Logistic regression 2

Percentage Accurate: 99.3% → 99.3%

Time: 8.9s

Alternatives: 8

Speedup: 1.8×

• could not determine a ground truth

  1. x = 3.282111321768906e+247
  2. y = -7.546751222945161e-23

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

\[\begin{array}{l} \\ \log \left(-1 + \left(2 + e^{x}\right)\right) - x \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (log (+ -1.0 (+ 2.0 (exp x)))) (* x y)))

C

double code(double x, double y) {
	return log((-1.0 + (2.0 + 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) + (2.0d0 + exp(x)))) - (x * y)
end function

Java

public static double code(double x, double y) {
	return Math.log((-1.0 + (2.0 + Math.exp(x)))) - (x * y);
}

Python

def code(x, y):
	return math.log((-1.0 + (2.0 + math.exp(x)))) - (x * y)

Julia

function code(x, y)
	return Float64(log(Float64(-1.0 + Float64(2.0 + exp(x)))) - Float64(x * y))
end

MATLAB

function tmp = code(x, y)
	tmp = log((-1.0 + (2.0 + exp(x)))) - (x * y);
end

Wolfram

code[x_, y_] := N[(N[Log[N[(-1.0 + N[(2.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 98.8%

      \[\leadsto \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{x}\right)\right)\right)} - x \cdot y \]

   2. expm1-undefine 98.8%

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

4. Applied egg-rr 98.8%

   \[\leadsto \log \color{blue}{\left(e^{\mathsf{log1p}\left(1 + e^{x}\right)} - 1\right)} - x \cdot y \]
5. Step-by-step derivation

   1. sub-neg 98.8%

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

   2. metadata-eval 98.8%

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

   3. +-commutative 98.8%

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

   4. log1p-undefine 98.8%

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

   5. rem-exp-log 100.0%

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

   6. associate-+r+ 100.0%

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

   7. metadata-eval 100.0%

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

6. Simplified 100.0%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 3: 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 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. Add Preprocessing

Alternative 4: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -105:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -105.0) (* x (- y)) (+ (log 2.0) (* x (- (+ 0.5 (* x 0.125)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -105.0) {
		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 <= (-105.0d0)) 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 <= -105.0) {
		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 <= -105.0:
		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 <= -105.0)
		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 <= -105.0)
		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, -105.0], 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 < -105

   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. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   5. Simplified 100.0%

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

   if -105 < 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 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -105:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -105:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -105.0) (* x (- y)) (+ (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -105.0) {
		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 <= (-105.0d0)) 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 <= -105.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) + (x * (0.5 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -105.0:
		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 <= -105.0)
		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 <= -105.0)
		tmp = x * -y;
	else
		tmp = log(2.0) + (x * (0.5 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -105.0], 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 < -105

   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. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   5. Simplified 100.0%

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

   if -105 < 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 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -105:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-46} \lor \neg \left(x \leq 2.6 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.05e-46) (not (<= x 2.6e-19))) (* x (- y)) (log 2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.05e-46) || !(x <= 2.6e-19)) {
		tmp = x * -y;
	} 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 <= (-3.05d-46)) .or. (.not. (x <= 2.6d-19))) then
        tmp = x * -y
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.05e-46) || !(x <= 2.6e-19)) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.05e-46) or not (x <= 2.6e-19):
		tmp = x * -y
	else:
		tmp = math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.05e-46) || !(x <= 2.6e-19))
		tmp = Float64(x * Float64(-y));
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.05e-46) || ~((x <= 2.6e-19)))
		tmp = x * -y;
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.05e-46], N[Not[LessEqual[x, 2.6e-19]], $MachinePrecision]], N[(x * (-y)), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.05000000000000018e-46 or 2.60000000000000013e-19 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 90.1%

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

      1. associate-*r* 90.1%

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

      2. neg-mul-1 90.1%

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

   5. Simplified 90.1%

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

   if -3.05000000000000018e-46 < x < 2.60000000000000013e-19

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. add-log-exp 83.4%

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

      4. sum-log 83.4%

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

      5. distribute-rgt-neg-in 83.4%

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

   4. Applied egg-rr 83.4%

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

   5. Taylor expanded in x around 0 81.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-46} \lor \neg \left(x \leq 2.6 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -105:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -105.0) (* x (- y)) (- (log1p 1.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -105.0) {
		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 <= -105.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log1p(1.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -105.0:
		tmp = x * -y
	else:
		tmp = math.log1p(1.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -105.0)
		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, -105.0], 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 < -105

   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. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   5. Simplified 100.0%

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

   if -105 < x

   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 97.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -105:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 100.0%

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

3. Taylor expanded in x around inf 52.1%

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

   1. associate-*r* 52.1%

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

   2. neg-mul-1 52.1%

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

5. Simplified 52.1%

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

6. Final simplification 52.1%

   \[\leadsto x \cdot \left(-y\right) \]
7. 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 2024111 
(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)))