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

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 8

Speedup: 1.5×

• 43.0% 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, 1.5× 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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

   5. lift-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-out N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 72.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot x + x \cdot 2\\ \mathbf{if}\;t\_0 \leq 10^{-65}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* x x) (* x 2.0))))
   (if (<= t_0 1e-65) (* y y) (if (<= t_0 4e-5) (* 2.0 x) (* x x)))))

C

double code(double x, double y) {
	double t_0 = (x * x) + (x * 2.0);
	double tmp;
	if (t_0 <= 1e-65) {
		tmp = y * y;
	} else if (t_0 <= 4e-5) {
		tmp = 2.0 * x;
	} 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 * x) + (x * 2.0d0)
    if (t_0 <= 1d-65) then
        tmp = y * y
    else if (t_0 <= 4d-5) then
        tmp = 2.0d0 * x
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * x) + (x * 2.0);
	double tmp;
	if (t_0 <= 1e-65) {
		tmp = y * y;
	} else if (t_0 <= 4e-5) {
		tmp = 2.0 * x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * x) + (x * 2.0)
	tmp = 0
	if t_0 <= 1e-65:
		tmp = y * y
	elif t_0 <= 4e-5:
		tmp = 2.0 * x
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) + Float64(x * 2.0))
	tmp = 0.0
	if (t_0 <= 1e-65)
		tmp = Float64(y * y);
	elseif (t_0 <= 4e-5)
		tmp = Float64(2.0 * x);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * x) + (x * 2.0);
	tmp = 0.0;
	if (t_0 <= 1e-65)
		tmp = y * y;
	elseif (t_0 <= 4e-5)
		tmp = 2.0 * x;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-65], N[(y * y), $MachinePrecision], If[LessEqual[t$95$0, 4e-5], N[(2.0 * x), $MachinePrecision], N[(x * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 9.99999999999999923e-66

   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 inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   if 9.99999999999999923e-66 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 4.00000000000000033e-5

   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. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto 2 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 78.6%

      \[\leadsto 2 \cdot x \]

   if 4.00000000000000033e-5 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x + x \cdot 2 \leq 10^{-65}:\\ \;\;\;\;y \cdot y\\ \mathbf{elif}\;x \cdot x + x \cdot 2 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x x) (* x 2.0)) 4e-5) (fma x 2.0 (* y y)) (fma y y (* x x))))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) + (x * 2.0)) <= 4e-5) {
		tmp = fma(x, 2.0, (y * y));
	} else {
		tmp = fma(y, y, (x * x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * x) + Float64(x * 2.0)) <= 4e-5)
		tmp = fma(x, 2.0, Float64(y * y));
	else
		tmp = fma(y, y, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], 4e-5], N[(x * 2.0 + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(y * y + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 4.00000000000000033e-5

   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 4.00000000000000033e-5 < (+.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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.7

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

   7. Applied rewrites 98.7%

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

4. Final simplification 98.7%

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

Alternative 4: 90.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x + x \cdot 2 \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x x) (* x 2.0)) 2e-35) (fma x 2.0 (* y y)) (fma x x (* x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) + (x * 2.0)) <= 2e-35) {
		tmp = fma(x, 2.0, (y * y));
	} else {
		tmp = fma(x, x, (x * 2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * x) + Float64(x * 2.0)) <= 2e-35)
		tmp = fma(x, 2.0, Float64(y * y));
	else
		tmp = fma(x, x, Float64(x * 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], 2e-35], N[(x * 2.0 + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x * 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)) < 2.00000000000000002e-35

   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 2.00000000000000002e-35 < (+.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 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. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval 82.5

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

   5. Applied rewrites 82.5%

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

   7. Applied rewrites 82.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x + x \cdot 2 \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x + x \cdot 2 \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - -2\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x x) (* x 2.0)) 2e-35) (fma x 2.0 (* y y)) (* (- x -2.0) x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) + (x * 2.0)) <= 2e-35) {
		tmp = fma(x, 2.0, (y * y));
	} else {
		tmp = (x - -2.0) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * x) + Float64(x * 2.0)) <= 2e-35)
		tmp = fma(x, 2.0, Float64(y * y));
	else
		tmp = Float64(Float64(x - -2.0) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], 2e-35], N[(x * 2.0 + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[(x - -2.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 2.00000000000000002e-35

   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 2.00000000000000002e-35 < (+.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 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. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval 82.5

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

   5. Applied rewrites 82.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x + x \cdot 2 \leq 2 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - -2\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x + x \cdot 2 \leq 10^{+36}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x x) (* x 2.0)) 1e+36) (* y y) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) + (x * 2.0)) <= 1e+36) {
		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 * x) + (x * 2.0d0)) <= 1d+36) 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 * x) + (x * 2.0)) <= 1e+36) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * x) + (x * 2.0)) <= 1e+36:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * x) + Float64(x * 2.0)) <= 1e+36)
		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 * x) + (x * 2.0)) <= 1e+36)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], 1e+36], 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)) < 1.00000000000000004e36

   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 inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.2

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

   5. Applied rewrites 61.2%

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

   if 1.00000000000000004e36 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x + x \cdot 2 \leq 10^{+36}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4.8 \cdot 10^{+15}:\\ \;\;\;\;\left(x - -2\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 4.8e+15) (* (- x -2.0) x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4.8e+15) {
		tmp = (x - -2.0) * 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) <= 4.8d+15) then
        tmp = (x - (-2.0d0)) * 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) <= 4.8e+15) {
		tmp = (x - -2.0) * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 4.8e+15:
		tmp = (x - -2.0) * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 4.8e+15)
		tmp = Float64(Float64(x - -2.0) * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 4.8e+15)
		tmp = (x - -2.0) * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 4.8e+15], N[(N[(x - -2.0), $MachinePrecision] * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.8e15

   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. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval 90.9

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

   5. Applied rewrites 90.9%

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

   if 4.8e15 < (*.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 y around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.6

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

   5. Applied rewrites 83.6%

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

Alternative 8: 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. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 42.2

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

5. Applied rewrites 42.2%

   \[\leadsto \color{blue}{x \cdot x} \]
6. 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 2024268 
(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)))