Logistic regression 2

Percentage Accurate: 99.3% → 99.3%

Time: 10.9s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. x = 2.888721965061796e+114
  2. y = 2.018875270077834e+307

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 99.2%

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

   1. log1p-define 99.6%

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

3. Simplified 99.6%

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

Alternative 2: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -150000000000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -150000000000.0)
   (* x (- y))
   (+ (log 2.0) (* x (- (+ 0.5 (* x 0.125)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -150000000000.0) {
		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 <= (-150000000000.0d0)) 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 <= -150000000000.0) {
		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 <= -150000000000.0:
		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 <= -150000000000.0)
		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 <= -150000000000.0)
		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, -150000000000.0], 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 < -1.5e11

   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. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

   if -1.5e11 < x

   1. Initial program 98.8%

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

      1. log1p-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -150000000000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.36:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.36) (* x (- y)) (+ (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.36) {
		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 <= (-1.36d0)) 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 <= -1.36) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) + (x * (0.5 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.36:
		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 <= -1.36)
		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 <= -1.36)
		tmp = x * -y;
	else
		tmp = log(2.0) + (x * (0.5 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.36], 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 < -1.3600000000000001

   1. Initial program 98.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. Taylor expanded in x around inf 98.9%

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

      1. associate-*r* 98.9%

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

      2. neg-mul-1 98.9%

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

   7. Simplified 98.9%

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

   if -1.3600000000000001 < x

   1. Initial program 99.4%

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

      1. log1p-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.36:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -150000000000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -150000000000.0) (* x (- y)) (- (log1p 1.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -150000000000.0) {
		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 <= -150000000000.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log1p(1.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -150000000000.0:
		tmp = x * -y
	else:
		tmp = math.log1p(1.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -150000000000.0)
		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, -150000000000.0], 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 < -1.5e11

   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. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

   if -1.5e11 < x

   1. Initial program 98.8%

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

      1. log1p-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 97.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -150000000000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.00088) (* x (- y)) (+ (log 2.0) (* x 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.00088) {
		tmp = x * -y;
	} else {
		tmp = log(2.0) + (x * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.00088d0)) then
        tmp = x * -y
    else
        tmp = log(2.0d0) + (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.00088) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) + (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.00088:
		tmp = x * -y
	else:
		tmp = math.log(2.0) + (x * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.00088)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(log(2.0) + Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.00088)
		tmp = x * -y;
	else
		tmp = log(2.0) + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.00088], N[(x * (-y)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.80000000000000031e-4

   1. Initial program 98.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. Taylor expanded in x around inf 98.9%

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

      1. associate-*r* 98.9%

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

      2. neg-mul-1 98.9%

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

   7. Simplified 98.9%

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

   if -8.80000000000000031e-4 < x

   1. Initial program 99.4%

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

      1. log1p-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.9%

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

   6. Taylor expanded in y around 0 76.5%

      \[\leadsto \color{blue}{\log 2 + 0.5 \cdot x} \]
   7. Step-by-step derivation

      1. +-commutative 76.5%

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

   8. Simplified 76.5%

      \[\leadsto \color{blue}{0.5 \cdot x + \log 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00088:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.2e-63) (* x (- y)) (log1p (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.2e-63) {
		tmp = x * -y;
	} else {
		tmp = log1p((x + 1.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.2e-63) {
		tmp = x * -y;
	} else {
		tmp = Math.log1p((x + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.2e-63:
		tmp = x * -y
	else:
		tmp = math.log1p((x + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.2e-63)
		tmp = Float64(x * Float64(-y));
	else
		tmp = log1p(Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.2e-63], N[(x * (-y)), $MachinePrecision], N[Log[1 + N[(x + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.19999999999999968e-63

