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

Percentage Accurate: 99.9% → 100.0%

Time: 4.5s

Alternatives: 7

Speedup: 1.5×

• 43.7% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(x + 2.0), $MachinePrecision] * x + 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. Step-by-step derivation

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. +-commutative N/A

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

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

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

   6. *-lowering-*.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot y + \left(x \cdot 2 + x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(2, x, y \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+17}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* y y) (+ (* x 2.0) (* x x)))))
   (if (<= t_0 5e-7)
     (fma 2.0 x (* y y))
     (if (<= t_0 5e+17) (* x (+ x 2.0)) (fma x x (* y y))))))

C

double code(double x, double y) {
	double t_0 = (y * y) + ((x * 2.0) + (x * x));
	double tmp;
	if (t_0 <= 5e-7) {
		tmp = fma(2.0, x, (y * y));
	} else if (t_0 <= 5e+17) {
		tmp = x * (x + 2.0);
	} else {
		tmp = fma(x, x, (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) + Float64(Float64(x * 2.0) + Float64(x * x)))
	tmp = 0.0
	if (t_0 <= 5e-7)
		tmp = fma(2.0, x, Float64(y * y));
	elseif (t_0 <= 5e+17)
		tmp = Float64(x * Float64(x + 2.0));
	else
		tmp = fma(x, x, Float64(y * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-7], N[(2.0 * x + N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+17], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) (*.f64 y y)) < 4.99999999999999977e-7

   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. accelerator-lowering-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. *-lowering-*.f64 98.5

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

   5. Simplified 98.5%

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

   if 4.99999999999999977e-7 < (+.f64 (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) (*.f64 y y)) < 5e17

   1. Initial program 99.8%

      \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
   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. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 97.6

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

   5. Simplified 97.6%

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

   if 5e17 < (+.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. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-commutative N/A

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

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

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

      6. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y + \left(x \cdot 2 + x \cdot x\right) \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(2, x, y \cdot y\right)\\ \mathbf{elif}\;y \cdot y + \left(x \cdot 2 + x \cdot x\right) \leq 5 \cdot 10^{+17}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 2 + x \cdot x\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-156}:\\ \;\;\;\;x \cdot 2\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+139}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* x 2.0) (* x x))))
   (if (<= t_0 -5e-156) (* x 2.0) (if (<= t_0 2e+139) (* y y) (* x x)))))

C

double code(double x, double y) {
	double t_0 = (x * 2.0) + (x * x);
	double tmp;
	if (t_0 <= -5e-156) {
		tmp = x * 2.0;
	} else if (t_0 <= 2e+139) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (x * 2.0d0) + (x * x)
    if (t_0 <= (-5d-156)) then
        tmp = x * 2.0d0
    else if (t_0 <= 2d+139) then
        tmp = y * y
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * 2.0) + (x * x);
	double tmp;
	if (t_0 <= -5e-156) {
		tmp = x * 2.0;
	} else if (t_0 <= 2e+139) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * 2.0) + (x * x)
	tmp = 0
	if t_0 <= -5e-156:
		tmp = x * 2.0
	elif t_0 <= 2e+139:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * 2.0) + Float64(x * x))
	tmp = 0.0
	if (t_0 <= -5e-156)
		tmp = Float64(x * 2.0);
	elseif (t_0 <= 2e+139)
		tmp = Float64(y * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * 2.0) + (x * x);
	tmp = 0.0;
	if (t_0 <= -5e-156)
		tmp = x * 2.0;
	elseif (t_0 <= 2e+139)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-156], N[(x * 2.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+139], N[(y * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < -5.00000000000000007e-156

   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. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 62.2

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

   5. Simplified 62.2%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 59.8%

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

   if -5.00000000000000007e-156 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 2.00000000000000007e139

   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. *-lowering-*.f64 69.5

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

   5. Simplified 69.5%

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

   if 2.00000000000000007e139 < (+.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. *-lowering-*.f64 90.6

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

   5. Simplified 90.6%

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

Alternative 4: 58.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y + \left(x \cdot 2 + x \cdot x\right) \leq 0.0001:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* y y) (+ (* x 2.0) (* x x))) 0.0001) (* x 2.0) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((y * y) + ((x * 2.0) + (x * x))) <= 0.0001) {
		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))) <= 0.0001d0) 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))) <= 0.0001) {
		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))) <= 0.0001:
		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))) <= 0.0001)
		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))) <= 0.0001)
		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], 0.0001], 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)) < 1.00000000000000005e-4

   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. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 84.1

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

   5. Simplified 84.1%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 80.0%

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

   if 1.00000000000000005e-4 < (+.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. *-lowering-*.f64 50.3

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

   5. Simplified 50.3%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y + \left(x \cdot 2 + x \cdot x\right) \leq 0.0001:\\ \;\;\;\;x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 4 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(2, x, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x 2.0) (* x x)) 4e+94) (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)) <= 4e+94) {
		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)) <= 4e+94)
		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], 4e+94], 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)) < 4.0000000000000001e94

   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. accelerator-lowering-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. *-lowering-*.f64 93.0

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

   5. Simplified 93.0%

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

   if 4.0000000000000001e94 < (+.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. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 89.1

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

   5. Simplified 89.1%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 4 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(2, x, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 9.5 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 9.5e+24) (* x (+ x 2.0)) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 9.5e+24) {
		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) <= 9.5d+24) 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) <= 9.5e+24) {
		tmp = x * (x + 2.0);
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 9.5e+24:
		tmp = x * (x + 2.0)
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 9.5e+24)
		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) <= 9.5e+24)
		tmp = x * (x + 2.0);
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 9.5e+24], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 9.5000000000000001e24

   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. *-lowering-*.f64 N/A

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

      11. +-lowering-+.f64 91.2

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

   5. Simplified 91.2%

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

   if 9.5000000000000001e24 < (*.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. *-lowering-*.f64 84.3

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

   5. Simplified 84.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 9.5 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 21.0% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
2. 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. *-lowering-*.f64 N/A

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

   11. +-lowering-+.f64 59.1

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

5. Simplified 59.1%

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

6. Taylor expanded in x around 0

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

8. Simplified 20.6%

   \[\leadsto x \cdot \color{blue}{2} \]
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 2024204 
(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)))