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

Percentage Accurate: 100.0% → 100.0%

Time: 3.7s

Alternatives: 6

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-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. Add Preprocessing

5. Final simplification 100.0%

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

Alternative 2: 75.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \lor \neg \left(y \leq 1.1 \cdot 10^{+70}\right) \land \left(y \leq 5.5 \cdot 10^{+98} \lor \neg \left(y \leq 4.6 \cdot 10^{+147}\right) \land y \leq 8.8 \cdot 10^{+253}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9.5)
         (and (not (<= y 1.1e+70))
              (or (<= y 5.5e+98) (and (not (<= y 4.6e+147)) (<= y 8.8e+253)))))
   (* y x)
   (+ y x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9.5) || (!(y <= 1.1e+70) && ((y <= 5.5e+98) || (!(y <= 4.6e+147) && (y <= 8.8e+253))))) {
		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 <= (-9.5d0)) .or. (.not. (y <= 1.1d+70)) .and. (y <= 5.5d+98) .or. (.not. (y <= 4.6d+147)) .and. (y <= 8.8d+253)) 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 <= -9.5) || (!(y <= 1.1e+70) && ((y <= 5.5e+98) || (!(y <= 4.6e+147) && (y <= 8.8e+253))))) {
		tmp = y * x;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9.5) or (not (y <= 1.1e+70) and ((y <= 5.5e+98) or (not (y <= 4.6e+147) and (y <= 8.8e+253)))):
		tmp = y * x
	else:
		tmp = y + x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9.5) || (!(y <= 1.1e+70) && ((y <= 5.5e+98) || (!(y <= 4.6e+147) && (y <= 8.8e+253)))))
		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 <= -9.5) || (~((y <= 1.1e+70)) && ((y <= 5.5e+98) || (~((y <= 4.6e+147)) && (y <= 8.8e+253)))))
		tmp = y * x;
	else
		tmp = y + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9.5], And[N[Not[LessEqual[y, 1.1e+70]], $MachinePrecision], Or[LessEqual[y, 5.5e+98], And[N[Not[LessEqual[y, 4.6e+147]], $MachinePrecision], LessEqual[y, 8.8e+253]]]]], N[(y * x), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.5 or 1.1e70 < y < 5.49999999999999946e98 or 4.5999999999999998e147 < y < 8.80000000000000022e253

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 99.0%

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

   4. Taylor expanded in x around inf 57.4%

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

      1. *-commutative 57.4%

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

   6. Simplified 57.4%

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

   if -9.5 < y < 1.1e70 or 5.49999999999999946e98 < y < 4.5999999999999998e147 or 8.80000000000000022e253 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 91.0%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \lor \neg \left(y \leq 1.1 \cdot 10^{+70}\right) \land \left(y \leq 5.5 \cdot 10^{+98} \lor \neg \left(y \leq 4.6 \cdot 10^{+147}\right) \land y \leq 8.8 \cdot 10^{+253}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.00056:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5) (* y x) (if (<= y 0.00056) (+ y x) (* y (+ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5) {
		tmp = y * x;
	} else if (y <= 0.00056) {
		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 <= (-9.5d0)) then
        tmp = y * x
    else if (y <= 0.00056d0) 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 <= -9.5) {
		tmp = y * x;
	} else if (y <= 0.00056) {
		tmp = y + x;
	} else {
		tmp = y * (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5:
		tmp = y * x
	elif y <= 0.00056:
		tmp = y + x
	else:
		tmp = y * (1.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5)
		tmp = Float64(y * x);
	elseif (y <= 0.00056)
		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 <= -9.5)
		tmp = y * x;
	elseif (y <= 0.00056)
		tmp = y + x;
	else
		tmp = y * (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.00056], N[(y + x), $MachinePrecision], N[(y * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5

   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 48.3%

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

      1. *-commutative 48.3%

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

   6. Simplified 48.3%

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

   if -9.5 < y < 5.5999999999999995e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.8%

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

   if 5.5999999999999995e-4 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 97.2%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.00056:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -340000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -340000.0) (not (<= x 1.0))) (* y x) y))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -340000.0) || !(x <= 1.0)) {
		tmp = y * 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 <= (-340000.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = y * x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -340000.0) || !(x <= 1.0)) {
		tmp = y * x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -340000.0) or not (x <= 1.0):
		tmp = y * x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -340000.0) || !(x <= 1.0))
		tmp = Float64(y * x);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -340000.0) || ~((x <= 1.0)))
		tmp = y * x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -340000.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(y * x), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -3.4e5 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 53.2%

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

   4. Taylor expanded in x around inf 53.1%

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

      1. *-commutative 53.1%

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

   6. Simplified 53.1%

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

   if -3.4e5 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 66.3%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -340000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 6: 39.0% 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 34.7%

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

4. Final simplification 34.7%

   \[\leadsto y \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024022 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))