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

Percentage Accurate: 100.0% → 100.0%

Time: 3.3s

Alternatives: 7

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

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

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: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4e-49) (* x (+ y 1.0)) (if (<= x 1.0) y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4e-49) {
		tmp = x * (y + 1.0);
	} 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 <= (-4d-49)) then
        tmp = x * (y + 1.0d0)
    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 <= -4e-49) {
		tmp = x * (y + 1.0);
	} else if (x <= 1.0) {
		tmp = y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4e-49:
		tmp = x * (y + 1.0)
	elif x <= 1.0:
		tmp = y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4e-49)
		tmp = Float64(x * Float64(y + 1.0));
	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 <= -4e-49)
		tmp = x * (y + 1.0);
	elseif (x <= 1.0)
		tmp = y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4e-49], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], y, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.99999999999999975e-49

   1. Initial program 99.9%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

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

      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. Taylor expanded in x around inf 87.8%

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

   if -3.99999999999999975e-49 < x < 1

   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. Taylor expanded in x around 0 82.6%

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

   if 1 < x

   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. Taylor expanded in x around inf 99.6%

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

   5. Taylor expanded in y around inf 35.7%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e-49) (* x (+ y 1.0)) (+ y (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e-49) {
		tmp = x * (y + 1.0);
	} 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 (x <= (-2d-49)) then
        tmp = x * (y + 1.0d0)
    else
        tmp = y + (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2e-49) {
		tmp = x * (y + 1.0);
	} else {
		tmp = y + (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2e-49:
		tmp = x * (y + 1.0)
	else:
		tmp = y + (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e-49)
		tmp = Float64(x * Float64(y + 1.0));
	else
		tmp = Float64(y + Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e-49)
		tmp = x * (y + 1.0);
	else
		tmp = y + (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e-49], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999987e-49

   1. Initial program 99.9%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

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

      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. Taylor expanded in x around inf 87.8%

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

   if -1.99999999999999987e-49 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 66.9%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y + y \cdot x\\ \end{array} \]

Alternative 4: 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. Final simplification 100.0%

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

Alternative 5: 62.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0) (* y x) (if (<= y 1.05e-54) x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = y * x;
	} else if (y <= 1.05e-54) {
		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 <= (-1.0d0)) then
        tmp = y * x
    else if (y <= 1.05d-54) 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 <= -1.0) {
		tmp = y * x;
	} else if (y <= 1.05e-54) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.0:
		tmp = y * x
	elif y <= 1.05e-54:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(y * x);
	elseif (y <= 1.05e-54)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = y * x;
	elseif (y <= 1.05e-54)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.05e-54], x, y]]

Derivation

1. Split input into 3 regimes

2. if y < -1

   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. Taylor expanded in x around inf 40.2%

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

   5. Taylor expanded in y around inf 40.2%

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

   if -1 < y < 1.05e-54

   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. Taylor expanded in y around 0 71.4%

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

   if 1.05e-54 < y

   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. Taylor expanded in x around 0 55.3%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-54}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6: 50.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -4.4e-49) x y))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-49) {
		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 (x <= (-4.4d-49)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-49) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.4e-49:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e-49)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.4e-49)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e-49], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -4.3999999999999998e-49

   1. Initial program 99.9%

      \[\left(x \cdot y + x\right) + y \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

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

      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. Taylor expanded in y around 0 44.0%

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

   if -4.3999999999999998e-49 < x

   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. Taylor expanded in x around 0 55.6%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7: 39.4% 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. 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. Taylor expanded in y around 0 37.7%

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

5. Final simplification 37.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))