   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 91.4%

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

      1. associate-*r* 91.4%

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

      2. neg-mul-1 91.4%

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

   7. Simplified 91.4%

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

   if -6.19999999999999968e-63 < x

   1. Initial program 99.4%

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

      1. log1p-define 99.4%

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

   3. Simplified 99.4%

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

      1. add-log-exp 99.4%

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

      2. add-sqr-sqrt 97.4%

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

      3. log-prod 98.1%

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

      4. log1p-undefine 98.1%

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

      5. rem-exp-log 98.1%

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

      6. +-commutative 98.1%

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

      7. log1p-undefine 98.1%

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

      8. rem-exp-log 98.1%

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

      9. +-commutative 98.1%

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

   6. Applied egg-rr 98.1%

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

      1. count-2 98.1%

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

      2. +-commutative 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in y around 0 77.9%

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

       1. unpow1/2 77.9%

          \[\leadsto 2 \cdot \log \color{blue}{\left({\left(1 + e^{x}\right)}^{0.5}\right)} \]

       2. exp-to-pow 77.9%

          \[\leadsto 2 \cdot \log \color{blue}{\left(e^{\log \left(1 + e^{x}\right) \cdot 0.5}\right)} \]

       3. log1p-define 77.9%

          \[\leadsto 2 \cdot \log \left(e^{\color{blue}{\mathsf{log1p}\left(e^{x}\right)} \cdot 0.5}\right) \]

       4. *-commutative 77.9%

          \[\leadsto 2 \cdot \log \left(e^{\color{blue}{0.5 \cdot \mathsf{log1p}\left(e^{x}\right)}}\right) \]

       5. log1p-expm1 77.9%

          \[\leadsto 2 \cdot \log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right)\right)\right)}}\right) \]

       6. log1p-undefine 77.9%

          \[\leadsto 2 \cdot \log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right)\right)\right)}}\right) \]

       7. rem-exp-log 77.9%

          \[\leadsto 2 \cdot \log \color{blue}{\left(1 + \mathsf{expm1}\left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right)\right)\right)} \]

       8. log1p-undefine 79.1%

          \[\leadsto 2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right)\right)\right)} \]

       9. log1p-expm1 79.1%

          \[\leadsto 2 \cdot \color{blue}{\left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right)\right)} \]

       10. count-2 79.1%

           \[\leadsto \color{blue}{0.5 \cdot \mathsf{log1p}\left(e^{x}\right) + 0.5 \cdot \mathsf{log1p}\left(e^{x}\right)} \]

       11. distribute-rgt-out 79.1%

           \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) \cdot \left(0.5 + 0.5\right)} \]

       12. metadata-eval 79.1%

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

       13. *-rgt-identity 79.1%

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

   11. Simplified 79.1%

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

   12. Taylor expanded in x around 0 79.2%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{1 + x}\right) \]
   13. Step-by-step derivation

       1. +-commutative 79.2%

          \[\leadsto \mathsf{log1p}\left(\color{blue}{x + 1}\right) \]

   14. Simplified 79.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -6.2e-63) (* x (- y)) (log 2.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.2e-63) {
		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 <= (-6.2d-63)) 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 <= -6.2e-63) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.2e-63:
		tmp = x * -y
	else:
		tmp = math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.2e-63)
		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 <= -6.2e-63)
		tmp = x * -y;
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.2e-63], N[(x * (-y)), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.19999999999999968e-63

   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 91.4%

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

      1. associate-*r* 91.4%

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

      2. neg-mul-1 91.4%

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

   7. Simplified 91.4%

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

   if -6.19999999999999968e-63 < x

   1. Initial program 99.4%

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

      1. log1p-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.4%

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

   6. Taylor expanded in x around 0 79.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.9% 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 99.2%

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

   1. log1p-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around inf 49.3%

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

   1. associate-*r* 49.3%

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

   2. neg-mul-1 49.3%

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

7. Simplified 49.3%

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

8. Final simplification 49.3%

   \[\leadsto x \cdot \left(-y\right) \]
9. Add Preprocessing

Alternative 9: 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 99.2%

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

   1. log1p-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 80.3%

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

6. Taylor expanded in x around inf 30.9%

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

7. Taylor expanded in y around 0 3.3%

   \[\leadsto x \cdot \color{blue}{0.5} \]
8. 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 2024135 
(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)))