Logistic regression 2

Percentage Accurate: 99.4% → 99.3%

Time: 8.2s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. x = 5.566316305017226e+198
  2. y = 2.9484676994584275e-158

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return Math.log((1.0 + Math.exp(x))) - (x * y);
}

Python

def code(x, y):
	return math.log((1.0 + math.exp(x))) - (x * y)

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return Math.log((1.0 + Math.exp(x))) - (x * y);
}

Python

def code(x, y):
	return math.log((1.0 + math.exp(x))) - (x * y)

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right) + \log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4) (- (* x y)) (+ (* x (- 0.5 y)) (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4) {
		tmp = -(x * y);
	} else {
		tmp = (x * (0.5 - y)) + log(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.4d0)) then
        tmp = -(x * y)
    else
        tmp = (x * (0.5d0 - y)) + log(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.4) {
		tmp = -(x * y);
	} else {
		tmp = (x * (0.5 - y)) + Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4:
		tmp = -(x * y)
	else:
		tmp = (x * (0.5 - y)) + math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(-Float64(x * y));
	else
		tmp = Float64(Float64(x * Float64(0.5 - y)) + log(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.4)
		tmp = -(x * y);
	else
		tmp = (x * (0.5 - y)) + log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4], (-N[(x * y), $MachinePrecision]), N[(N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999

   1. Initial program 99.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Step-by-step derivation

      1. log1p-def 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]

   4. Taylor expanded in x around inf 99.0%

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

      1. associate-*r* 99.0%

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

      2. *-commutative 99.0%

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

      3. mul-1-neg 99.0%

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

   6. Simplified 99.0%

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

   if -1.3999999999999999 < x

   1. Initial program 98.8%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Step-by-step derivation

      1. log1p-def 98.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]

   4. Taylor expanded in x around 0 99.5%

      \[\leadsto \color{blue}{\left(0.5 - y\right) \cdot x + \log 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right) + \log 2\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, -y, \mathsf{log1p}\left(e^{x}\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma x (- y) (log1p (exp x))))

C

double code(double x, double y) {
	return fma(x, -y, log1p(exp(x)));
}

Julia

function code(x, y)
	return fma(x, Float64(-y), log1p(exp(x)))
end

Wolfram

code[x_, y_] := N[(x * (-y) + N[Log[1 + N[Exp[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
2. Step-by-step derivation

   1. sub-neg 99.2%

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

   2. +-commutative 99.2%

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

   3. distribute-rgt-neg-in 99.2%

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

   4. fma-def 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -y, \log \left(1 + e^{x}\right)\right)} \]

   5. log1p-def 99.3%

      \[\leadsto \mathsf{fma}\left(x, -y, \color{blue}{\mathsf{log1p}\left(e^{x}\right)}\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, -y, \mathsf{log1p}\left(e^{x}\right)\right)} \]

4. Final simplification 99.3%

   \[\leadsto \mathsf{fma}\left(x, -y, \mathsf{log1p}\left(e^{x}\right)\right) \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(e^{x}\right) - x \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (log1p (exp x)) (* x y)))

C

double code(double x, double y) {
	return log1p(exp(x)) - (x * y);
}

Java

public static double code(double x, double y) {
	return Math.log1p(Math.exp(x)) - (x * y);
}

Python

def code(x, y):
	return math.log1p(math.exp(x)) - (x * y)

Julia

function code(x, y)
	return Float64(log1p(exp(x)) - Float64(x * y))
end

Wolfram

code[x_, y_] := N[(N[Log[1 + N[Exp[x], $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
2. Step-by-step derivation

   1. log1p-def 99.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

3. Simplified 99.2%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]

4. Final simplification 99.2%

   \[\leadsto \mathsf{log1p}\left(e^{x}\right) - x \cdot y \]

Alternative 4: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -70:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -70.0) (- (* x y)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -70.0) {
		tmp = -(x * y);
	} else {
		tmp = log(2.0) - (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 <= (-70.0d0)) then
        tmp = -(x * y)
    else
        tmp = log(2.0d0) - (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -70.0) {
		tmp = -(x * y);
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -70.0:
		tmp = -(x * y)
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -70.0)
		tmp = Float64(-Float64(x * y));
	else
		tmp = Float64(log(2.0) - Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -70.0)
		tmp = -(x * y);
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -70.0], (-N[(x * y), $MachinePrecision]), N[(N[Log[2.0], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -70

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Step-by-step derivation

      1. log1p-def 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]

   4. Taylor expanded in x around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -70 < x

   1. Initial program 98.7%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Step-by-step derivation

      1. log1p-def 98.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]

