Logistic regression 2

Percentage Accurate: 99.3% → 99.2%

Time: 7.6s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. x = 8.810933271620642e+153
  2. y = 5.988775552655872e+293

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: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -35:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right) + \log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -35.0) (* y (- x)) (+ (* x (- 0.5 y)) (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -35.0) {
		tmp = y * -x;
	} 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 <= (-35.0d0)) then
        tmp = y * -x
    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 <= -35.0) {
		tmp = y * -x;
	} else {
		tmp = (x * (0.5 - y)) + Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -35.0:
		tmp = y * -x
	else:
		tmp = (x * (0.5 - y)) + math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -35.0)
		tmp = Float64(y * Float64(-x));
	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 <= -35.0)
		tmp = y * -x;
	else
		tmp = (x * (0.5 - y)) + log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -35.0], N[(y * (-x)), $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 < -35

   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. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

   6. Simplified 100.0%

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

   if -35 < x

   1. Initial program 98.2%

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

      1. log1p-def 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in x around 0 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -35:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right) + \log 2\\ \end{array} \]

Alternative 2: 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 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. Final simplification 98.8%

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

Alternative 3: 82.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-77}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-8}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.5 - y\right) + x \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.1e-77)
   (* y (- x))
   (if (<= x 1.7e-8) (+ (log 2.0) (* x 0.5)) (* x (+ (- 0.5 y) (* x 0.125))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.1e-77) {
		tmp = y * -x;
	} else if (x <= 1.7e-8) {
		tmp = log(2.0) + (x * 0.5);
	} 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 <= (-3.1d-77)) then
        tmp = y * -x
    else if (x <= 1.7d-8) then
        tmp = log(2.0d0) + (x * 0.5d0)
    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 <= -3.1e-77) {
		tmp = y * -x;
	} else if (x <= 1.7e-8) {
		tmp = Math.log(2.0) + (x * 0.5);
	} else {
		tmp = x * ((0.5 - y) + (x * 0.125));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.1e-77:
		tmp = y * -x
	elif x <= 1.7e-8:
		tmp = math.log(2.0) + (x * 0.5)
	else:
		tmp = x * ((0.5 - y) + (x * 0.125))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.1e-77)
		tmp = Float64(y * Float64(-x));
	elseif (x <= 1.7e-8)
		tmp = Float64(log(2.0) + Float64(x * 0.5));
	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 <= -3.1e-77)
		tmp = y * -x;
	elseif (x <= 1.7e-8)
		tmp = log(2.0) + (x * 0.5);
	else
		tmp = x * ((0.5 - y) + (x * 0.125));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.1e-77], N[(y * (-x)), $MachinePrecision], If[LessEqual[x, 1.7e-8], N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $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 < -3.10000000000000008e-77

   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 93.7%

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

      1. mul-1-neg 93.7%

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

      2. distribute-rgt-neg-out 93.7%

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

   6. Simplified 93.7%

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

   if -3.10000000000000008e-77 < x < 1.7e-8

   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 0 100.0%

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

   5. Taylor expanded in y around 0 78.7%

      \[\leadsto \color{blue}{0.5 \cdot x} + \log 2 \]

   if 1.7e-8 < x

   1. Initial program 84.5%

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

      1. log1p-def 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in x around 0 98.5%

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

   5. Taylor expanded in x around inf 90.9%

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

      1. *-commutative 90.9%

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

      2. *-commutative 90.9%

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

      3. unpow2 90.9%

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

      4. associate-*r* 90.9%

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

      5. distribute-lft-out 90.9%

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

   7. Simplified 90.9%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-77}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-8}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.5 - y\right) + x \cdot 0.125\right)\\ \end{array} \]

Alternative 4: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -200.0) (* y (- x)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -200.0) {
		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 <= (-200.0d0)) 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 <= -200.0) {
		tmp = y * -x;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -200.0:
		tmp = y * -x
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -200.0)
		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 <= -200.0)
		tmp = y * -x;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -200.0], 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 < -200

   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. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

   6. Simplified 100.0%

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

   if -200 < x

   1. Initial program 98.2%

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

      1. log1p-def 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in x around 0 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 5: 82.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-77}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-12}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.5 - y\right) + x \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e-77)
   (* y (- x))
   (if (<= x 3.4e-12) (log 2.0) (* x (+ (- 0.5 y) (* x 0.125))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.4e-77) {
		tmp = y * -x;
	} else if (x <= 3.4e-12) {
		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 <= (-3.4d-77)) then
        tmp = y * -x
    else if (x <= 3.4d-12) 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 <= -3.4e-77) {
		tmp = y * -x;
	} else if (x <= 3.4e-12) {
		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 <= -3.4e-77:
		tmp = y * -x
	elif x <= 3.4e-12:
		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 <= -3.4e-77)
		tmp = Float64(y * Float64(-x));
	elseif (x <= 3.4e-12)
		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 <= -3.4e-77)
		tmp = y * -x;
	elseif (x <= 3.4e-12)
		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, -3.4e-77], N[(y * (-x)), $MachinePrecision], If[LessEqual[x, 3.4e-12], 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 < -3.39999999999999983e-77

   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 93.7%

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

      1. mul-1-neg 93.7%

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

      2. distribute-rgt-neg-out 93.7%

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

   6. Simplified 93.7%

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

   if -3.39999999999999983e-77 < x < 3.4000000000000001e-12

   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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(e^{x}\right)\right)\right)} - x \cdot y \]

      2. expm1-udef 98.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{log1p}\left(e^{x}\right)\right)} - 1\right)} - x \cdot y \]

      3. log1p-udef 98.8%

         \[\leadsto \left(e^{\color{blue}{\log \left(1 + \mathsf{log1p}\left(e^{x}\right)\right)}} - 1\right) - x \cdot y \]

      4. add-exp-log 98.8%

         \[\leadsto \left(\color{blue}{\left(1 + \mathsf{log1p}\left(e^{x}\right)\right)} - 1\right) - x \cdot y \]

   5. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\left(\left(1 + \mathsf{log1p}\left(e^{x}\right)\right) - 1\right)} - x \cdot y \]

   6. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in x around 0 78.5%

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

   if 3.4000000000000001e-12 < x

   1. Initial program 84.5%

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

      1. log1p-def 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in x around 0 98.5%

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

   5. Taylor expanded in x around inf 90.9%

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

      1. *-commutative 90.9%

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

      2. *-commutative 90.9%

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

      3. unpow2 90.9%

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

      4. associate-*r* 90.9%

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

      5. distribute-lft-out 90.9%

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

   7. Simplified 90.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-77}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-12}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.5 - y\right) + x \cdot 0.125\right)\\ \end{array} \]

Alternative 6: 52.8% 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 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 inf 58.1%

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

   1. mul-1-neg 58.1%

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

   2. distribute-rgt-neg-out 58.1%

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

6. Simplified 58.1%

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

7. Final simplification 58.1%

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

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 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 83.7%

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

5. Taylor expanded in y around 0 44.4%

   \[\leadsto \color{blue}{0.5 \cdot x} + \log 2 \]

6. Taylor expanded in x around inf 3.5%

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

7. Final simplification 3.5%

   \[\leadsto x \cdot 0.5 \]

Developer target: 100.0% 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 2023207 
(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)))