Logistic regression 2

Percentage Accurate: 99.3% → 99.4%

Time: 9.4s

Alternatives: 10

Speedup: 1.8×

• could not determine a ground truth

  1. x = 6.44388400205341e+152
  2. y = 3.508560567941444e+242

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, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -150:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -150.0) (* y (- x)) (+ (log 2.0) (* x (- (+ 0.5 (* x 0.125)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -150.0) {
		tmp = y * -x;
	} 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 <= (-150.0d0)) then
        tmp = y * -x
    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 <= -150.0) {
		tmp = y * -x;
	} else {
		tmp = Math.log(2.0) + (x * ((0.5 + (x * 0.125)) - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -150.0:
		tmp = y * -x
	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 <= -150.0)
		tmp = Float64(y * Float64(-x));
	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 <= -150.0)
		tmp = y * -x;
	else
		tmp = log(2.0) + (x * ((0.5 + (x * 0.125)) - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -150.0], N[(y * (-x)), $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 < -150

   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. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   5. Simplified 100.0%

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

   if -150 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -150:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, -x, \mathsf{log1p}\left(e^{x}\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma y (- x) (log1p (exp x))))

C

double code(double x, double y) {
	return fma(y, -x, log1p(exp(x)));
}

Julia

function code(x, y)
	return fma(y, Float64(-x), log1p(exp(x)))
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. cancel-sign-sub-inv 99.2%

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

   2. +-commutative 99.2%

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

   3. *-commutative 99.2%

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

   4. fma-define 99.2%

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

   5. log1p-define 99.2%

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

3. Simplified 99.2%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Final simplification 99.2%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\frac{\mathsf{log1p}\left(e^{x}\right)}{x} - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (- (/ (log1p (exp x)) x) y)))

C

double code(double x, double y) {
	return x * ((log1p(exp(x)) / x) - y);
}

Java

public static double code(double x, double y) {
	return x * ((Math.log1p(Math.exp(x)) / x) - y);
}

Python

def code(x, y):
	return x * ((math.log1p(math.exp(x)) / x) - y)

Julia

function code(x, y)
	return Float64(x * Float64(Float64(log1p(exp(x)) / x) - y))
end

Wolfram

code[x_, y_] := N[(x * N[(N[(N[Log[1 + N[Exp[x], $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around inf 99.2%

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

   1. sub-neg 99.2%

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

   2. sub-neg 99.2%

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

   3. log1p-define 99.2%

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

5. Simplified 99.2%

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

Alternative 5: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4], N[(y * (-x)), $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 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. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   5. Simplified 100.0%

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

   if -1.3999999999999999 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0 99.1%

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

Alternative 6: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -95:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -95.0) (* y (- x)) (- (log 2.0) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -95.0) {
		tmp = y * -x;
	} else {
		tmp = log(2.0) - (y * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-95.0d0)) then
        tmp = y * -x
    else
        tmp = log(2.0d0) - (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -95.0) {
		tmp = y * -x;
	} else {
		tmp = Math.log(2.0) - (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -95.0:
		tmp = y * -x
	else:
		tmp = math.log(2.0) - (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -95.0)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(log(2.0) - Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -95.0)
		tmp = y * -x;
	else
		tmp = log(2.0) - (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -95.0], N[(y * (-x)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -95

   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. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   5. Simplified 100.0%

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

   if -95 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -95:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-80}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.5e-80) (* y (- x)) (+ (log 2.0) (* x 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.5e-80) {
		tmp = y * -x;
	} 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 <= (-2.5d-80)) then
        tmp = y * -x
    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 <= -2.5e-80) {
		tmp = y * -x;
	} else {
		tmp = Math.log(2.0) + (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.5e-80:
		tmp = y * -x
	else:
		tmp = math.log(2.0) + (x * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.5e-80)
		tmp = Float64(y * Float64(-x));
	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 <= -2.5e-80)
		tmp = y * -x;
	else
		tmp = log(2.0) + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.5e-80], N[(y * (-x)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5e-80

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. *-commutative 92.9%

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

      3. distribute-rgt-neg-in 92.9%

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

   5. Simplified 92.9%

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

   if -2.5e-80 < x

   1. Initial program 98.6%

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

   3. Taylor expanded in x around 0 99.0%

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

   4. Taylor expanded in y around 0 77.3%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-80}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-80}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -3e-80) (* y (- x)) (log 2.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3e-80) {
		tmp = y * -x;
	} 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 <= (-3d-80)) then
        tmp = y * -x
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3e-80) {
		tmp = y * -x;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3e-80:
		tmp = y * -x
	else:
		tmp = math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3e-80)
		tmp = Float64(y * Float64(-x));
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3e-80)
		tmp = y * -x;
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3e-80], N[(y * (-x)), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.00000000000000007e-80

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. *-commutative 92.9%

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

      3. distribute-rgt-neg-in 92.9%

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

   5. Simplified 92.9%

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

   if -3.00000000000000007e-80 < x

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf 98.5%

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

      1. sub-neg 98.5%

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

      2. sub-neg 98.5%

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

      3. log1p-define 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around 0 98.3%

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

   7. Taylor expanded in x around 0 76.8%

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

Alternative 9: 50.8% accurate, 51.8× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* y (- x)))

C

double code(double x, double y) {
	return y * -x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * -x
end function

Java

public static double code(double x, double y) {
	return y * -x;
}

Python

def code(x, y):
	return y * -x

Julia

function code(x, y)
	return Float64(y * Float64(-x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in x around inf 53.2%

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

   1. mul-1-neg 53.2%

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

   2. *-commutative 53.2%

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

   3. distribute-rgt-neg-in 53.2%

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

5. Simplified 53.2%

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

Alternative 10: 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 99.2%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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 0 47.8%

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

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 2024096 
(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)))