Logistic regression 2

Percentage Accurate: 99.5% → 99.5%

Time: 9.5s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. x = 5.518844934622226e+238
  2. y = 2.412452860218434e-187

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: 99.5% 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: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. sub-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

   4. fma-def 99.6%

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

   5. log1p-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.5% 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 99.5%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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. Final simplification 99.6%

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

Alternative 3: 83.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -1.52 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-55}:\\ \;\;\;\;\log 2\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (- x))))
   (if (<= x -1.52e-37)
     t_0
     (if (<= x 2.9e-55)
       (log 2.0)
       (if (<= x 3.3e-49)
         t_0
         (if (<= x 4.5e-7) (+ (log 2.0) (* x 0.5)) (* x (- 0.5 y))))))))

C

double code(double x, double y) {
	double t_0 = y * -x;
	double tmp;
	if (x <= -1.52e-37) {
		tmp = t_0;
	} else if (x <= 2.9e-55) {
		tmp = log(2.0);
	} else if (x <= 3.3e-49) {
		tmp = t_0;
	} else if (x <= 4.5e-7) {
		tmp = log(2.0) + (x * 0.5);
	} 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 <= (-1.52d-37)) then
        tmp = t_0
    else if (x <= 2.9d-55) then
        tmp = log(2.0d0)
    else if (x <= 3.3d-49) then
        tmp = t_0
    else if (x <= 4.5d-7) then
        tmp = log(2.0d0) + (x * 0.5d0)
    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 <= -1.52e-37) {
		tmp = t_0;
	} else if (x <= 2.9e-55) {
		tmp = Math.log(2.0);
	} else if (x <= 3.3e-49) {
		tmp = t_0;
	} else if (x <= 4.5e-7) {
		tmp = Math.log(2.0) + (x * 0.5);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * -x
	tmp = 0
	if x <= -1.52e-37:
		tmp = t_0
	elif x <= 2.9e-55:
		tmp = math.log(2.0)
	elif x <= 3.3e-49:
		tmp = t_0
	elif x <= 4.5e-7:
		tmp = math.log(2.0) + (x * 0.5)
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(-x))
	tmp = 0.0
	if (x <= -1.52e-37)
		tmp = t_0;
	elseif (x <= 2.9e-55)
		tmp = log(2.0);
	elseif (x <= 3.3e-49)
		tmp = t_0;
	elseif (x <= 4.5e-7)
		tmp = Float64(log(2.0) + Float64(x * 0.5));
	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 <= -1.52e-37)
		tmp = t_0;
	elseif (x <= 2.9e-55)
		tmp = log(2.0);
	elseif (x <= 3.3e-49)
		tmp = t_0;
	elseif (x <= 4.5e-7)
		tmp = log(2.0) + (x * 0.5);
	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, -1.52e-37], t$95$0, If[LessEqual[x, 2.9e-55], N[Log[2.0], $MachinePrecision], If[LessEqual[x, 3.3e-49], t$95$0, If[LessEqual[x, 4.5e-7], N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.52e-37 or 2.9e-55 < x < 3.3e-49

   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. Taylor expanded in x around inf 96.5%

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

      1. associate-*r* 96.5%

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

      2. *-commutative 96.5%

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

      3. mul-1-neg 96.5%

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

   6. Simplified 96.5%

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

   if -1.52e-37 < x < 2.9e-55

   1. Initial program 99.9%

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

      1. log1p-def 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 99.9%

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

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(e^{x}\right)\right)\right)} - x \cdot y \]
   6. Step-by-step derivation

      1. expm1-udef 98.6%

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

      2. log1p-expm1-u 98.6%

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

      3. log1p-udef 98.7%

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

      4. add-exp-log 98.7%

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

      5. +-commutative 98.7%

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

      6. expm1-log1p-u 98.7%

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

   7. Applied egg-rr 98.7%

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

   8. Taylor expanded in x around 0 98.7%

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

      1. +-commutative 98.7%

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

   10. Simplified 98.7%

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

   11. Taylor expanded in x around 0 77.2%

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

   if 3.3e-49 < x < 4.4999999999999998e-7

   1. Initial program 99.8%

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

      1. log1p-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.9%

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

   5. Taylor expanded in y around 0 89.3%

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

   if 4.4999999999999998e-7 < x

   1. Initial program 93.2%

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

      1. log1p-def 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in x around 0 84.9%

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

   5. Taylor expanded in x around inf 77.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52 \cdot 10^{-37}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-55}:\\ \;\;\;\;\log 2\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-49}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 4: 83.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -4.55 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-55}:\\ \;\;\;\;\log 2\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (- x))))
   (if (<= x -4.55e-38)
     t_0
     (if (<= x 3.2e-55)
       (log 2.0)
       (if (<= x 3.3e-49) t_0 (if (<= x 3.5e-7) (log 2.0) (* x (- 0.5 y))))))))