   4. Taylor expanded in x around 0 98.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -70:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 5: 82.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-97}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-27}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.5 - y\right) + x \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.95e-97)
   (- (* x y))
   (if (<= x 2.4e-27) (log 2.0) (* x (+ (- 0.5 y) (* x 0.125))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.95e-97) {
		tmp = -(x * y);
	} else if (x <= 2.4e-27) {
		tmp = log(2.0);
	} else {
		tmp = x * ((0.5 - y) + (x * 0.125));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.95d-97)) then
        tmp = -(x * y)
    else if (x <= 2.4d-27) then
        tmp = log(2.0d0)
    else
        tmp = x * ((0.5d0 - y) + (x * 0.125d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.95e-97) {
		tmp = -(x * y);
	} else if (x <= 2.4e-27) {
		tmp = Math.log(2.0);
	} else {
		tmp = x * ((0.5 - y) + (x * 0.125));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.95e-97:
		tmp = -(x * y)
	elif x <= 2.4e-27:
		tmp = math.log(2.0)
	else:
		tmp = x * ((0.5 - y) + (x * 0.125))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.95e-97)
		tmp = Float64(-Float64(x * y));
	elseif (x <= 2.4e-27)
		tmp = log(2.0);
	else
		tmp = Float64(x * Float64(Float64(0.5 - y) + Float64(x * 0.125)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.95e-97)
		tmp = -(x * y);
	elseif (x <= 2.4e-27)
		tmp = log(2.0);
	else
		tmp = x * ((0.5 - y) + (x * 0.125));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.95e-97], (-N[(x * y), $MachinePrecision]), If[LessEqual[x, 2.4e-27], N[Log[2.0], $MachinePrecision], N[(x * N[(N[(0.5 - y), $MachinePrecision] + N[(x * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9499999999999999e-97

   1. Initial program 99.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Step-by-step derivation

      1. log1p-def 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]

   4. Taylor expanded in x around inf 94.3%

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

      1. associate-*r* 94.3%

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

      2. *-commutative 94.3%

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

      3. mul-1-neg 94.3%

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

   6. Simplified 94.3%

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

   if -1.9499999999999999e-97 < x < 2.40000000000000002e-27

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -y, \log \left(1 + e^{x}\right)\right)} \]

      5. log1p-def 100.0%

         \[\leadsto \mathsf{fma}\left(x, -y, \color{blue}{\mathsf{log1p}\left(e^{x}\right)}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -y, \mathsf{log1p}\left(e^{x}\right)\right)} \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{fma}\left(x, -y, \color{blue}{\log 2}\right) \]

   5. Taylor expanded in x around 0 80.7%

      \[\leadsto \color{blue}{\log 2} \]

   if 2.40000000000000002e-27 < x

   1. Initial program 92.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Step-by-step derivation

      1. log1p-def 92.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

   3. Simplified 92.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]

   4. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{\left(0.5 - y\right) \cdot x + \left(0.125 \cdot {x}^{2} + \log 2\right)} \]

   5. Taylor expanded in x around inf 79.7%

      \[\leadsto \color{blue}{\left(0.5 - y\right) \cdot x + 0.125 \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 79.7%

         \[\leadsto \left(0.5 - y\right) \cdot x + 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]

      2. associate-*r* 79.7%

         \[\leadsto \left(0.5 - y\right) \cdot x + \color{blue}{\left(0.125 \cdot x\right) \cdot x} \]

      3. distribute-rgt-out 79.7%

         \[\leadsto \color{blue}{x \cdot \left(\left(0.5 - y\right) + 0.125 \cdot x\right)} \]

   7. Simplified 79.7%

      \[\leadsto \color{blue}{x \cdot \left(\left(0.5 - y\right) + 0.125 \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-97}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-27}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.5 - y\right) + x \cdot 0.125\right)\\ \end{array} \]

Alternative 6: 54.8% accurate, 51.8× speedup

Math

\[\begin{array}{l} \\ -x \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
2. Step-by-step derivation

   1. log1p-def 99.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

3. Simplified 99.2%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]

4. Taylor expanded in x around inf 58.0%

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

   1. associate-*r* 58.0%

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

   2. *-commutative 58.0%

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

   3. mul-1-neg 58.0%

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

6. Simplified 58.0%

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

7. Final simplification 58.0%

   \[\leadsto -x \cdot y \]

Alternative 7: 3.6% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x 0.5))

C

double code(double x, double y) {
	return x * 0.5;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 0.5d0
end function

Java

public static double code(double x, double y) {
	return x * 0.5;
}

Python

def code(x, y):
	return x * 0.5

Julia

function code(x, y)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x, y)
	tmp = x * 0.5;
end

Wolfram

code[x_, y_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
2. Step-by-step derivation

   1. log1p-def 99.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

3. Simplified 99.2%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]

4. Taylor expanded in x around 0 80.8%

   \[\leadsto \color{blue}{\left(0.5 - y\right) \cdot x + \log 2} \]

5. Taylor expanded in x around inf 39.7%

   \[\leadsto \color{blue}{\left(0.5 - y\right) \cdot x} \]

6. Taylor expanded in y around 0 3.8%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

7. Final simplification 3.8%

   \[\leadsto x \cdot 0.5 \]

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0)
   (- (log (+ 1.0 (exp x))) (* x y))
   (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0) {
		tmp = log((1.0 + exp(x))) - (x * y);
	} else {
		tmp = log((1.0 + exp(-x))) - (-x * (1.0 - 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 <= 0.0d0) then
        tmp = log((1.0d0 + exp(x))) - (x * y)
    else
        tmp = log((1.0d0 + exp(-x))) - (-x * (1.0d0 - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.0) {
		tmp = Math.log((1.0 + Math.exp(x))) - (x * y);
	} else {
		tmp = Math.log((1.0 + Math.exp(-x))) - (-x * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.0:
		tmp = math.log((1.0 + math.exp(x))) - (x * y)
	else:
		tmp = math.log((1.0 + math.exp(-x))) - (-x * (1.0 - y))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0], N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[((-x) * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023240 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1.0 (exp x))) (* x y)) (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y))))

  (- (log (+ 1.0 (exp x))) (* x y)))