Logistic regression 2

Percentage Accurate: 99.3% → 99.4%

Time: 11.7s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. x = 6.658268932984624e+292
  2. y = -13.459808415457083

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. log1p-def 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. Final simplification 99.6%

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

Alternative 2: 99.2% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4) (* y (- x)) (- (+ (log 2.0) (* x 0.5)) (* x 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)) - (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 <= (-1.4d0)) then
        tmp = y * -x
    else
        tmp = (log(2.0d0) + (x * 0.5d0)) - (x * 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)) - (x * 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)) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(Float64(log(2.0) + Float64(x * 0.5)) - Float64(x * 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)) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4], N[(y * (-x)), $MachinePrecision], N[(N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[(x * y), $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. Step-by-step derivation

      1. log1p-def 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 -1.3999999999999999 < x

   1. Initial program 99.4%

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

      1. log1p-def 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 98.4%

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

4. Final simplification 98.8%

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

Alternative 3: 81.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.85 \cdot 10^{-39}:\\ \;\;\;\;\log 2\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-23}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (- x))))
   (if (<= x -1.05e-9)
     t_0
     (if (<= x -2.85e-39)
       (log 2.0)
       (if (<= x -6e-88)
         t_0
         (if (<= x 1.06e-23) (log 2.0) (* x (- 0.5 y))))))))

C

double code(double x, double y) {
	double t_0 = y * -x;
	double tmp;
	if (x <= -1.05e-9) {
		tmp = t_0;
	} else if (x <= -2.85e-39) {
		tmp = log(2.0);
	} else if (x <= -6e-88) {
		tmp = t_0;
	} else if (x <= 1.06e-23) {
		tmp = log(2.0);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * -x
    if (x <= (-1.05d-9)) then
        tmp = t_0
    else if (x <= (-2.85d-39)) then
        tmp = log(2.0d0)
    else if (x <= (-6d-88)) then
        tmp = t_0
    else if (x <= 1.06d-23) then
        tmp = log(2.0d0)
    else
        tmp = x * (0.5d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * -x;
	double tmp;
	if (x <= -1.05e-9) {
		tmp = t_0;
	} else if (x <= -2.85e-39) {
		tmp = Math.log(2.0);
	} else if (x <= -6e-88) {
		tmp = t_0;
	} else if (x <= 1.06e-23) {
		tmp = Math.log(2.0);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * -x
	tmp = 0
	if x <= -1.05e-9:
		tmp = t_0
	elif x <= -2.85e-39:
		tmp = math.log(2.0)
	elif x <= -6e-88:
		tmp = t_0
	elif x <= 1.06e-23:
		tmp = math.log(2.0)
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(-x))
	tmp = 0.0
	if (x <= -1.05e-9)
		tmp = t_0;
	elseif (x <= -2.85e-39)
		tmp = log(2.0);
	elseif (x <= -6e-88)
		tmp = t_0;
	elseif (x <= 1.06e-23)
		tmp = log(2.0);
	else
		tmp = Float64(x * Float64(0.5 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * -x;
	tmp = 0.0;
	if (x <= -1.05e-9)
		tmp = t_0;
	elseif (x <= -2.85e-39)
		tmp = log(2.0);
	elseif (x <= -6e-88)
		tmp = t_0;
	elseif (x <= 1.06e-23)
		tmp = log(2.0);
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * (-x)), $MachinePrecision]}, If[LessEqual[x, -1.05e-9], t$95$0, If[LessEqual[x, -2.85e-39], N[Log[2.0], $MachinePrecision], If[LessEqual[x, -6e-88], t$95$0, If[LessEqual[x, 1.06e-23], N[Log[2.0], $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0500000000000001e-9 or -2.8499999999999998e-39 < x < -5.9999999999999999e-88

   1. Initial program 100.0%

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

      1. log1p-def 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 96.7%

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

      1. associate-*r* 96.7%

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

      2. neg-mul-1 96.7%

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

   7. Simplified 96.7%

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

   if -1.0500000000000001e-9 < x < -2.8499999999999998e-39 or -5.9999999999999999e-88 < x < 1.05999999999999994e-23

   1. Initial program 100.0%

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

      1. log1p-def 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 99.8%

      \[\leadsto \color{blue}{\log 2} - x \cdot y \]

   6. Taylor expanded in x around 0 80.8%

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

   if 1.05999999999999994e-23 < x

   1. Initial program 89.1%

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

      1. log1p-def 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in x around 0 67.3%

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

   6. Taylor expanded in x around inf 63.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -2.85 \cdot 10^{-39}:\\ \;\;\;\;\log 2\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-88}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-23}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-42}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-23}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (- x))))
   (if (<= x -4.3e-8)
     t_0
     (if (<= x -6.6e-42)
       (+ (log 2.0) (* x 0.5))
       (if (<= x -6.5e-88)
         t_0
         (if (<= x 2.9e-23) (log 2.0) (* x (- 0.5 y))))))))

