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

Percentage Accurate: 100.0% → 100.0%

Time: 2.2s

Alternatives: 6

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 60.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 32500000000000:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+49} \lor \neg \left(y \leq 3.5 \cdot 10^{+104}\right) \land \left(y \leq 2.85 \cdot 10^{+152} \lor \neg \left(y \leq 5.6 \cdot 10^{+161}\right) \land \left(y \leq 4.7 \cdot 10^{+214} \lor \neg \left(y \leq 4.2 \cdot 10^{+247}\right) \land y \leq 3.8 \cdot 10^{+260}\right)\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (* x y)
   (if (<= y 1.9e-96)
     x
     (if (<= y 32500000000000.0)
       y
       (if (or (<= y 1.15e+49)
               (and (not (<= y 3.5e+104))
                    (or (<= y 2.85e+152)
                        (and (not (<= y 5.6e+161))
                             (or (<= y 4.7e+214)
                                 (and (not (<= y 4.2e+247))
                                      (<= y 3.8e+260)))))))
         (* x y)
         y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 1.9e-96) {
		tmp = x;
	} else if (y <= 32500000000000.0) {
		tmp = y;
	} else if ((y <= 1.15e+49) || (!(y <= 3.5e+104) && ((y <= 2.85e+152) || (!(y <= 5.6e+161) && ((y <= 4.7e+214) || (!(y <= 4.2e+247) && (y <= 3.8e+260))))))) {
		tmp = x * y;
	} 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 (y <= (-1.0d0)) then
        tmp = x * y
    else if (y <= 1.9d-96) then
        tmp = x
    else if (y <= 32500000000000.0d0) then
        tmp = y
    else if ((y <= 1.15d+49) .or. (.not. (y <= 3.5d+104)) .and. (y <= 2.85d+152) .or. (.not. (y <= 5.6d+161)) .and. (y <= 4.7d+214) .or. (.not. (y <= 4.2d+247)) .and. (y <= 3.8d+260)) then
        tmp = x * y
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = x * y;
	} else if (y <= 1.9e-96) {
		tmp = x;
	} else if (y <= 32500000000000.0) {
		tmp = y;
	} else if ((y <= 1.15e+49) || (!(y <= 3.5e+104) && ((y <= 2.85e+152) || (!(y <= 5.6e+161) && ((y <= 4.7e+214) || (!(y <= 4.2e+247) && (y <= 3.8e+260))))))) {
		tmp = x * y;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = x * y
	elif y <= 1.9e-96:
		tmp = x
	elif y <= 32500000000000.0:
		tmp = y
	elif (y <= 1.15e+49) or (not (y <= 3.5e+104) and ((y <= 2.85e+152) or (not (y <= 5.6e+161) and ((y <= 4.7e+214) or (not (y <= 4.2e+247) and (y <= 3.8e+260)))))):
		tmp = x * y
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(x * y);
	elseif (y <= 1.9e-96)
		tmp = x;
	elseif (y <= 32500000000000.0)
		tmp = y;
	elseif ((y <= 1.15e+49) || (!(y <= 3.5e+104) && ((y <= 2.85e+152) || (!(y <= 5.6e+161) && ((y <= 4.7e+214) || (!(y <= 4.2e+247) && (y <= 3.8e+260)))))))
		tmp = Float64(x * y);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = x * y;
	elseif (y <= 1.9e-96)
		tmp = x;
	elseif (y <= 32500000000000.0)
		tmp = y;
	elseif ((y <= 1.15e+49) || (~((y <= 3.5e+104)) && ((y <= 2.85e+152) || (~((y <= 5.6e+161)) && ((y <= 4.7e+214) || (~((y <= 4.2e+247)) && (y <= 3.8e+260)))))))
		tmp = x * y;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.9e-96], x, If[LessEqual[y, 32500000000000.0], y, If[Or[LessEqual[y, 1.15e+49], And[N[Not[LessEqual[y, 3.5e+104]], $MachinePrecision], Or[LessEqual[y, 2.85e+152], And[N[Not[LessEqual[y, 5.6e+161]], $MachinePrecision], Or[LessEqual[y, 4.7e+214], And[N[Not[LessEqual[y, 4.2e+247]], $MachinePrecision], LessEqual[y, 3.8e+260]]]]]]], N[(x * y), $MachinePrecision], y]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1 or 3.25e13 < y < 1.15000000000000001e49 or 3.5000000000000002e104 < y < 2.85000000000000003e152 or 5.60000000000000041e161 < y < 4.7e214 or 4.2e247 < y < 3.7999999999999997e260

