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

Percentage Accurate: 99.9% → 99.9%

Time: 5.1s

Alternatives: 8

Speedup: 1.2×

• 43.8% 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: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(x * 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. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. fma-define 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 90.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+38}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+25}:\\ \;\;\;\;x \cdot 2 + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8e+38)
   (* x x)
   (if (<= x 3.1e+25) (+ (* x 2.0) (* y y)) (* x (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8e+38) {
		tmp = x * x;
	} else if (x <= 3.1e+25) {
		tmp = (x * 2.0) + (y * y);
	} else {
		tmp = x * (x + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8d+38)) then
        tmp = x * x
    else if (x <= 3.1d+25) then
        tmp = (x * 2.0d0) + (y * y)
    else
        tmp = x * (x + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8e+38) {
		tmp = x * x;
	} else if (x <= 3.1e+25) {
		tmp = (x * 2.0) + (y * y);
	} else {
		tmp = x * (x + 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8e+38:
		tmp = x * x
	elif x <= 3.1e+25:
		tmp = (x * 2.0) + (y * y)
	else:
		tmp = x * (x + 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8e+38)
		tmp = Float64(x * x);
	elseif (x <= 3.1e+25)
		tmp = Float64(Float64(x * 2.0) + Float64(y * y));
	else
		tmp = Float64(x * Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8e+38)
		tmp = x * x;
	elseif (x <= 3.1e+25)
		tmp = (x * 2.0) + (y * y);
	else
		tmp = x * (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8e+38], N[(x * x), $MachinePrecision], If[LessEqual[x, 3.1e+25], N[(N[(x * 2.0), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.99999999999999982e38

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-define 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 88.3%

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

      1. unpow2 88.3%

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

      2. associate-*l* 88.3%

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

      3. distribute-rgt-in 88.3%

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

      4. *-lft-identity 88.3%

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

      5. +-commutative 88.3%

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

      6. associate-*l* 88.3%

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

      7. lft-mult-inverse 88.3%

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

      8. metadata-eval 88.3%

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

   9. Simplified 88.3%

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

   10. Taylor expanded in x around inf 88.3%

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

   if -7.99999999999999982e38 < x < 3.0999999999999998e25

   1. Initial program 100.0%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 96.5%

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

   if 3.0999999999999998e25 < x

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-define 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 82.1%

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

      1. unpow2 82.2%

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

      2. associate-*l* 82.2%

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

      3. distribute-rgt-in 82.2%

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

      4. *-lft-identity 82.2%

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

      5. +-commutative 82.2%

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

      6. associate-*l* 82.2%

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

      7. lft-mult-inverse 82.2%

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

      8. metadata-eval 82.2%

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

   9. Simplified 82.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+38}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+25}:\\ \;\;\;\;x \cdot 2 + y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5.6 \cdot 10^{-198}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5.6e-198) (* x (+ x 2.0)) (+ (* y y) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5.6e-198) {
		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) <= 5.6d-198) 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) <= 5.6e-198) {
		tmp = x * (x + 2.0);
	} else {
		tmp = (y * y) + (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5.6e-198:
		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) <= 5.6e-198)
		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) <= 5.6e-198)
		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], 5.6e-198], 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) < 5.5999999999999998e-198

   1. Initial program 100.0%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-define 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 80.0%

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

      1. unpow2 80.0%

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

      2. associate-*l* 98.7%

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

      3. distribute-rgt-in 98.6%

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

      4. *-lft-identity 98.6%

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

      5. +-commutative 98.6%

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

      6. associate-*l* 98.6%

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

      7. lft-mult-inverse 98.7%

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

      8. metadata-eval 98.7%

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

   9. Simplified 98.7%

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

   if 5.5999999999999998e-198 < (*.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. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 97.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5.6 \cdot 10^{-198}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.0) (not (<= x 2.0))) (* x x) (* x 2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.0) || !(x <= 2.0)) {
		tmp = x * x;
	} else {
		tmp = x * 2.0;
	}
	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)) .or. (.not. (x <= 2.0d0))) then
        tmp = x * x
    else
        tmp = x * 2.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if (x <= -2.0) or not (x <= 2.0):
		tmp = x * x
	else:
		tmp = x * 2.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.0) || !(x <= 2.0))
		tmp = Float64(x * x);
	else
		tmp = Float64(x * 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.0) || ~((x <= 2.0)))
		tmp = x * x;
	else
		tmp = x * 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.0], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(x * x), $MachinePrecision], N[(x * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2 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. distribute-lft-out 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. +-commutative 100.0%

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

      3. fma-define 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 80.2%

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

      1. unpow2 80.2%

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

      2. associate-*l* 80.2%

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

      3. distribute-rgt-in 80.2%

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

      4. *-lft-identity 80.2%

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

      5. +-commutative 80.2%

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

      6. associate-*l* 80.2%

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

      7. lft-mult-inverse 80.2%

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

      8. metadata-eval 80.2%

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

   9. Simplified 80.2%

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

   10. Taylor expanded in x around inf 79.5%

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

   if -2 < x < 2

   1. Initial program 100.0%

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

      1. distribute-lft-out 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 98.5%

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

   6. Taylor expanded in x around inf 36.0%

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

4. Final simplification 58.9%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 6: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + 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. Step-by-step derivation

   1. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 7: 60.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(x + 2.0), $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. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. +-commutative 100.0%

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

   3. fma-define 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 54.1%

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

   1. unpow2 54.2%

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

   2. associate-*l* 59.9%

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

   3. distribute-rgt-in 59.9%

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

   4. *-lft-identity 59.9%

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

   5. +-commutative 59.9%

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

   6. associate-*l* 59.9%

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

   7. lft-mult-inverse 60.0%

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

   8. metadata-eval 60.0%

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

9. Simplified 60.0%

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

10. Final simplification 60.0%

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

Alternative 8: 20.4% 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. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 65.5%

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

6. Taylor expanded in x around inf 19.1%

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

7. Final simplification 19.1%

   \[\leadsto x \cdot 2 \]
8. 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 2024123 
(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)))