Logistic regression 2

Percentage Accurate: 99.3% → 99.4%

Time: 8.5s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. x = 1.5689499969395696e+209
  2. y = -1.8619482417941356e+208

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: 99.4% 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 99.9%

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

   1. log1p-define 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. Add Preprocessing
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -122000000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -122000000.0)
   (* x (- y))
   (+ (log 2.0) (* x (- (+ 0.5 (* x 0.125)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -122000000.0) {
		tmp = x * -y;
	} else {
		tmp = log(2.0) + (x * ((0.5 + (x * 0.125)) - 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 <= (-122000000.0d0)) then
        tmp = x * -y
    else
        tmp = log(2.0d0) + (x * ((0.5d0 + (x * 0.125d0)) - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -122000000.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) + (x * ((0.5 + (x * 0.125)) - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -122000000.0:
		tmp = x * -y
	else:
		tmp = math.log(2.0) + (x * ((0.5 + (x * 0.125)) - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -122000000.0)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(log(2.0) + Float64(x * Float64(Float64(0.5 + Float64(x * 0.125)) - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -122000000.0)
		tmp = x * -y;
	else
		tmp = log(2.0) + (x * ((0.5 + (x * 0.125)) - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -122000000.0], N[(x * (-y)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(N[(0.5 + N[(x * 0.125), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.22e8

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

   5. Simplified 100.0%

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

   if -1.22e8 < x

   1. Initial program 99.8%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -122000000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-43}:\\ \;\;\;\;\log \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.65e-8)
   (* x (- y))
   (if (<= x 8e-43) (log (+ x 2.0)) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.65e-8) {
		tmp = x * -y;
	} else if (x <= 8e-43) {
		tmp = log((x + 2.0));
	} else {
		tmp = 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 <= (-2.65d-8)) then
        tmp = x * -y
    else if (x <= 8d-43) then
        tmp = log((x + 2.0d0))
    else
        tmp = x * (0.5d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.65e-8) {
		tmp = x * -y;
	} else if (x <= 8e-43) {
		tmp = Math.log((x + 2.0));
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.65e-8:
		tmp = x * -y
	elif x <= 8e-43:
		tmp = math.log((x + 2.0))
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.65e-8)
		tmp = Float64(x * Float64(-y));
	elseif (x <= 8e-43)
		tmp = log(Float64(x + 2.0));
	else
		tmp = Float64(x * Float64(0.5 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.65e-8)
		tmp = x * -y;
	elseif (x <= 8e-43)
		tmp = log((x + 2.0));
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.65e-8], N[(x * (-y)), $MachinePrecision], If[LessEqual[x, 8e-43], N[Log[N[(x + 2.0), $MachinePrecision]], $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6499999999999999e-8

   1. Initial program 99.6%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. distribute-rgt-neg-out 98.8%

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

   5. Simplified 98.8%

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

   if -2.6499999999999999e-8 < x < 8.00000000000000062e-43

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 82.2%

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

   4. Taylor expanded in x around 0 82.2%

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

   if 8.00000000000000062e-43 < x

   1. Initial program 99.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 84.2%

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

   4. Taylor expanded in y around -inf 84.3%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(x + -1 \cdot \frac{\log 2 + 0.5 \cdot x}{y}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 84.3%

         \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(x + -1 \cdot \frac{\log 2 + 0.5 \cdot x}{y}\right)} \]

      2. neg-mul-1 84.3%

         \[\leadsto \color{blue}{\left(-y\right)} \cdot \left(x + -1 \cdot \frac{\log 2 + 0.5 \cdot x}{y}\right) \]

      3. mul-1-neg 84.3%

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

   6. Simplified 84.3%

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

   7. Taylor expanded in x around inf 66.8%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(y \cdot \left(1 - 0.5 \cdot \frac{1}{y}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 66.8%

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(y \cdot \left(1 - 0.5 \cdot \frac{1}{y}\right)\right)} \]

      2. neg-mul-1 66.8%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \left(y \cdot \left(1 - 0.5 \cdot \frac{1}{y}\right)\right) \]

      3. associate-*r/ 66.8%

         \[\leadsto \left(-x\right) \cdot \left(y \cdot \left(1 - \color{blue}{\frac{0.5 \cdot 1}{y}}\right)\right) \]

      4. metadata-eval 66.8%

         \[\leadsto \left(-x\right) \cdot \left(y \cdot \left(1 - \frac{\color{blue}{0.5}}{y}\right)\right) \]

   9. Simplified 66.8%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \left(y \cdot \left(1 - \frac{0.5}{y}\right)\right)} \]

   10. Taylor expanded in x around 0 66.8%

       \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(y \cdot \left(1 - 0.5 \cdot \frac{1}{y}\right)\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 66.8%

