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

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 5

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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 2: 74.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+286} \lor \neg \left(x \leq -8 \cdot 10^{+259} \lor \neg \left(x \leq -2.7 \cdot 10^{+248}\right) \land \left(x \leq -1.32 \cdot 10^{+222} \lor \neg \left(x \leq -8.8 \cdot 10^{+153}\right) \land x \leq 350\right)\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.15e+286)
         (not
          (or (<= x -8e+259)
              (and (not (<= x -2.7e+248))
                   (or (<= x -1.32e+222)
                       (and (not (<= x -8.8e+153)) (<= x 350.0)))))))
   (* x y)
   (+ x y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.15e+286) || !((x <= -8e+259) || (!(x <= -2.7e+248) && ((x <= -1.32e+222) || (!(x <= -8.8e+153) && (x <= 350.0)))))) {
		tmp = x * 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 <= (-1.15d+286)) .or. (.not. (x <= (-8d+259)) .or. (.not. (x <= (-2.7d+248))) .and. (x <= (-1.32d+222)) .or. (.not. (x <= (-8.8d+153))) .and. (x <= 350.0d0))) then
        tmp = x * y
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.15e+286) || !((x <= -8e+259) || (!(x <= -2.7e+248) && ((x <= -1.32e+222) || (!(x <= -8.8e+153) && (x <= 350.0)))))) {
		tmp = x * y;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.15e+286) or not ((x <= -8e+259) or (not (x <= -2.7e+248) and ((x <= -1.32e+222) or (not (x <= -8.8e+153) and (x <= 350.0))))):
		tmp = x * y
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.15e+286) || !((x <= -8e+259) || (!(x <= -2.7e+248) && ((x <= -1.32e+222) || (!(x <= -8.8e+153) && (x <= 350.0))))))
		tmp = Float64(x * y);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.15e+286) || ~(((x <= -8e+259) || (~((x <= -2.7e+248)) && ((x <= -1.32e+222) || (~((x <= -8.8e+153)) && (x <= 350.0)))))))
		tmp = x * y;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.15e+286], N[Not[Or[LessEqual[x, -8e+259], And[N[Not[LessEqual[x, -2.7e+248]], $MachinePrecision], Or[LessEqual[x, -1.32e+222], And[N[Not[LessEqual[x, -8.8e+153]], $MachinePrecision], LessEqual[x, 350.0]]]]]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1500000000000001e286 or -7.999999999999999e259 < x < -2.69999999999999989e248 or -1.31999999999999997e222 < x < -8.7999999999999998e153 or 350 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 63.4%

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

   4. Taylor expanded in x around inf 61.4%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 61.4%

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

   6. Simplified 61.4%

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

   if -1.1500000000000001e286 < x < -7.999999999999999e259 or -2.69999999999999989e248 < x < -1.31999999999999997e222 or -8.7999999999999998e153 < x < 350

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 93.8%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+286} \lor \neg \left(x \leq -8 \cdot 10^{+259} \lor \neg \left(x \leq -2.7 \cdot 10^{+248}\right) \land \left(x \leq -1.32 \cdot 10^{+222} \lor \neg \left(x \leq -8.8 \cdot 10^{+153}\right) \land x \leq 350\right)\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -72000000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -72000000.0) (* x y) (if (<= y 2.3e-5) (+ x y) (* y (+ x 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -72000000.0) {
		tmp = x * y;
	} else if (y <= 2.3e-5) {
		tmp = x + y;
	} 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 <= (-72000000.0d0)) then
        tmp = x * y
    else if (y <= 2.3d-5) then
        tmp = x + y
    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 <= -72000000.0) {
		tmp = x * y;
	} else if (y <= 2.3e-5) {
		tmp = x + y;
	} else {
		tmp = y * (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -72000000.0:
		tmp = x * y
	elif y <= 2.3e-5:
		tmp = x + y
	else:
		tmp = y * (x + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -72000000.0)
		tmp = Float64(x * y);
	elseif (y <= 2.3e-5)
		tmp = Float64(x + y);
	else
		tmp = Float64(y * Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -72000000.0)
		tmp = x * y;
	elseif (y <= 2.3e-5)
		tmp = x + y;
	else
		tmp = y * (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -72000000.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 2.3e-5], N[(x + y), $MachinePrecision], N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.2e7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 98.7%

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

   4. Taylor expanded in x around inf 51.3%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 51.3%

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

   6. Simplified 51.3%

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

   if -7.2e7 < y < 2.3e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.4%

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

   if 2.3e-5 < y

   1. Initial program 100.0%

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

   3. 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 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -72000000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-5}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -480000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -480000.0) (not (<= x 1.0))) (* x y) y))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -480000.0) || !(x <= 1.0)) {
		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 ((x <= (-480000.0d0)) .or. (.not. (x <= 1.0d0))) 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 ((x <= -480000.0) || !(x <= 1.0)) {
		tmp = x * y;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -480000.0) or not (x <= 1.0):
		tmp = x * y
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -480000.0) || !(x <= 1.0))
		tmp = Float64(x * y);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -480000.0) || ~((x <= 1.0)))
		tmp = x * y;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -480000.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(x * y), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -4.8e5 or 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 57.3%

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

   4. Taylor expanded in x around inf 55.6%

      \[\leadsto \color{blue}{x \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 55.6%

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

   6. Simplified 55.6%

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

   if -4.8e5 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 73.1%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -480000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.6% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 41.6%

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

4. Final simplification 41.6%

   \[\leadsto y \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))