Logistic regression 2

Percentage Accurate: 99.3% → 99.3%

Time: 8.1s

Alternatives: 6

Speedup: 1.0×

• could not determine a ground truth

  1. x = 138531713373080.6
  2. y = -2.7217580278420154e+186

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: 99.3% 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, 1.0× 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 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. Final simplification 100.0%

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

Alternative 2: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -32:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 - y\right) + x \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -32.0) (* y (- x)) (+ (log 2.0) (* x (+ (- 0.5 y) (* x 0.125))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -32.0) {
		tmp = y * -x;
	} else {
		tmp = log(2.0) + (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 <= (-32.0d0)) then
        tmp = y * -x
    else
        tmp = log(2.0d0) + (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 <= -32.0) {
		tmp = y * -x;
	} else {
		tmp = Math.log(2.0) + (x * ((0.5 - y) + (x * 0.125)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -32.0:
		tmp = y * -x
	else:
		tmp = math.log(2.0) + (x * ((0.5 - y) + (x * 0.125)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -32.0)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(log(2.0) + 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 <= -32.0)
		tmp = y * -x;
	else
		tmp = log(2.0) + (x * ((0.5 - y) + (x * 0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -32.0], N[(y * (-x)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(N[(0.5 - y), $MachinePrecision] + N[(x * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -32

   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%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   6. Simplified 100.0%

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

   if -32 < x

   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 0 99.5%

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

      1. +-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. unpow2 99.5%

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

      4. associate-*l* 99.5%

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

      5. distribute-lft-out 99.5%

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

   6. Simplified 99.5%

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\left(0.5 - y\right) + x \cdot 0.125\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -32:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 - y\right) + x \cdot 0.125\right)\\ \end{array} \]

Alternative 3: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4) {
		tmp = y * -x;
	} else {
		tmp = log(2.0) + (x * (0.5 - 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 <= (-1.4d0)) then
        tmp = y * -x
    else
        tmp = log(2.0d0) + (x * (0.5d0 - y))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999

   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%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   6. Simplified 100.0%

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

   if -1.3999999999999999 < x

   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 0 99.3%

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

4. Final simplification 99.5%

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

Alternative 4: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -120:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -120.0) (* y (- x)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -120.0) {
		tmp = y * -x;
	} 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 <= (-120.0d0)) then
        tmp = y * -x
    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 <= -120.0) {
		tmp = y * -x;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -120.0:
		tmp = y * -x
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -120.0)
		tmp = Float64(y * Float64(-x));
	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 <= -120.0)
		tmp = y * -x;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -120.0], N[(y * (-x)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -120

   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%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   6. Simplified 100.0%

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

   if -120 < x

   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 0 99.3%

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

   5. Taylor expanded in y around inf 99.1%

      \[\leadsto \log 2 + \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto \log 2 + \color{blue}{\left(-1 \cdot x\right) \cdot y} \]

      2. *-commutative 99.1%

         \[\leadsto \log 2 + \color{blue}{y \cdot \left(-1 \cdot x\right)} \]

      3. neg-mul-1 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in y around 0 99.1%

      \[\leadsto \color{blue}{\log 2 + -1 \cdot \left(x \cdot y\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 99.1%

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

      2. sub-neg 99.1%

         \[\leadsto \color{blue}{\log 2 - x \cdot y} \]

   10. Simplified 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -120:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 5: 83.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-50} \lor \neg \left(x \leq 3 \cdot 10^{-26}\right):\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.1e-50) (not (<= x 3e-26))) (* y (- x)) (log 2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.1e-50) || !(x <= 3e-26)) {
		tmp = y * -x;
	} else {
		tmp = 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 <= (-1.1d-50)) .or. (.not. (x <= 3d-26))) then
        tmp = y * -x
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.1e-50) || !(x <= 3e-26)) {
		tmp = y * -x;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.1e-50) or not (x <= 3e-26):
		tmp = y * -x
	else:
		tmp = math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.1e-50) || !(x <= 3e-26))
		tmp = Float64(y * Float64(-x));
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.1e-50) || ~((x <= 3e-26)))
		tmp = y * -x;
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.1e-50], N[Not[LessEqual[x, 3e-26]], $MachinePrecision]], N[(y * (-x)), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0999999999999999e-50 or 3.00000000000000012e-26 < x

   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 96.4%

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

      1. mul-1-neg 96.4%

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

      2. *-commutative 96.4%

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

      3. distribute-rgt-neg-in 96.4%

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

   6. Simplified 96.4%

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

   if -1.0999999999999999e-50 < x < 3.00000000000000012e-26

   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 0 76.9%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-50} \lor \neg \left(x \leq 3 \cdot 10^{-26}\right):\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 6: 53.0% accurate, 51.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 55.3%

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

   1. mul-1-neg 55.3%

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

   2. *-commutative 55.3%

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

   3. distribute-rgt-neg-in 55.3%

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

6. Simplified 55.3%

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

7. Final simplification 55.3%

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

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 2023310 
(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)))