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

Percentage Accurate: 100.0% → 100.0%

Time: 1.9s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(y + N[(x * y + 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%

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

   2. fma-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+203}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+100} \lor \neg \left(x \leq -9.2 \cdot 10^{+79}\right) \land x \leq 10500:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.45e+203)
   (* y x)
   (if (or (<= x -1e+100) (and (not (<= x -9.2e+79)) (<= x 10500.0)))
     (+ y x)
     (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.45e+203) {
		tmp = y * x;
	} else if ((x <= -1e+100) || (!(x <= -9.2e+79) && (x <= 10500.0))) {
		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 <= (-1.45d+203)) then
        tmp = y * x
    else if ((x <= (-1d+100)) .or. (.not. (x <= (-9.2d+79))) .and. (x <= 10500.0d0)) 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 <= -1.45e+203) {
		tmp = y * x;
	} else if ((x <= -1e+100) || (!(x <= -9.2e+79) && (x <= 10500.0))) {
		tmp = y + x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.45e+203:
		tmp = y * x
	elif (x <= -1e+100) or (not (x <= -9.2e+79) and (x <= 10500.0)):
		tmp = y + x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.45e+203)
		tmp = Float64(y * x);
	elseif ((x <= -1e+100) || (!(x <= -9.2e+79) && (x <= 10500.0)))
		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 <= -1.45e+203)
		tmp = y * x;
	elseif ((x <= -1e+100) || (~((x <= -9.2e+79)) && (x <= 10500.0)))
		tmp = y + x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.45e+203], N[(y * x), $MachinePrecision], If[Or[LessEqual[x, -1e+100], And[N[Not[LessEqual[x, -9.2e+79]], $MachinePrecision], LessEqual[x, 10500.0]]], N[(y + x), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45000000000000005e203 or -1.00000000000000002e100 < x < -9.2000000000000002e79 or 10500 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 60.9%

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

   3. Taylor expanded in x around inf 59.7%

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

      1. *-commutative 59.7%

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

   5. Simplified 59.7%

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

   if -1.45000000000000005e203 < x < -1.00000000000000002e100 or -9.2000000000000002e79 < x < 10500

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 90.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+203}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+100} \lor \neg \left(x \leq -9.2 \cdot 10^{+79}\right) \land x \leq 10500:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -940:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.125:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -940.0) (* y x) (if (<= y 0.125) (+ y x) (+ y (* y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -940.0) {
		tmp = y * x;
	} else if (y <= 0.125) {
		tmp = y + x;
	} else {
		tmp = y + (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 <= (-940.0d0)) then
        tmp = y * x
    else if (y <= 0.125d0) then
        tmp = y + x
    else
        tmp = y + (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -940.0) {
		tmp = y * x;
	} else if (y <= 0.125) {
		tmp = y + x;
	} else {
		tmp = y + (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -940.0:
		tmp = y * x
	elif y <= 0.125:
		tmp = y + x
	else:
		tmp = y + (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -940.0)
		tmp = Float64(y * x);
	elseif (y <= 0.125)
		tmp = Float64(y + x);
	else
		tmp = Float64(y + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -940.0)
		tmp = y * x;
	elseif (y <= 0.125)
		tmp = y + x;
	else
		tmp = y + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -940.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.125], N[(y + x), $MachinePrecision], N[(y + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -940

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 98.9%

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

   3. Taylor expanded in x around inf 50.1%

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

      1. *-commutative 50.1%

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

   5. Simplified 50.1%

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

   if -940 < y < 0.125

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.1%

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

   if 0.125 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 98.2%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -940:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.125:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \]

Alternative 4: 61.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-10}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.8e-10) (* y x) (if (<= x 1.0) y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.8e-10) {
		tmp = y * 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 <= (-6.8d-10)) then
        tmp = y * 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 <= -6.8e-10) {
		tmp = y * x;
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.8e-10:
		tmp = y * x
	elif x <= 1.0:
		tmp = y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.8e-10)
		tmp = Float64(y * 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 <= -6.8e-10)
		tmp = y * x;
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.8e-10], N[(y * x), $MachinePrecision], If[LessEqual[x, 1.0], y, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.8000000000000003e-10 or 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 53.7%

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

   3. Taylor expanded in x around inf 51.8%

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

      1. *-commutative 51.8%

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

   5. Simplified 51.8%

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

   if -6.8000000000000003e-10 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 74.0%

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

   3. Taylor expanded in x around 0 73.4%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-10}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 6: 38.9% 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. Taylor expanded in y around inf 63.8%

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

3. Taylor expanded in x around 0 38.4%

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

4. Final simplification 38.4%

   \[\leadsto y \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))