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

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 8

Speedup: 1.5×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 72.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 2 + x \cdot x\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-71}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+51}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* x 2.0) (* x x))))
   (if (<= t_0 -1e-71) (* 2.0 x) (if (<= t_0 2e+51) (* y y) (* x x)))))

C

double code(double x, double y) {
	double t_0 = (x * 2.0) + (x * x);
	double tmp;
	if (t_0 <= -1e-71) {
		tmp = 2.0 * x;
	} else if (t_0 <= 2e+51) {
		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 <= (-1d-71)) then
        tmp = 2.0d0 * x
    else if (t_0 <= 2d+51) 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 <= -1e-71) {
		tmp = 2.0 * x;
	} else if (t_0 <= 2e+51) {
		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 <= -1e-71:
		tmp = 2.0 * x
	elif t_0 <= 2e+51:
		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 <= -1e-71)
		tmp = Float64(2.0 * x);
	elseif (t_0 <= 2e+51)
		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 <= -1e-71)
		tmp = 2.0 * x;
	elseif (t_0 <= 2e+51)
		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, -1e-71], N[(2.0 * x), $MachinePrecision], If[LessEqual[t$95$0, 2e+51], 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)) < -9.9999999999999992e-72

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.0%

      \[\leadsto 2 \cdot x \]

   if -9.9999999999999992e-72 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 2e51

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if 2e51 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.6

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

   7. Applied rewrites 90.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq -1 \cdot 10^{-71}:\\ \;\;\;\;2 \cdot x\\ \mathbf{elif}\;x \cdot 2 + x \cdot x \leq 2 \cdot 10^{+51}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, y, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x 2.0) (* x x)) 2e+51) (+ (fma y y x) x) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * 2.0) + (x * x)) <= 2e+51) {
		tmp = fma(y, y, x) + x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * 2.0) + Float64(x * x)) <= 2e+51)
		tmp = Float64(fma(y, y, x) + x);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], 2e+51], N[(N[(y * y + x), $MachinePrecision] + x), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x)) < 2e51

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 96.6

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

   5. Applied rewrites 96.6%

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

   7. Applied rewrites 96.6%

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

   if 2e51 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.6

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

   7. Applied rewrites 90.6%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, y, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 2 \cdot 10^{+51}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* x 2.0) (* x x)) 2e+51) (* y y) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * 2.0) + (x * x)) <= 2e+51) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * 2.0d0) + (x * x)) <= 2d+51) then
        tmp = y * y
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * 2.0) + (x * x)) <= 2e+51) {
		tmp = y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * 2.0) + (x * x)) <= 2e+51:
		tmp = y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * 2.0) + Float64(x * x)) <= 2e+51)
		tmp = Float64(y * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * 2.0) + (x * x)) <= 2e+51)
		tmp = y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * 2.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], 2e+51], 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)) < 2e51

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

   if 2e51 < (+.f64 (*.f64 x #s(literal 2 binary64)) (*.f64 x x))

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.6

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

   7. Applied rewrites 90.6%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 + x \cdot x \leq 2 \cdot 10^{+51}:\\ \;\;\;\;y \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2800000:\\ \;\;\;\;\mathsf{fma}\left(x, x, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2800000.0) (+ (fma x x x) x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2800000.0) {
		tmp = fma(x, x, x) + x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2800000.0)
		tmp = Float64(fma(x, x, x) + x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2800000.0], N[(N[(x * x + x), $MachinePrecision] + x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 74.8

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

   5. Applied rewrites 74.8%

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

   7. Applied rewrites 74.8%

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

   if 2.8e6 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 89.6

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

   5. Applied rewrites 89.6%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2800000:\\ \;\;\;\;\mathsf{fma}\left(x, x, x\right) + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2800000:\\ \;\;\;\;\left(2 + x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2800000.0) (* (+ 2.0 x) x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2800000.0) {
		tmp = (2.0 + x) * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2800000.0d0) then
        tmp = (2.0d0 + x) * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2800000.0) {
		tmp = (2.0 + x) * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2800000.0:
		tmp = (2.0 + x) * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2800000.0)
		tmp = Float64(Float64(2.0 + x) * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2800000.0)
		tmp = (2.0 + x) * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2800000.0], N[(N[(2.0 + x), $MachinePrecision] * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8e6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if 2.8e6 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 89.6

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

   5. Applied rewrites 89.6%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2800000:\\ \;\;\;\;\left(2 + x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(2.0 + x), $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. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 8: 42.6% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 45.0

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

7. Applied rewrites 45.0%

   \[\leadsto \color{blue}{x \cdot x} \]
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 2024342 
(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)))