          \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(y \cdot \left(1 - 0.5 \cdot \frac{1}{y}\right)\right)} \]

       2. *-commutative 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \color{blue}{\left(\left(1 - 0.5 \cdot \frac{1}{y}\right) \cdot y\right)} \]

       3. sub-neg 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\color{blue}{\left(1 + \left(-0.5 \cdot \frac{1}{y}\right)\right)} \cdot y\right) \]

       4. associate-*r/ 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\left(1 + \left(-\color{blue}{\frac{0.5 \cdot 1}{y}}\right)\right) \cdot y\right) \]

       5. metadata-eval 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\left(1 + \left(-\frac{\color{blue}{0.5}}{y}\right)\right) \cdot y\right) \]

       6. distribute-neg-frac 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\left(1 + \color{blue}{\frac{-0.5}{y}}\right) \cdot y\right) \]

       7. metadata-eval 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\left(1 + \frac{\color{blue}{-0.5}}{y}\right) \cdot y\right) \]

       8. +-commutative 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\color{blue}{\left(\frac{-0.5}{y} + 1\right)} \cdot y\right) \]

       9. distribute-lft1-in 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \color{blue}{\left(\frac{-0.5}{y} \cdot y + y\right)} \]

       10. *-commutative 66.8%

           \[\leadsto \left(-1 \cdot x\right) \cdot \left(\color{blue}{y \cdot \frac{-0.5}{y}} + y\right) \]

       11. distribute-lft-in 66.8%

           \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(y \cdot \frac{-0.5}{y}\right) + \left(-1 \cdot x\right) \cdot y} \]

       12. associate-*r* 66.8%

           \[\leadsto \left(-1 \cdot x\right) \cdot \left(y \cdot \frac{-0.5}{y}\right) + \color{blue}{-1 \cdot \left(x \cdot y\right)} \]

       13. mul-1-neg 66.8%

           \[\leadsto \left(-1 \cdot x\right) \cdot \left(y \cdot \frac{-0.5}{y}\right) + \color{blue}{\left(-x \cdot y\right)} \]

   12. Simplified 66.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-43}:\\ \;\;\;\;\log \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y):
	tmp = 0
	if x <= -1.4:
		tmp = x * -y
	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(x * Float64(-y));
	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 = x * -y;
	else
		tmp = log(2.0) + (x * (0.5 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4], N[(x * (-y)), $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 99.6%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-out 98.7%

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

   5. Simplified 98.7%

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

   if -1.3999999999999999 < x

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.8%

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

Alternative 5: 82.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-31}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-42}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e-31)
   (* x (- y))
   (if (<= x 1.5e-42) (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-31) {
		tmp = x * -y;
	} else if (x <= 1.5e-42) {
		tmp = log(2.0);
	} else {
		tmp = 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 <= (-2.8d-31)) then
        tmp = x * -y
    else if (x <= 1.5d-42) then
        tmp = log(2.0d0)
    else
        tmp = x * (0.5d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.8e-31) {
		tmp = x * -y;
	} else if (x <= 1.5e-42) {
		tmp = Math.log(2.0);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.8e-31:
		tmp = x * -y
	elif x <= 1.5e-42:
		tmp = math.log(2.0)
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.8e-31)
		tmp = Float64(x * Float64(-y));
	elseif (x <= 1.5e-42)
		tmp = log(2.0);
	else
		tmp = Float64(x * Float64(0.5 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.8e-31)
		tmp = x * -y;
	elseif (x <= 1.5e-42)
		tmp = log(2.0);
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.8e-31], N[(x * (-y)), $MachinePrecision], If[LessEqual[x, 1.5e-42], N[Log[2.0], $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7999999999999999e-31

   1. Initial program 99.6%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 95.4%

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

      1. mul-1-neg 95.4%

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

      2. distribute-rgt-neg-out 95.4%

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

   5. Simplified 95.4%

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

   if -2.7999999999999999e-31 < x < 1.50000000000000014e-42

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 83.3%

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

   4. Taylor expanded in x around 0 83.3%

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

   if 1.50000000000000014e-42 < x

   1. Initial program 99.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 84.2%

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

   4. Taylor expanded in y around -inf 84.3%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(x + -1 \cdot \frac{\log 2 + 0.5 \cdot x}{y}\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 84.3%

         \[\leadsto \color{blue}{\left(-1 \cdot y\right) \cdot \left(x + -1 \cdot \frac{\log 2 + 0.5 \cdot x}{y}\right)} \]

      2. neg-mul-1 84.3%

         \[\leadsto \color{blue}{\left(-y\right)} \cdot \left(x + -1 \cdot \frac{\log 2 + 0.5 \cdot x}{y}\right) \]

