Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y + 1, x, y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (+ y 1.0) x y))

C

double code(double x, double y) {
	return fma((y + 1.0), x, y);
}

Julia

function code(x, y)
	return fma(Float64(y + 1.0), x, y)
end

Wolfram

code[x_, y_] := N[(N[(y + 1.0), $MachinePrecision] * x + y), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. distribute-lft1-in 100.0%

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

   3. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y + 1, x, y\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y + 1, x, y\right)} \]

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(y + 1, x, y\right) \]

Alternative 2: 60.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+230}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{+203}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+179}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{+102}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{+27}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-119}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3e+230)
   (* y x)
   (if (<= x -9.5e+203)
     x
     (if (<= x -1.45e+179)
       (* y x)
       (if (<= x -4.8e+102)
         x
         (if (<= x -4.1e+27)
           (* y x)
           (if (<= x -4.9e-119) x (if (<= x 1.0) y (* y x)))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3e+230) {
		tmp = y * x;
	} else if (x <= -9.5e+203) {
		tmp = x;
	} else if (x <= -1.45e+179) {
		tmp = y * x;
	} else if (x <= -4.8e+102) {
		tmp = x;
	} else if (x <= -4.1e+27) {
		tmp = y * x;
	} else if (x <= -4.9e-119) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = y * 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 <= (-3d+230)) then
        tmp = y * x
    else if (x <= (-9.5d+203)) then
        tmp = x
    else if (x <= (-1.45d+179)) then
        tmp = y * x
    else if (x <= (-4.8d+102)) then
        tmp = x
    else if (x <= (-4.1d+27)) then
        tmp = y * x
    else if (x <= (-4.9d-119)) then
        tmp = x
    else if (x <= 1.0d0) then
        tmp = y
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3e+230) {
		tmp = y * x;
	} else if (x <= -9.5e+203) {
		tmp = x;
	} else if (x <= -1.45e+179) {
		tmp = y * x;
	} else if (x <= -4.8e+102) {
		tmp = x;
	} else if (x <= -4.1e+27) {
		tmp = y * x;
	} else if (x <= -4.9e-119) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3e+230:
		tmp = y * x
	elif x <= -9.5e+203:
		tmp = x
	elif x <= -1.45e+179:
		tmp = y * x
	elif x <= -4.8e+102:
		tmp = x
	elif x <= -4.1e+27:
		tmp = y * x
	elif x <= -4.9e-119:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3e+230)
		tmp = Float64(y * x);
	elseif (x <= -9.5e+203)
		tmp = x;
	elseif (x <= -1.45e+179)
		tmp = Float64(y * x);
	elseif (x <= -4.8e+102)
		tmp = x;
	elseif (x <= -4.1e+27)
		tmp = Float64(y * x);
	elseif (x <= -4.9e-119)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3e+230)
		tmp = y * x;
	elseif (x <= -9.5e+203)
		tmp = x;
	elseif (x <= -1.45e+179)
		tmp = y * x;
	elseif (x <= -4.8e+102)
		tmp = x;
	elseif (x <= -4.1e+27)
		tmp = y * x;
	elseif (x <= -4.9e-119)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3e+230], N[(y * x), $MachinePrecision], If[LessEqual[x, -9.5e+203], x, If[LessEqual[x, -1.45e+179], N[(y * x), $MachinePrecision], If[LessEqual[x, -4.8e+102], x, If[LessEqual[x, -4.1e+27], N[(y * x), $MachinePrecision], If[LessEqual[x, -4.9e-119], x, If[LessEqual[x, 1.0], y, N[(y * x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.00000000000000008e230 or -9.4999999999999995e203 < x < -1.45000000000000009e179 or -4.79999999999999989e102 < x < -4.1000000000000002e27 or 1 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. add-cube-cbrt 99.4%

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

      8. unpow2 99.4%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \sqrt[3]{y}\right) \cdot \left(1 + x\right) + x \]

      9. associate-*l* 99.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{y}\right)}^{2} \cdot \left(\sqrt[3]{y} \cdot \left(1 + x\right)\right)} + x \]

      10. fma-def 99.4%

          \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

   5. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

   6. Taylor expanded in x around inf 99.2%

      \[\leadsto \color{blue}{x \cdot \left(1 + {1}^{0.3333333333333333} \cdot y\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 99.2%

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

      2. *-lft-identity 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in y around inf 58.0%

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

   if -3.00000000000000008e230 < x < -9.4999999999999995e203 or -1.45000000000000009e179 < x < -4.79999999999999989e102 or -4.1000000000000002e27 < x < -4.9e-119

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. add-cube-cbrt 99.4%

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

      8. unpow2 99.4%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \sqrt[3]{y}\right) \cdot \left(1 + x\right) + x \]

