Logistic regression 2

Percentage Accurate: 99.3% → 99.3%

Time: 10.9s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. x = 1.4779399089253479e+128
  2. y = 1.398415742840406e+276

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 98.2%

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

   1. log1p-define 98.5%

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

3. Simplified 98.5%

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

Alternative 2: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -45000000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -45000000.0)
   (* x (- y))
   (+ (log 2.0) (* x (- (+ 0.5 (* x 0.125)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -45000000.0) {
		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 <= (-45000000.0d0)) 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 <= -45000000.0) {
		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 <= -45000000.0:
		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 <= -45000000.0)
		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 <= -45000000.0)
		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, -45000000.0], 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 < -4.5e7

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   if -4.5e7 < x

   1. Initial program 97.5%

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

      1. log1p-define 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in x around 0 96.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -45000000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 99.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. Taylor expanded in x around inf 97.4%

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

      1. mul-1-neg 97.4%

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

      2. distribute-rgt-neg-out 97.4%

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

   7. Simplified 97.4%

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

   if -1.3999999999999999 < x

   1. Initial program 97.9%

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

      1. log1p-define 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in x around 0 97.5%

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

Alternative 4: 81.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-20}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-69}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e-20)
   (* x (- y))
   (if (<= x 1.85e-69) (log 2.0) (- (* x 0.5) (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.4e-20) {
		tmp = x * -y;
	} else if (x <= 1.85e-69) {
		tmp = log(2.0);
	} else {
		tmp = (x * 0.5) - (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 <= (-3.4d-20)) then
        tmp = x * -y
    else if (x <= 1.85d-69) then
        tmp = log(2.0d0)
    else
        tmp = (x * 0.5d0) - (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.4e-20) {
		tmp = x * -y;
	} else if (x <= 1.85e-69) {
		tmp = Math.log(2.0);
	} else {
		tmp = (x * 0.5) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.4e-20:
		tmp = x * -y
	elif x <= 1.85e-69:
		tmp = math.log(2.0)
	else:
		tmp = (x * 0.5) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.4e-20)
		tmp = Float64(x * Float64(-y));
	elseif (x <= 1.85e-69)
		tmp = log(2.0);
	else
		tmp = Float64(Float64(x * 0.5) - Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.4e-20)
		tmp = x * -y;
	elseif (x <= 1.85e-69)
		tmp = log(2.0);
	else
		tmp = (x * 0.5) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.4e-20], N[(x * (-y)), $MachinePrecision], If[LessEqual[x, 1.85e-69], N[Log[2.0], $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3999999999999997e-20

   1. Initial program 99.1%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. Taylor expanded in x around inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. distribute-rgt-neg-out 94.2%

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

   7. Simplified 94.2%

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

   if -3.3999999999999997e-20 < x < 1.8500000000000001e-69

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. Taylor expanded in x around 0 85.7%

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

   if 1.8500000000000001e-69 < x

   1. Initial program 89.4%

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

      1. log1p-define 89.4%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in x around 0 89.2%

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

   6. Taylor expanded in y around -inf 89.0%

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

      1. associate-*r* 89.0%

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

      2. mul-1-neg 89.0%

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

      3. mul-1-neg 89.0%

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

      4. *-commutative 89.0%

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

   8. Simplified 89.0%

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

   9. Taylor expanded in x around -inf 57.3%

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

       1. *-commutative 57.3%

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

       2. *-commutative 57.3%

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

       3. associate-*l* 57.2%

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

       4. sub-neg 57.2%

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

       5. metadata-eval 57.2%

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

       6. +-commutative 57.2%

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

       7. associate-*r/ 57.2%

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

       8. metadata-eval 57.2%

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

       9. *-commutative 57.2%

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

   11. Simplified 57.2%

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

   12. Taylor expanded in y around 0 57.3%

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

       1. +-commutative 57.3%

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

       2. *-commutative 57.3%

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

       3. mul-1-neg 57.3%

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

       4. unsub-neg 57.3%

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

       5. *-commutative 57.3%

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

   14. Simplified 57.3%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-20}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-69}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5 - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -45000000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -45000000.0) (* x (- y)) (- (log1p 1.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -45000000.0) {
		tmp = x * -y;
	} else {
		tmp = log1p(1.0) - (x * y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -45000000.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log1p(1.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -45000000.0:
		tmp = x * -y
	else:
		tmp = math.log1p(1.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -45000000.0)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(log1p(1.0) - Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -45000000.0], N[(x * (-y)), $MachinePrecision], N[(N[Log[1 + 1.0], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5e7

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   if -4.5e7 < x

   1. Initial program 97.5%

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

      1. log1p-define 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in x around 0 96.0%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{1}\right) - x \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 51.6% 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 98.2%

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

   1. log1p-define 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in x around inf 46.8%

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

   1. mul-1-neg 46.8%

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

   2. distribute-rgt-neg-out 46.8%

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

7. Simplified 46.8%

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

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 98.2%

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

   1. log1p-define 98.5%

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

3. Simplified 98.5%

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

5. Taylor expanded in x around 0 83.8%

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

6. Taylor expanded in y around -inf 83.7%

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

   1. associate-*r* 83.7%

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

   2. mul-1-neg 83.7%

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

   3. mul-1-neg 83.7%

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

   4. *-commutative 83.7%

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

8. Simplified 83.7%

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

9. Taylor expanded in x around -inf 33.7%

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

    1. *-commutative 33.7%

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

    2. *-commutative 33.7%

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

    3. associate-*l* 33.7%

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

    4. sub-neg 33.7%

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

    5. metadata-eval 33.7%

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

    6. +-commutative 33.7%

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

    7. associate-*r/ 33.7%

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

    8. metadata-eval 33.7%

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

    9. *-commutative 33.7%

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

11. Simplified 33.7%

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

12. Taylor expanded in y around 0 3.5%

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

13. Final simplification 3.5%

    \[\leadsto x \cdot 0.5 \]
14. Add Preprocessing

Developer Target 1: 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 2024157 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (<= x 0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y)))))

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