      3. mul-1-neg 84.3%

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

   6. Simplified 84.3%

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

   7. Taylor expanded in x around inf 66.8%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(y \cdot \left(1 - 0.5 \cdot \frac{1}{y}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 66.8%

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(y \cdot \left(1 - 0.5 \cdot \frac{1}{y}\right)\right)} \]

      2. neg-mul-1 66.8%

         \[\leadsto \color{blue}{\left(-x\right)} \cdot \left(y \cdot \left(1 - 0.5 \cdot \frac{1}{y}\right)\right) \]

      3. associate-*r/ 66.8%

         \[\leadsto \left(-x\right) \cdot \left(y \cdot \left(1 - \color{blue}{\frac{0.5 \cdot 1}{y}}\right)\right) \]

      4. metadata-eval 66.8%

         \[\leadsto \left(-x\right) \cdot \left(y \cdot \left(1 - \frac{\color{blue}{0.5}}{y}\right)\right) \]

   9. Simplified 66.8%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \left(y \cdot \left(1 - \frac{0.5}{y}\right)\right)} \]

   10. Taylor expanded in x around 0 66.8%

       \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(y \cdot \left(1 - 0.5 \cdot \frac{1}{y}\right)\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 66.8%

          \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(y \cdot \left(1 - 0.5 \cdot \frac{1}{y}\right)\right)} \]

       2. *-commutative 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \color{blue}{\left(\left(1 - 0.5 \cdot \frac{1}{y}\right) \cdot y\right)} \]

       3. sub-neg 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\color{blue}{\left(1 + \left(-0.5 \cdot \frac{1}{y}\right)\right)} \cdot y\right) \]

       4. associate-*r/ 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\left(1 + \left(-\color{blue}{\frac{0.5 \cdot 1}{y}}\right)\right) \cdot y\right) \]

       5. metadata-eval 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\left(1 + \left(-\frac{\color{blue}{0.5}}{y}\right)\right) \cdot y\right) \]

       6. distribute-neg-frac 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\left(1 + \color{blue}{\frac{-0.5}{y}}\right) \cdot y\right) \]

       7. metadata-eval 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\left(1 + \frac{\color{blue}{-0.5}}{y}\right) \cdot y\right) \]

       8. +-commutative 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \left(\color{blue}{\left(\frac{-0.5}{y} + 1\right)} \cdot y\right) \]

       9. distribute-lft1-in 66.8%

          \[\leadsto \left(-1 \cdot x\right) \cdot \color{blue}{\left(\frac{-0.5}{y} \cdot y + y\right)} \]

       10. *-commutative 66.8%

           \[\leadsto \left(-1 \cdot x\right) \cdot \left(\color{blue}{y \cdot \frac{-0.5}{y}} + y\right) \]

       11. distribute-lft-in 66.8%

           \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(y \cdot \frac{-0.5}{y}\right) + \left(-1 \cdot x\right) \cdot y} \]

       12. associate-*r* 66.8%

           \[\leadsto \left(-1 \cdot x\right) \cdot \left(y \cdot \frac{-0.5}{y}\right) + \color{blue}{-1 \cdot \left(x \cdot y\right)} \]

       13. mul-1-neg 66.8%

           \[\leadsto \left(-1 \cdot x\right) \cdot \left(y \cdot \frac{-0.5}{y}\right) + \color{blue}{\left(-x \cdot y\right)} \]

   12. Simplified 66.8%

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

Alternative 6: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -122000000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -122000000.0) (* x (- y)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -122000000.0) {
		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 <= (-122000000.0d0)) 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 <= -122000000.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -122000000.0:
		tmp = x * -y
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -122000000.0)
		tmp = Float64(x * Float64(-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 <= -122000000.0)
		tmp = x * -y;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -122000000.0], 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 < -1.22e8

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

   5. Simplified 100.0%

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

   if -1.22e8 < x

   1. Initial program 99.8%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 97.7%

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

Alternative 7: 51.8% accurate, 51.8× speedup

Math

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

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(x * Float64(-y))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
2. Add Preprocessing

3. Taylor expanded in x around inf 46.0%

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

   1. mul-1-neg 46.0%

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

   2. distribute-rgt-neg-out 46.0%

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

5. Simplified 46.0%

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

Alternative 8: 3.5% 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 99.9%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
2. Add Preprocessing

3. Taylor expanded in x around 0 83.1%

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

4. Taylor expanded in y around 0 54.9%

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

5. Taylor expanded in x around inf 3.5%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
6. Step-by-step derivation

   1. *-commutative 3.5%

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

7. Simplified 3.5%

   \[\leadsto \color{blue}{x \cdot 0.5} \]
8. Add Preprocessing

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 2024106 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :alt
  (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)))