      9. associate-*l* 99.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{y}\right)}^{2} \cdot \left(\sqrt[3]{y} \cdot \left(1 + x\right)\right)} + x \]

      10. fma-def 99.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

   5. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

   6. Taylor expanded in y around 0 61.1%

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

   if -4.9e-119 < x < 1

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in x around 0 73.4%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+230}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{+203}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+179}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{+102}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{+27}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-119}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1500000 \lor \neg \left(y \leq 1.8 \cdot 10^{+121}\right) \land \left(y \leq 5 \cdot 10^{+177} \lor \neg \left(y \leq 1.05 \cdot 10^{+281}\right)\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1500000.0)
         (and (not (<= y 1.8e+121)) (or (<= y 5e+177) (not (<= y 1.05e+281)))))
   (* y x)
   (+ y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1500000.0) || (!(y <= 1.8e+121) && ((y <= 5e+177) || !(y <= 1.05e+281)))) {
		tmp = y * x;
	} else {
		tmp = y + 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 <= (-1500000.0d0)) .or. (.not. (y <= 1.8d+121)) .and. (y <= 5d+177) .or. (.not. (y <= 1.05d+281))) then
        tmp = y * x
    else
        tmp = y + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1500000.0) || (!(y <= 1.8e+121) && ((y <= 5e+177) || !(y <= 1.05e+281)))) {
		tmp = y * x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1500000.0) or (not (y <= 1.8e+121) and ((y <= 5e+177) or not (y <= 1.05e+281))):
		tmp = y * x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1500000.0) || (!(y <= 1.8e+121) && ((y <= 5e+177) || !(y <= 1.05e+281))))
		tmp = Float64(y * x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1500000.0) || (~((y <= 1.8e+121)) && ((y <= 5e+177) || ~((y <= 1.05e+281)))))
		tmp = y * x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1500000.0], And[N[Not[LessEqual[y, 1.8e+121]], $MachinePrecision], Or[LessEqual[y, 5e+177], N[Not[LessEqual[y, 1.05e+281]], $MachinePrecision]]]], N[(y * x), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5e6 or 1.79999999999999991e121 < y < 5.0000000000000003e177 or 1.05000000000000003e281 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. add-cube-cbrt 98.7%

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

      8. unpow2 98.7%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \sqrt[3]{y}\right) \cdot \left(1 + x\right) + x \]

      9. associate-*l* 98.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{y}\right)}^{2} \cdot \left(\sqrt[3]{y} \cdot \left(1 + x\right)\right)} + x \]

      10. fma-def 98.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

   5. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

   6. Taylor expanded in x around inf 58.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {1}^{0.3333333333333333} \cdot y\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 58.0%

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

      2. *-lft-identity 58.0%

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

   8. Simplified 58.0%

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

   9. Taylor expanded in y around inf 57.6%

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

   if -1.5e6 < y < 1.79999999999999991e121 or 5.0000000000000003e177 < y < 1.05000000000000003e281

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in y around 0 88.1%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1500000 \lor \neg \left(y \leq 1.8 \cdot 10^{+121}\right) \land \left(y \leq 5 \cdot 10^{+177} \lor \neg \left(y \leq 1.05 \cdot 10^{+281}\right)\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 4: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;y \leq 0.55:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.6e-7) (* (+ y 1.0) x) (if (<= y 0.55) (+ y x) (* y (+ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.6e-7) {
		tmp = (y + 1.0) * x;
	} else if (y <= 0.55) {
		tmp = y + x;
	} else {
		tmp = y * (1.0 + 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 <= (-2.6d-7)) then
        tmp = (y + 1.0d0) * x
    else if (y <= 0.55d0) then
        tmp = y + x
    else
        tmp = y * (1.0d0 + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.6e-7) {
		tmp = (y + 1.0) * x;
	} else if (y <= 0.55) {
		tmp = y + x;
	} else {
		tmp = y * (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.6e-7:
		tmp = (y + 1.0) * x
	elif y <= 0.55:
		tmp = y + x
	else:
		tmp = y * (1.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.6e-7)
		tmp = Float64(Float64(y + 1.0) * x);
	elseif (y <= 0.55)
		tmp = Float64(y + x);
	else
		tmp = Float64(y * Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.6e-7)
		tmp = (y + 1.0) * x;
	elseif (y <= 0.55)
		tmp = y + x;
	else
		tmp = y * (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.6e-7], N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 0.55], N[(y + x), $MachinePrecision], N[(y * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.59999999999999999e-7

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. add-cube-cbrt 98.7%

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

      8. unpow2 98.7%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \sqrt[3]{y}\right) \cdot \left(1 + x\right) + x \]

