%

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

   if 4.9999999999999997e111 < (+.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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.3

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

   7. Applied rewrites 90.3%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(2, x, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(y, y, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x 2.0) (* x x)) 5e+111) (+ (fma y y x) x) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * 2.0) + (x * x)) <= 5e+111) {
		tmp = fma(y, y, x) + x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * 2.0) + Float64(x * x)) <= 5e+111)
		tmp = Float64(fma(y, y, x) + x);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], 5e+111], N[(N[(y * y + x), $MachinePrecision] + x), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 4.9999999999999997e111

   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 93.7

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

   5. Applied rewrites 93.7%

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

   7. Applied rewrites 93.7%

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

   if 4.9999999999999997e111 < (+.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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.3

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

   7. Applied rewrites 90.3%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(y, y, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 5 \cdot 10^{+111}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x 2.0) (* x x)) 5e+111) (* y y) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * 2.0) + (x * x)) <= 5e+111) {
		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)) <= 5d+111) 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)) <= 5e+111) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * 2.0) + (x * x)) <= 5e+111:
		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)) <= 5e+111)
		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)) <= 5e+111)
		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], 5e+111], 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)) < 4.9999999999999997e111

   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 61.8

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

   5. Applied rewrites 61.8%

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

   if 4.9999999999999997e111 < (+.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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.3

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

   7. Applied rewrites 90.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 5 \cdot 10^{+111}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(x, x, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1.65e-13) (+ (fma x x x) x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.65e-13) {
		tmp = fma(x, x, x) + x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1.65e-13)
		tmp = Float64(fma(x, x, x) + x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1.65e-13], N[(N[(x * x + x), $MachinePrecision] + x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.65e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 92.9

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

   5. Applied rewrites 92.9%

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

   7. Applied rewrites 92.9%

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

   if 1.65e-13 < (*.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 82.8

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

   5. Applied rewrites 82.8%

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

Alternative 6: 85.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;\left(2 + x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1.65e-13) (* (+ 2.0 x) x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.65e-13) {
		tmp = (2.0 + x) * x;
	} 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) <= 1.65d-13) then
        tmp = (2.0d0 + x) * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.65e-13) {
		tmp = (2.0 + x) * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1.65e-13:
		tmp = (2.0 + x) * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1.65e-13)
		tmp = Float64(Float64(2.0 + x) * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 1.65e-13)
		tmp = (2.0 + x) * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1.65e-13], N[(N[(2.0 + x), $MachinePrecision] * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.65e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 92.9

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

   5. Applied rewrites 92.9%

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

   if 1.65e-13 < (*.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 82.8

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

   5. Applied rewrites 82.8%

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

Alternative 7: 42.0% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 44.8

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

7. Applied rewrites 44.8%

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

8. Final simplification 44.8%

   \[\leadsto x \cdot x \]
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 2024327 
(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)))