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

Percentage Accurate: 100.0% → 100.0%

Time: 11.8s

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: 100.0% 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, 1.0× 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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 90.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.205 \lor \neg \left(x \leq 3600000\right):\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.205) (not (<= x 3600000.0)))
   (* x (+ x 2.0))
   (+ (* x 2.0) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.205) || !(x <= 3600000.0)) {
		tmp = x * (x + 2.0);
	} else {
		tmp = (x * 2.0) + (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 ((x <= (-0.205d0)) .or. (.not. (x <= 3600000.0d0))) then
        tmp = x * (x + 2.0d0)
    else
        tmp = (x * 2.0d0) + (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -0.205) || !(x <= 3600000.0)) {
		tmp = x * (x + 2.0);
	} else {
		tmp = (x * 2.0) + (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -0.205) or not (x <= 3600000.0):
		tmp = x * (x + 2.0)
	else:
		tmp = (x * 2.0) + (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.205) || !(x <= 3600000.0))
		tmp = Float64(x * Float64(x + 2.0));
	else
		tmp = Float64(Float64(x * 2.0) + Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -0.205) || ~((x <= 3600000.0)))
		tmp = x * (x + 2.0);
	else
		tmp = (x * 2.0) + (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.205], N[Not[LessEqual[x, 3600000.0]], $MachinePrecision]], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.204999999999999988 or 3.6e6 < 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. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 80.3%

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

   if -0.204999999999999988 < x < 3.6e6

   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 98.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.205 \lor \neg \left(x \leq 3600000\right):\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 7 \cdot 10^{-205}:\\ \;\;\;\;x \cdot \left(-1 + \left(x + 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 7e-205) (* x (+ -1.0 (+ x 3.0))) (+ (* x x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 7e-205) {
		tmp = x * (-1.0 + (x + 3.0));
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 7d-205) then
        tmp = x * ((-1.0d0) + (x + 3.0d0))
    else
        tmp = (x * x) + (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 7e-205) {
		tmp = x * (-1.0 + (x + 3.0));
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 7e-205:
		tmp = x * (-1.0 + (x + 3.0))
	else:
		tmp = (x * x) + (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 7e-205)
		tmp = Float64(x * Float64(-1.0 + Float64(x + 3.0)));
	else
		tmp = Float64(Float64(x * x) + Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 7e-205)
		tmp = x * (-1.0 + (x + 3.0));
	else
		tmp = (x * x) + (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 7e-205], N[(x * N[(-1.0 + N[(x + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 7.00000000000000001e-205

   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. +-commutative 99.9%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 97.9%

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

      1. expm1-log1p-u 73.5%

         \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 + x\right)\right)} \]

      2. expm1-undefine 73.5%

         \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 + x\right)} - 1\right)} \]

      3. +-commutative 73.5%

         \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{x + 2}\right)} - 1\right) \]

   9. Applied egg-rr 73.5%

      \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x + 2\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. sub-neg 73.5%

          \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x + 2\right)} + \left(-1\right)\right)} \]

       2. metadata-eval 73.5%

          \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(x + 2\right)} + \color{blue}{-1}\right) \]

       3. +-commutative 73.5%

          \[\leadsto x \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(x + 2\right)}\right)} \]

       4. log1p-undefine 73.5%

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

       5. rem-exp-log 97.9%

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

       6. +-commutative 97.9%

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

       7. associate-+r+ 98.0%

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

       8. metadata-eval 98.0%

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

   11. Simplified 98.0%

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

   if 7.00000000000000001e-205 < (*.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 96.8%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 7 \cdot 10^{-205}:\\ \;\;\;\;x \cdot \left(-1 + \left(x + 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.8% 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. +-commutative 100.0%

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

      2. fma-define 100.0%

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

      3. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 79.3%

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

   8. Taylor expanded in x around inf 77.6%

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

   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 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 99.1%

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

   6. Taylor expanded in x around inf 42.4%

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

4. Final simplification 60.0%

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

Alternative 5: 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 6: 59.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(-1 + \left(x + 3\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (+ -1.0 (+ x 3.0))))

C

double code(double x, double y) {
	return x * (-1.0 + (x + 3.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * ((-1.0d0) + (x + 3.0d0))
end function

Java

public static double code(double x, double y) {
	return x * (-1.0 + (x + 3.0));
}

Python

def code(x, y):
	return x * (-1.0 + (x + 3.0))

Julia

function code(x, y)
	return Float64(x * Float64(-1.0 + Float64(x + 3.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (-1.0 + (x + 3.0));
end

Wolfram

code[x_, y_] := N[(x * N[(-1.0 + N[(x + 3.0), $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. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. fma-define 100.0%

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

   3. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in y around 0 61.3%

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

   1. expm1-log1p-u 41.2%

      \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 + x\right)\right)} \]

   2. expm1-undefine 41.2%

      \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 + x\right)} - 1\right)} \]

   3. +-commutative 41.2%

      \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{x + 2}\right)} - 1\right) \]

9. Applied egg-rr 41.2%

   \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x + 2\right)} - 1\right)} \]
10. Step-by-step derivation

    1. sub-neg 41.2%

       \[\leadsto x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x + 2\right)} + \left(-1\right)\right)} \]

    2. metadata-eval 41.2%

       \[\leadsto x \cdot \left(e^{\mathsf{log1p}\left(x + 2\right)} + \color{blue}{-1}\right) \]

    3. +-commutative 41.2%

       \[\leadsto x \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(x + 2\right)}\right)} \]

    4. log1p-undefine 41.2%

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

    5. rem-exp-log 61.2%

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

    6. +-commutative 61.2%

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

    7. associate-+r+ 61.3%

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

    8. metadata-eval 61.3%

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

11. Simplified 61.3%

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

12. Final simplification 61.3%

    \[\leadsto x \cdot \left(-1 + \left(x + 3\right)\right) \]
13. Add Preprocessing

Alternative 7: 59.9% 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. +-commutative 100.0%

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

   2. fma-define 100.0%

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

   3. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in y around 0 61.3%

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

8. Final simplification 61.3%

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

Alternative 8: 19.9% 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 67.9%

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

6. Taylor expanded in x around inf 23.1%

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

7. Final simplification 23.1%

   \[\leadsto x \cdot 2 \]
8. Add Preprocessing

Developer Target 1: 100.0% 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 2024143 
(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)))