      9. associate-*l* 98.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{y}\right)}^{2} \cdot \left(\sqrt[3]{y} \cdot \left(1 + x\right)\right)} + x \]

      10. fma-def 98.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

   5. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

   6. Taylor expanded in x around inf 53.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + {1}^{0.3333333333333333} \cdot y\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 53.9%

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

      2. *-lft-identity 53.9%

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

   8. Simplified 53.9%

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

   if -2.59999999999999999e-7 < y < 0.55000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in y around 0 99.9%

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

   if 0.55000000000000004 < y

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in y around inf 100.0%

      \[\leadsto \color{blue}{y \cdot \left(1 + x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;y \leq 0.55:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \]

Alternative 5: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -160000000:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;x \leq 650000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -160000000.0) (* (+ y 1.0) x) (if (<= x 650000.0) (+ y x) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -160000000.0) {
		tmp = (y + 1.0) * x;
	} else if (x <= 650000.0) {
		tmp = y + x;
	} else {
		tmp = y * 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 <= (-160000000.0d0)) then
        tmp = (y + 1.0d0) * x
    else if (x <= 650000.0d0) then
        tmp = y + x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -160000000.0) {
		tmp = (y + 1.0) * x;
	} else if (x <= 650000.0) {
		tmp = y + x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -160000000.0:
		tmp = (y + 1.0) * x
	elif x <= 650000.0:
		tmp = y + x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -160000000.0)
		tmp = Float64(Float64(y + 1.0) * x);
	elseif (x <= 650000.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -160000000.0)
		tmp = (y + 1.0) * x;
	elseif (x <= 650000.0)
		tmp = y + x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -160000000.0], N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 650000.0], N[(y + x), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6e8

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. add-cube-cbrt 99.4%

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

      8. unpow2 99.4%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \sqrt[3]{y}\right) \cdot \left(1 + x\right) + x \]

      9. associate-*l* 99.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{y}\right)}^{2} \cdot \left(\sqrt[3]{y} \cdot \left(1 + x\right)\right)} + x \]

      10. fma-def 99.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

   5. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

   6. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + {1}^{0.3333333333333333} \cdot y\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 99.6%

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

      2. *-lft-identity 99.6%

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

   8. Simplified 99.6%

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

   if -1.6e8 < x < 6.5e5

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in y around 0 96.8%

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

   if 6.5e5 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. add-cube-cbrt 99.5%

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

      8. unpow2 99.5%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \sqrt[3]{y}\right) \cdot \left(1 + x\right) + x \]

      9. associate-*l* 99.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{y}\right)}^{2} \cdot \left(\sqrt[3]{y} \cdot \left(1 + x\right)\right)} + x \]

      10. fma-def 99.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

   6. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{x \cdot \left(1 + {1}^{0.3333333333333333} \cdot y\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 98.7%

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

      2. *-lft-identity 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in y around inf 53.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -160000000:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;x \leq 650000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 6: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(y + N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. distribute-lft1-in 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto y + \left(y + 1\right) \cdot x \]

Alternative 7: 48.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-119}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -4.9e-119) x y))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.9e-119) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.9d-119)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.9e-119) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.9e-119:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.9e-119)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.9e-119)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.9e-119], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -4.9e-119

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. add-cube-cbrt 99.3%

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

      8. unpow2 99.3%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \sqrt[3]{y}\right) \cdot \left(1 + x\right) + x \]

      9. associate-*l* 99.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{y}\right)}^{2} \cdot \left(\sqrt[3]{y} \cdot \left(1 + x\right)\right)} + x \]

      10. fma-def 99.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

   5. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

   6. Taylor expanded in y around 0 49.4%

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

   if -4.9e-119 < x

   1. Initial program 100.0%

      \[\left(x \cdot y + x\right) + y \]

   2. Taylor expanded in x around 0 46.7%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{-119}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 8: 37.8% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. distribute-lft1-in 100.0%

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

3. Applied egg-rr 100.0%

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

   1. +-commutative 100.0%

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

   2. distribute-lft1-in 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-+r+ 100.0%

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

   5. *-un-lft-identity 100.0%

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

   6. distribute-rgt-in 100.0%

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

   7. add-cube-cbrt 99.0%

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

   8. unpow2 99.0%

      \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \sqrt[3]{y}\right) \cdot \left(1 + x\right) + x \]

   9. associate-*l* 99.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{y}\right)}^{2} \cdot \left(\sqrt[3]{y} \cdot \left(1 + x\right)\right)} + x \]

   10. fma-def 99.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

5. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y}\right)}^{2}, \sqrt[3]{y} \cdot \left(1 + x\right), x\right)} \]

6. Taylor expanded in y around 0 40.5%

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

7. Final simplification 40.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023337 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))