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 61.9%

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

      1. +-commutative 61.9%

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

   4. Simplified 61.9%

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

   5. Taylor expanded in y around inf 61.3%

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

   if -1 < y < 1.9e-96

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 74.3%

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

   if 1.9e-96 < y < 3.25e13 or 1.15000000000000001e49 < y < 3.5000000000000002e104 or 2.85000000000000003e152 < y < 5.60000000000000041e161 or 4.7e214 < y < 4.2e247 or 3.7999999999999997e260 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 66.3%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 32500000000000:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+49} \lor \neg \left(y \leq 3.5 \cdot 10^{+104}\right) \land \left(y \leq 2.85 \cdot 10^{+152} \lor \neg \left(y \leq 5.6 \cdot 10^{+161}\right) \land \left(y \leq 4.7 \cdot 10^{+214} \lor \neg \left(y \leq 4.2 \cdot 10^{+247}\right) \land y \leq 3.8 \cdot 10^{+260}\right)\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 3: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-44}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-142}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.2e-14)
   (* x (+ y 1.0))
   (if (<= x -2.05e-44) y (if (<= x -3.4e-142) x (if (<= x 1.0) y (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.2e-14) {
		tmp = x * (y + 1.0);
	} else if (x <= -2.05e-44) {
		tmp = y;
	} else if (x <= -3.4e-142) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * 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 <= (-5.2d-14)) then
        tmp = x * (y + 1.0d0)
    else if (x <= (-2.05d-44)) then
        tmp = y
    else if (x <= (-3.4d-142)) then
        tmp = x
    else if (x <= 1.0d0) then
        tmp = y
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.2e-14) {
		tmp = x * (y + 1.0);
	} else if (x <= -2.05e-44) {
		tmp = y;
	} else if (x <= -3.4e-142) {
		tmp = x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.2e-14:
		tmp = x * (y + 1.0)
	elif x <= -2.05e-44:
		tmp = y
	elif x <= -3.4e-142:
		tmp = x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.2e-14)
		tmp = Float64(x * Float64(y + 1.0));
	elseif (x <= -2.05e-44)
		tmp = y;
	elseif (x <= -3.4e-142)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.2e-14)
		tmp = x * (y + 1.0);
	elseif (x <= -2.05e-44)
		tmp = y;
	elseif (x <= -3.4e-142)
		tmp = x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.2e-14], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.05e-44], y, If[LessEqual[x, -3.4e-142], x, If[LessEqual[x, 1.0], y, N[(x * y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.19999999999999993e-14

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 97.1%

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

      1. +-commutative 97.1%

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

   4. Simplified 97.1%

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

   if -5.19999999999999993e-14 < x < -2.04999999999999996e-44 or -3.40000000000000029e-142 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 82.7%

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

   if -2.04999999999999996e-44 < x < -3.40000000000000029e-142

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 49.7%

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

   if 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 99.9%

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

      1. +-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around inf 55.8%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{-44}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-142}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 4: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-96}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.35e-96) (* x (+ y 1.0)) (* y (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.35e-96) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.35d-96) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y * (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.35e-96) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.35e-96:
		tmp = x * (y + 1.0)
	else:
		tmp = y * (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.35e-96)
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = Float64(y * Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.35e-96)
		tmp = x * (y + 1.0);
	else
		tmp = y * (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.35e-96], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e-96

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 69.6%

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

      1. +-commutative 69.6%

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

   4. Simplified 69.6%

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

   if 1.35e-96 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 92.3%

      \[\leadsto \color{blue}{y \cdot \left(1 + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-96}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 5: 48.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.8e-96) x y))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-96) {
		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 (y <= 1.8d-96) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-96) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.8e-96:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.8e-96)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e-96)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.8e-96], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 1.80000000000000004e-96

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 47.0%

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

   if 1.80000000000000004e-96 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 52.8%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6: 38.1% 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. Taylor expanded in y around 0 35.1%

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

3. Final simplification 35.1%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))