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

Percentage Accurate: 100.0% → 100.0%

Time: 3.6s

Alternatives: 8

Speedup: 0.1×

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-define 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. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 2: 60.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+235}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{+174}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{+137}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-142}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.5e+235)
   (* y x)
   (if (<= x -6.5e+174)
     x
     (if (<= x -1.5e+137)
       (* y x)
       (if (<= x -7.4e-142) x (if (<= x 1.0) y (* y x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.5e+235) {
		tmp = y * x;
	} else if (x <= -6.5e+174) {
		tmp = x;
	} else if (x <= -1.5e+137) {
		tmp = y * x;
	} else if (x <= -7.4e-142) {
		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 <= (-6.5d+235)) then
        tmp = y * x
    else if (x <= (-6.5d+174)) then
        tmp = x
    else if (x <= (-1.5d+137)) then
        tmp = y * x
    else if (x <= (-7.4d-142)) 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 <= -6.5e+235) {
		tmp = y * x;
	} else if (x <= -6.5e+174) {
		tmp = x;
	} else if (x <= -1.5e+137) {
		tmp = y * x;
	} else if (x <= -7.4e-142) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.5e+235:
		tmp = y * x
	elif x <= -6.5e+174:
		tmp = x
	elif x <= -1.5e+137:
		tmp = y * x
	elif x <= -7.4e-142:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.5e+235)
		tmp = Float64(y * x);
	elseif (x <= -6.5e+174)
		tmp = x;
	elseif (x <= -1.5e+137)
		tmp = Float64(y * x);
	elseif (x <= -7.4e-142)
		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 <= -6.5e+235)
		tmp = y * x;
	elseif (x <= -6.5e+174)
		tmp = x;
	elseif (x <= -1.5e+137)
		tmp = y * x;
	elseif (x <= -7.4e-142)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.5e+235], N[(y * x), $MachinePrecision], If[LessEqual[x, -6.5e+174], x, If[LessEqual[x, -1.5e+137], N[(y * x), $MachinePrecision], If[LessEqual[x, -7.4e-142], x, If[LessEqual[x, 1.0], y, N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5000000000000001e235 or -6.5000000000000001e174 < x < -1.5e137 or 1 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 99.7%

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

   6. Taylor expanded in y around inf 59.1%

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

   if -6.5000000000000001e235 < x < -6.5000000000000001e174 or -1.5e137 < x < -7.39999999999999972e-142

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-lft1-in 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 50.2%

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

   if -7.39999999999999972e-142 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 85.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+235}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{+174}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{+137}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-142}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{+235}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{+174}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{+128} \lor \neg \left(x \leq 6800000000\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.8e+235)
   (* y x)
   (if (<= x -6.5e+174)
     x
     (if (or (<= x -1.6e+128) (not (<= x 6800000000.0))) (* y x) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.8e+235) {
		tmp = y * x;
	} else if (x <= -6.5e+174) {
		tmp = x;
	} else if ((x <= -1.6e+128) || !(x <= 6800000000.0)) {
		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 <= (-9.8d+235)) then
        tmp = y * x
    else if (x <= (-6.5d+174)) then
        tmp = x
    else if ((x <= (-1.6d+128)) .or. (.not. (x <= 6800000000.0d0))) 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 <= -9.8e+235) {
		tmp = y * x;
	} else if (x <= -6.5e+174) {
		tmp = x;
	} else if ((x <= -1.6e+128) || !(x <= 6800000000.0)) {
		tmp = y * x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.8e+235:
		tmp = y * x
	elif x <= -6.5e+174:
		tmp = x
	elif (x <= -1.6e+128) or not (x <= 6800000000.0):
		tmp = y * x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.8e+235)
		tmp = Float64(y * x);
	elseif (x <= -6.5e+174)
		tmp = x;
	elseif ((x <= -1.6e+128) || !(x <= 6800000000.0))
		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 <= -9.8e+235)
		tmp = y * x;
	elseif (x <= -6.5e+174)
		tmp = x;
	elseif ((x <= -1.6e+128) || ~((x <= 6800000000.0)))
		tmp = y * x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.8e+235], N[(y * x), $MachinePrecision], If[LessEqual[x, -6.5e+174], x, If[Or[LessEqual[x, -1.6e+128], N[Not[LessEqual[x, 6800000000.0]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(y + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.7999999999999995e235 or -6.5000000000000001e174 < x < -1.59999999999999993e128 or 6.8e9 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 99.7%