C

double code(double x, double y) {
	double t_0 = y * -x;
	double tmp;
	if (x <= -4.3e-8) {
		tmp = t_0;
	} else if (x <= -6.6e-42) {
		tmp = log(2.0) + (x * 0.5);
	} else if (x <= -6.5e-88) {
		tmp = t_0;
	} else if (x <= 2.9e-23) {
		tmp = log(2.0);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * -x
    if (x <= (-4.3d-8)) then
        tmp = t_0
    else if (x <= (-6.6d-42)) then
        tmp = log(2.0d0) + (x * 0.5d0)
    else if (x <= (-6.5d-88)) then
        tmp = t_0
    else if (x <= 2.9d-23) then
        tmp = log(2.0d0)
    else
        tmp = x * (0.5d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * -x;
	double tmp;
	if (x <= -4.3e-8) {
		tmp = t_0;
	} else if (x <= -6.6e-42) {
		tmp = Math.log(2.0) + (x * 0.5);
	} else if (x <= -6.5e-88) {
		tmp = t_0;
	} else if (x <= 2.9e-23) {
		tmp = Math.log(2.0);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * -x
	tmp = 0
	if x <= -4.3e-8:
		tmp = t_0
	elif x <= -6.6e-42:
		tmp = math.log(2.0) + (x * 0.5)
	elif x <= -6.5e-88:
		tmp = t_0
	elif x <= 2.9e-23:
		tmp = math.log(2.0)
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(-x))
	tmp = 0.0
	if (x <= -4.3e-8)
		tmp = t_0;
	elseif (x <= -6.6e-42)
		tmp = Float64(log(2.0) + Float64(x * 0.5));
	elseif (x <= -6.5e-88)
		tmp = t_0;
	elseif (x <= 2.9e-23)
		tmp = log(2.0);
	else
		tmp = Float64(x * Float64(0.5 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * -x;
	tmp = 0.0;
	if (x <= -4.3e-8)
		tmp = t_0;
	elseif (x <= -6.6e-42)
		tmp = log(2.0) + (x * 0.5);
	elseif (x <= -6.5e-88)
		tmp = t_0;
	elseif (x <= 2.9e-23)
		tmp = log(2.0);
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * (-x)), $MachinePrecision]}, If[LessEqual[x, -4.3e-8], t$95$0, If[LessEqual[x, -6.6e-42], N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.5e-88], t$95$0, If[LessEqual[x, 2.9e-23], N[Log[2.0], $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.3000000000000001e-8 or -6.6000000000000005e-42 < x < -6.50000000000000006e-88

   1. Initial program 100.0%

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

      1. log1p-def 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 96.7%

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

      1. associate-*r* 96.7%

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

      2. neg-mul-1 96.7%

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

   7. Simplified 96.7%

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

   if -4.3000000000000001e-8 < x < -6.6000000000000005e-42

   1. Initial program 99.9%

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

      1. log1p-def 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 99.9%

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

   6. Taylor expanded in y around 0 73.1%

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

      1. *-commutative 73.1%

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

   8. Simplified 73.1%

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

   if -6.50000000000000006e-88 < x < 2.9000000000000002e-23

   1. Initial program 100.0%

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

      1. log1p-def 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 0 100.0%

      \[\leadsto \color{blue}{\log 2} - x \cdot y \]

   6. Taylor expanded in x around 0 81.4%

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

   if 2.9000000000000002e-23 < x

   1. Initial program 89.1%

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

      1. log1p-def 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in x around 0 67.3%

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

   6. Taylor expanded in x around inf 63.3%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-8}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-42}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-88}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-23}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.9× 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. Step-by-step derivation

      1. log1p-def 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 -1.3999999999999999 < x

   1. Initial program 99.4%

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

      1. log1p-def 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 98.4%

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

4. Final simplification 98.8%

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

Alternative 6: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -56:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -56.0) (* y (- x)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -56.0) {
		tmp = y * -x;
	} else {
		tmp = log(2.0) - (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-56.0d0)) then
        tmp = y * -x
    else
        tmp = log(2.0d0) - (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -56.0) {
		tmp = y * -x;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -56.0:
		tmp = y * -x
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -56.0)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(log(2.0) - Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -56.0)
		tmp = y * -x;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -56.0], N[(y * (-x)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -56

   1. Initial program 100.0%

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

      1. log1p-def 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 -56 < x

   1. Initial program 99.4%

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

      1. log1p-def 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 98.1%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -56:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.5% 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 99.6%

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

   1. log1p-def 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 46.4%

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

   1. associate-*r* 46.4%

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

   2. neg-mul-1 46.4%

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

7. Simplified 46.4%

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

8. Final simplification 46.4%

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

Alternative 8: 3.5% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x 0.5))

C

double code(double x, double y) {
	return x * 0.5;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 0.5d0
end function

Java

public static double code(double x, double y) {
	return x * 0.5;
}

Python

def code(x, y):
	return x * 0.5

Julia

function code(x, y)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x, y)
	tmp = x * 0.5;
end

Wolfram

code[x_, y_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. log1p-def 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 86.5%

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

6. Taylor expanded in x around inf 34.5%

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

7. Taylor expanded in y around 0 3.7%

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

   1. *-commutative 3.7%

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

9. Simplified 3.7%

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

10. Final simplification 3.7%

    \[\leadsto x \cdot 0.5 \]
11. 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 2024013 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (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)))