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

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 8

Speedup: 1.2×

• 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: 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-out 100.0%

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

   2. fma-define 100.0%

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

   3. +-commutative 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 90.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(\left(x + 3\right) + -1\right)\\ \mathbf{elif}\;x \leq 10000:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.55e-11)
   (* x (+ (+ x 3.0) -1.0))
   (if (<= x 10000.0) (+ (* y y) (* x 2.0)) (* x (+ x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.55e-11) {
		tmp = x * ((x + 3.0) + -1.0);
	} else if (x <= 10000.0) {
		tmp = (y * y) + (x * 2.0);
	} else {
		tmp = x * (x + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.55d-11)) then
        tmp = x * ((x + 3.0d0) + (-1.0d0))
    else if (x <= 10000.0d0) then
        tmp = (y * y) + (x * 2.0d0)
    else
        tmp = x * (x + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.55e-11) {
		tmp = x * ((x + 3.0) + -1.0);
	} else if (x <= 10000.0) {
		tmp = (y * y) + (x * 2.0);
	} else {
		tmp = x * (x + 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.55e-11:
		tmp = x * ((x + 3.0) + -1.0)
	elif x <= 10000.0:
		tmp = (y * y) + (x * 2.0)
	else:
		tmp = x * (x + 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.55e-11)
		tmp = Float64(x * Float64(Float64(x + 3.0) + -1.0));
	elseif (x <= 10000.0)
		tmp = Float64(Float64(y * y) + Float64(x * 2.0));
	else
		tmp = Float64(x * Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.55e-11)
		tmp = x * ((x + 3.0) + -1.0);
	elseif (x <= 10000.0)
		tmp = (y * y) + (x * 2.0);
	else
		tmp = x * (x + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.55e-11], N[(x * N[(N[(x + 3.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 10000.0], N[(N[(y * y), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000014e-11

   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 83.7%

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

   8. Taylor expanded in x around inf 83.7%

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

      1. associate-*r/ 83.7%

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

      2. metadata-eval 83.7%

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

   10. Simplified 83.7%

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

       1. expm1-log1p-u 3.6%

          \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(1 + \frac{2}{x}\right)\right)\right)} \]

       2. expm1-undefine 3.6%

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

       3. +-commutative 3.6%

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

       4. distribute-lft-in 3.6%

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

       5. *-rgt-identity 3.6%

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

       6. fma-define 3.6%

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

   12. Applied egg-rr 3.6%

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

       1. sub-neg 3.6%

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

       2. log1p-undefine 3.6%

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

       3. rem-exp-log 83.7%

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

       4. fma-undefine 83.7%

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

       5. associate-+r+ 83.7%

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

       6. metadata-eval 83.7%

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

       7. associate-*l/ 83.7%

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

       8. associate-*r* 83.7%

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

       9. *-commutative 83.7%

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

       10. lft-mult-inverse 83.7%

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

       11. metadata-eval 83.7%

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

       12. metadata-eval 83.7%

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

       13. metadata-eval 83.7%

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

   14. Simplified 83.7%

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

   if -1.55000000000000014e-11 < x < 1e4

   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 0 100.0%

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

   if 1e4 < 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 81.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \left(\left(x + 3\right) + -1\right)\\ \mathbf{elif}\;x \leq 10000:\\ \;\;\;\;y \cdot y + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2.35 \cdot 10^{-145}:\\ \;\;\;\;x \cdot \left(\left(x + 3\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2.35e-145) (* x (+ (+ x 3.0) -1.0)) (+ (* y y) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2.35e-145) {
		tmp = x * ((x + 3.0) + -1.0);
	} else {
		tmp = (y * y) + (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 2.35d-145) then
        tmp = x * ((x + 3.0d0) + (-1.0d0))
    else
        tmp = (y * y) + (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2.35e-145) {
		tmp = x * ((x + 3.0) + -1.0);
	} else {
		tmp = (y * y) + (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2.35e-145:
		tmp = x * ((x + 3.0) + -1.0)
	else:
		tmp = (y * y) + (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2.35e-145)
		tmp = Float64(x * Float64(Float64(x + 3.0) + -1.0));
	else
		tmp = Float64(Float64(y * y) + Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 2.35e-145)
		tmp = x * ((x + 3.0) + -1.0);
	else
		tmp = (y * y) + (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2.35e-145], N[(x * N[(N[(x + 3.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.3500000000000001e-145

   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 96.4%

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

   8. Taylor expanded in x around inf 96.2%

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

      1. associate-*r/ 96.2%

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

      2. metadata-eval 96.2%

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

   10. Simplified 96.2%

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

       1. expm1-log1p-u 67.7%

          \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(1 + \frac{2}{x}\right)\right)\right)} \]

       2. expm1-undefine 67.7%

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

       3. +-commutative 67.7%

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

       4. distribute-lft-in 67.6%

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

       5. *-rgt-identity 67.6%

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

       6. fma-define 67.7%

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

   12. Applied egg-rr 67.7%

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

       1. sub-neg 67.7%

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

       2. log1p-undefine 67.7%

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

       3. rem-exp-log 96.3%

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

       4. fma-undefine 96.4%

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

       5. associate-+r+ 96.4%

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

       6. metadata-eval 96.4%

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

       7. associate-*l/ 96.4%

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

       8. associate-*r* 96.4%

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

       9. *-commutative 96.4%

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

       10. lft-mult-inverse 96.4%

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

       11. metadata-eval 96.4%

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

       12. metadata-eval 96.4%

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

       13. metadata-eval 96.4%

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

   14. Simplified 96.4%

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

   if 2.3500000000000001e-145 < (*.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 98.2%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2.35 \cdot 10^{-145}:\\ \;\;\;\;x \cdot \left(\left(x + 3\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.4% 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 81.8%

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

   8. Taylor expanded in x around inf 81.0%

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

   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 98.7%

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

   6. Taylor expanded in x around inf 43.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2\\ \end{array} \]
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: 60.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(N[(x + 3.0), $MachinePrecision] + -1.0), $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 64.5%

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

8. Taylor expanded in x around inf 64.4%

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

   1. associate-*r/ 64.4%

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

   2. metadata-eval 64.4%

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

10. Simplified 64.4%

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

    1. expm1-log1p-u 37.9%

       \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(1 + \frac{2}{x}\right)\right)\right)} \]

    2. expm1-undefine 38.0%

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

    3. +-commutative 38.0%

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

    4. distribute-lft-in 37.9%

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

    5. *-rgt-identity 37.9%

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

    6. fma-define 38.0%

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

12. Applied egg-rr 38.0%

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

    1. sub-neg 38.0%

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

    2. log1p-undefine 38.0%

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

    3. rem-exp-log 64.5%

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

    4. fma-undefine 64.5%

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

    5. associate-+r+ 64.5%

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

    6. metadata-eval 64.5%

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

    7. associate-*l/ 64.5%

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

    8. associate-*r* 64.5%

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

    9. *-commutative 64.5%

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

    10. lft-mult-inverse 64.5%

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

    11. metadata-eval 64.5%

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

    12. metadata-eval 64.5%

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

    13. metadata-eval 64.5%

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

14. Simplified 64.5%

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

15. Final simplification 64.5%

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

Alternative 7: 60.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

   1. +-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 64.5%

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

8. Final simplification 64.5%

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

Alternative 8: 20.3% 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 63.4%

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

6. Taylor expanded in x around inf 21.5%

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

7. Final simplification 21.5%

   \[\leadsto x \cdot 2 \]
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 2024141 
(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)))