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

Percentage Accurate: 100.0% → 100.0%

Time: 2.7s

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, 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. Add Preprocessing

Alternative 2: 74.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -52000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+41}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+237}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+283}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -52000000.0)
   (* y x)
   (if (<= y 4.8e+23)
     (+ y x)
     (if (<= y 1.8e+41)
       (* y x)
       (if (<= y 7.2e+237) (+ y x) (if (<= y 3.1e+283) (* y x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -52000000.0) {
		tmp = y * x;
	} else if (y <= 4.8e+23) {
		tmp = y + x;
	} else if (y <= 1.8e+41) {
		tmp = y * x;
	} else if (y <= 7.2e+237) {
		tmp = y + x;
	} else if (y <= 3.1e+283) {
		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 (y <= (-52000000.0d0)) then
        tmp = y * x
    else if (y <= 4.8d+23) then
        tmp = y + x
    else if (y <= 1.8d+41) then
        tmp = y * x
    else if (y <= 7.2d+237) then
        tmp = y + x
    else if (y <= 3.1d+283) 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 (y <= -52000000.0) {
		tmp = y * x;
	} else if (y <= 4.8e+23) {
		tmp = y + x;
	} else if (y <= 1.8e+41) {
		tmp = y * x;
	} else if (y <= 7.2e+237) {
		tmp = y + x;
	} else if (y <= 3.1e+283) {
		tmp = y * x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -52000000.0:
		tmp = y * x
	elif y <= 4.8e+23:
		tmp = y + x
	elif y <= 1.8e+41:
		tmp = y * x
	elif y <= 7.2e+237:
		tmp = y + x
	elif y <= 3.1e+283:
		tmp = y * x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -52000000.0)
		tmp = Float64(y * x);
	elseif (y <= 4.8e+23)
		tmp = Float64(y + x);
	elseif (y <= 1.8e+41)
		tmp = Float64(y * x);
	elseif (y <= 7.2e+237)
		tmp = Float64(y + x);
	elseif (y <= 3.1e+283)
		tmp = Float64(y * x);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -52000000.0)
		tmp = y * x;
	elseif (y <= 4.8e+23)
		tmp = y + x;
	elseif (y <= 1.8e+41)
		tmp = y * x;
	elseif (y <= 7.2e+237)
		tmp = y + x;
	elseif (y <= 3.1e+283)
		tmp = y * x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -52000000.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 4.8e+23], N[(y + x), $MachinePrecision], If[LessEqual[y, 1.8e+41], N[(y * x), $MachinePrecision], If[LessEqual[y, 7.2e+237], N[(y + x), $MachinePrecision], If[LessEqual[y, 3.1e+283], N[(y * x), $MachinePrecision], y]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.2e7 or 4.8e23 < y < 1.80000000000000013e41 or 7.20000000000000029e237 < y < 3.09999999999999988e283

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 56.2%

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

      1. *-commutative 56.2%

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

   8. Simplified 56.2%

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

   if -5.2e7 < y < 4.8e23 or 1.80000000000000013e41 < y < 7.20000000000000029e237

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 89.3%

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

   if 3.09999999999999988e283 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 61.8%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -52000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+41}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+237}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+283}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -340000000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.45:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -340000000000.0) (* y x) (if (<= y 0.45) (+ y x) (* y (+ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -340000000000.0) {
		tmp = y * x;
	} else if (y <= 0.45) {
		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 <= (-340000000000.0d0)) then
        tmp = y * x
    else if (y <= 0.45d0) 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 <= -340000000000.0) {
		tmp = y * x;
	} else if (y <= 0.45) {
		tmp = y + x;
	} else {
		tmp = y * (1.0 + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -340000000000.0:
		tmp = y * x
	elif y <= 0.45:
		tmp = y + x
	else:
		tmp = y * (1.0 + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -340000000000.0)
		tmp = Float64(y * x);
	elseif (y <= 0.45)
		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 <= -340000000000.0)
		tmp = y * x;
	elseif (y <= 0.45)
		tmp = y + x;
	else
		tmp = y * (1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -340000000000.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 0.45], N[(y + x), $MachinePrecision], N[(y * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.4e11

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 53.6%

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

      1. *-commutative 53.6%

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

   8. Simplified 53.6%

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

   if -3.4e11 < y < 0.450000000000000011

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 97.9%

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

   if 0.450000000000000011 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 99.4%

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

      1. +-commutative 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -340000000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 0.45:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-8} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.15e-8) (not (<= x 1.0))) (* y x) y))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.15e-8) || !(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 <= (-1.15d-8)) .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 <= -1.15e-8) || !(x <= 1.0)) {
		tmp = y * x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.15e-8) or not (x <= 1.0):
		tmp = y * x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.15e-8) || !(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 <= -1.15e-8) || ~((x <= 1.0)))
		tmp = y * x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.15e-8], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(y * x), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e-8 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 53.1%

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

      1. +-commutative 53.1%

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

   5. Simplified 53.1%

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

   6. Taylor expanded in x around inf 52.8%

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

      1. *-commutative 52.8%

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

   8. Simplified 52.8%

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

   if -1.15e-8 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-8} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 6: 39.6% 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 39.4%

   \[\leadsto \color{blue}{y} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024091 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))