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

   6. Taylor expanded in y around inf 59.4%

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

   if -9.7999999999999995e235 < x < -6.5000000000000001e174

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 74.3%

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

   if -1.59999999999999993e128 < x < 6.8e9

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 90.7%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{+235}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{+174}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{+128} \lor \neg \left(x \leq 6800000000\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-11}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.15e-11)
   (* (+ y 1.0) x)
   (if (<= y 1.6e-5) (+ y x) (* y (+ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.15e-11) {
		tmp = (y + 1.0) * x;
	} else if (y <= 1.6e-5) {
		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 <= (-1.15d-11)) then
        tmp = (y + 1.0d0) * x
    else if (y <= 1.6d-5) 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 <= -1.15e-11) {
		tmp = (y + 1.0) * x;
	} else if (y <= 1.6e-5) {
		tmp = y + x;
	} else {
		tmp = y * (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.15e-11:
		tmp = (y + 1.0) * x
	elif y <= 1.6e-5:
		tmp = y + x
	else:
		tmp = y * (1.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.15e-11)
		tmp = Float64(Float64(y + 1.0) * x);
	elseif (y <= 1.6e-5)
		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 <= -1.15e-11)
		tmp = (y + 1.0) * x;
	elseif (y <= 1.6e-5)
		tmp = y + x;
	else
		tmp = y * (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.15e-11], N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 1.6e-5], N[(y + x), $MachinePrecision], N[(y * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15000000000000007e-11

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-lft1-in 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 51.3%

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

   if -1.15000000000000007e-11 < y < 1.59999999999999993e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   if 1.59999999999999993e-5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 98.4%

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

      1. +-commutative 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-11}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+20}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;x \leq 22000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.5e+20)
   (* (+ y 1.0) x)
   (if (<= x 22000000000.0) (+ y x) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.5e+20) {
		tmp = (y + 1.0) * x;
	} else if (x <= 22000000000.0) {
		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 <= (-8.5d+20)) then
        tmp = (y + 1.0d0) * x
    else if (x <= 22000000000.0d0) 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 <= -8.5e+20) {
		tmp = (y + 1.0) * x;
	} else if (x <= 22000000000.0) {
		tmp = y + x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.5e+20:
		tmp = (y + 1.0) * x
	elif x <= 22000000000.0:
		tmp = y + x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.5e+20)
		tmp = Float64(Float64(y + 1.0) * x);
	elseif (x <= 22000000000.0)
		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 <= -8.5e+20)
		tmp = (y + 1.0) * x;
	elseif (x <= 22000000000.0)
		tmp = y + x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.5e+20], N[(N[(y + 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 22000000000.0], N[(y + x), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5e20

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   if -8.5e20 < x < 2.2e10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 97.5%

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

   if 2.2e10 < x

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-lft1-in 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 99.6%

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

   6. Taylor expanded in y around inf 54.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+20}:\\ \;\;\;\;\left(y + 1\right) \cdot x\\ \mathbf{elif}\;x \leq 22000000000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. distribute-lft1-in 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

   \[\leadsto y + \left(y + 1\right) \cdot x \]
6. Add Preprocessing

Alternative 7: 48.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-142}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -7.4e-142) x y))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.4e-142) {
		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 <= (-7.4d-142)) 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 <= -7.4e-142) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.4e-142:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.4e-142)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.4e-142)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.4e-142], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -7.39999999999999972e-142

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-lft1-in 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 45.6%

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

   if -7.39999999999999972e-142 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 54.8%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-142}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.2% 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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. distribute-lft1-in 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in y around 0 34.3%

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

6. Final simplification 34.3%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))