C

double code(double x, double y) {
	double t_0 = y * -x;
	double tmp;
	if (x <= -4.55e-38) {
		tmp = t_0;
	} else if (x <= 3.2e-55) {
		tmp = log(2.0);
	} else if (x <= 3.3e-49) {
		tmp = t_0;
	} else if (x <= 3.5e-7) {
		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 <= (-4.55d-38)) then
        tmp = t_0
    else if (x <= 3.2d-55) then
        tmp = log(2.0d0)
    else if (x <= 3.3d-49) then
        tmp = t_0
    else if (x <= 3.5d-7) 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 <= -4.55e-38) {
		tmp = t_0;
	} else if (x <= 3.2e-55) {
		tmp = Math.log(2.0);
	} else if (x <= 3.3e-49) {
		tmp = t_0;
	} else if (x <= 3.5e-7) {
		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 <= -4.55e-38:
		tmp = t_0
	elif x <= 3.2e-55:
		tmp = math.log(2.0)
	elif x <= 3.3e-49:
		tmp = t_0
	elif x <= 3.5e-7:
		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 <= -4.55e-38)
		tmp = t_0;
	elseif (x <= 3.2e-55)
		tmp = log(2.0);
	elseif (x <= 3.3e-49)
		tmp = t_0;
	elseif (x <= 3.5e-7)
		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 <= -4.55e-38)
		tmp = t_0;
	elseif (x <= 3.2e-55)
		tmp = log(2.0);
	elseif (x <= 3.3e-49)
		tmp = t_0;
	elseif (x <= 3.5e-7)
		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, -4.55e-38], t$95$0, If[LessEqual[x, 3.2e-55], N[Log[2.0], $MachinePrecision], If[LessEqual[x, 3.3e-49], t$95$0, If[LessEqual[x, 3.5e-7], N[Log[2.0], $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.55000000000000006e-38 or 3.2000000000000001e-55 < x < 3.3e-49

   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. Taylor expanded in x around inf 96.5%

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

      1. associate-*r* 96.5%

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

      2. *-commutative 96.5%

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

      3. mul-1-neg 96.5%

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

   6. Simplified 96.5%

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

   if -4.55000000000000006e-38 < x < 3.2000000000000001e-55 or 3.3e-49 < x < 3.49999999999999984e-7

   1. Initial program 99.9%

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

      1. log1p-def 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 99.9%

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

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(e^{x}\right)\right)\right)} - x \cdot y \]
   6. Step-by-step derivation

      1. expm1-udef 98.6%

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

      2. log1p-expm1-u 98.6%

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

      3. log1p-udef 98.7%

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

      4. add-exp-log 98.7%

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

      5. +-commutative 98.7%

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

      6. expm1-log1p-u 98.7%

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

   7. Applied egg-rr 98.7%

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

   8. Taylor expanded in x around 0 98.3%

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

      1. +-commutative 98.3%

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

   10. Simplified 98.3%

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

   11. Taylor expanded in x around 0 77.9%

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

   if 3.49999999999999984e-7 < x

   1. Initial program 93.2%

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

      1. log1p-def 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in x around 0 84.9%

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

   5. Taylor expanded in x around inf 77.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.55 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-55}:\\ \;\;\;\;\log 2\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-49}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 5: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8200000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right) + \log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8200000.0) (* y (- x)) (+ (* x (- 0.5 y)) (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8200000.0) {
		tmp = y * -x;
	} else {
		tmp = (x * (0.5 - y)) + 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 <= (-8200000.0d0)) then
        tmp = y * -x
    else
        tmp = (x * (0.5d0 - y)) + log(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8200000.0) {
		tmp = y * -x;
	} else {
		tmp = (x * (0.5 - y)) + Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8200000.0:
		tmp = y * -x
	else:
		tmp = (x * (0.5 - y)) + math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8200000.0)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(Float64(x * Float64(0.5 - y)) + log(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8200000.0)
		tmp = y * -x;
	else
		tmp = (x * (0.5 - y)) + log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8200000.0], N[(y * (-x)), $MachinePrecision], N[(N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.2e6

   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. Taylor expanded in x around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -8.2e6 < x

   1. Initial program 99.3%

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

      1. log1p-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 97.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8200000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right) + \log 2\\ \end{array} \]

Alternative 6: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8200000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8200000.0) (* y (- x)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8200000.0) {
		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 <= (-8200000.0d0)) 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 <= -8200000.0) {
		tmp = y * -x;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8200000.0:
		tmp = y * -x
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8200000.0)
		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 <= -8200000.0)
		tmp = y * -x;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8200000.0], 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 < -8.2e6

   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. Taylor expanded in x around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   if -8.2e6 < x

   1. Initial program 99.3%

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

      1. log1p-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 96.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8200000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 7: 53.7% 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 99.5%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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. Taylor expanded in x around inf 57.6%

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

   1. associate-*r* 57.6%

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

   2. *-commutative 57.6%

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

   3. mul-1-neg 57.6%

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

6. Simplified 57.6%

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

7. Final simplification 57.6%

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

Alternative 8: 3.7% 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 99.5%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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. Taylor expanded in x around 0 77.8%

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

5. Taylor expanded in y around 0 43.1%

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

6. Taylor expanded in x around inf 3.7%

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

7. Final simplification 3.7%

   \[\leadsto x \cdot 0.5 \]

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 2023258 
(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)))