Logistic regression 2

Percentage Accurate: 99.3% → 99.4%

Time: 9.0s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. x = 2.262055787222261e+284
  2. y = 1.3300319191780256e-189

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, 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 98.8%

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

   1. sub-neg 98.8%

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

   2. +-commutative 98.8%

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

   3. distribute-rgt-neg-in 98.8%

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

   4. fma-def 98.8%

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

   5. log1p-def 98.9%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 2: 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 98.8%

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

   1. log1p-def 98.8%

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

3. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 3: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right) + \log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35) (* y (- x)) (+ (* x (- 0.5 y)) (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35) {
		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 <= (-1.35d0)) 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 <= -1.35) {
		tmp = y * -x;
	} else {
		tmp = (x * (0.5 - y)) + Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35:
		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 <= -1.35)
		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 <= -1.35)
		tmp = y * -x;
	else
		tmp = (x * (0.5 - y)) + log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35], 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 < -1.3500000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot x\right)} \]
   3. 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%

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

   4. Simplified 100.0%

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

   if -1.3500000000000001 < x

   1. Initial program 98.3%

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

   2. 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 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right) + \log 2\\ \end{array} \]

Alternative 4: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -48:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -48.0) (* y (- x)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -48.0) {
		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 <= (-48.0d0)) 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 <= -48.0) {
		tmp = y * -x;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -48.0:
		tmp = y * -x
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -48.0)
		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 <= -48.0)
		tmp = y * -x;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -48.0], 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 < -48

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot x\right)} \]
   3. 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%

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

   4. Simplified 100.0%

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

   if -48 < x

   1. Initial program 98.3%

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

   2. Taylor expanded in x around 0 96.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -48:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 5: 83.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-30}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 3.75 \cdot 10^{-31}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e-30)
   (* y (- x))
   (if (<= x 3.75e-31) (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-30) {
		tmp = y * -x;
	} else if (x <= 3.75e-31) {
		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) :: tmp
    if (x <= (-2.2d-30)) then
        tmp = y * -x
    else if (x <= 3.75d-31) 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 tmp;
	if (x <= -2.2e-30) {
		tmp = y * -x;
	} else if (x <= 3.75e-31) {
		tmp = Math.log(2.0);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.2e-30:
		tmp = y * -x
	elif x <= 3.75e-31:
		tmp = math.log(2.0)
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.2e-30)
		tmp = Float64(y * Float64(-x));
	elseif (x <= 3.75e-31)
		tmp = log(2.0);
	else
		tmp = Float64(x * Float64(0.5 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e-30)
		tmp = y * -x;
	elseif (x <= 3.75e-31)
		tmp = log(2.0);
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.2e-30], N[(y * (-x)), $MachinePrecision], If[LessEqual[x, 3.75e-31], N[Log[2.0], $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.19999999999999983e-30

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 96.0%

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

      1. associate-*r* 96.0%

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

      2. *-commutative 96.0%

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

      3. mul-1-neg 96.0%

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

   4. Simplified 96.0%

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

   if -2.19999999999999983e-30 < x < 3.74999999999999987e-31

   1. Initial program 100.0%

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

      1. log1p-udef 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. add-sqr-sqrt 52.7%

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

      4. sqrt-unprod 76.9%

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

      5. sqr-neg 76.9%

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

      6. sqrt-unprod 38.8%

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

      7. add-sqr-sqrt 80.6%

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

      8. distribute-rgt-neg-in 80.6%

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

      9. neg-mul-1 80.6%

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

      10. add-sqr-sqrt 41.8%

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

      11. sqrt-unprod 76.0%

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

      12. sqr-neg 76.0%

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

      13. sqrt-unprod 47.1%

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

      14. add-sqr-sqrt 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. prod-diff 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-neg 100.0%

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

      2. *-lft-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. mul-1-neg 100.0%

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

      7. cancel-sign-sub 100.0%

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

      8. *-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-def 99.9%

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

      11. fma-udef 99.9%

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

      12. distribute-lft-out 99.9%

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

      13. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x, y, \mathsf{log1p}\left(e^{x}\right)\right) + \left(x \cdot y\right) \cdot \color{blue}{-2} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, \mathsf{log1p}\left(e^{x}\right)\right) + \left(x \cdot y\right) \cdot -2} \]

   6. Taylor expanded in x around 0 81.3%

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

   if 3.74999999999999987e-31 < x

   1. Initial program 86.1%

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

   2. Taylor expanded in x around 0 84.5%

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

   3. Taylor expanded in x around inf 67.5%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-30}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 3.75 \cdot 10^{-31}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 6: 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 98.8%

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

2. Taylor expanded in x around inf 51.8%

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

   1. associate-*r* 51.8%

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

   2. *-commutative 51.8%

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

   3. mul-1-neg 51.8%

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

4. Simplified 51.8%

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

5. Final simplification 51.8%

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

Alternative 7: 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 98.8%

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

2. Taylor expanded in x around 0 79.0%

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

3. Taylor expanded in x around inf 32.7%

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

4. Taylor expanded in y around 0 3.8%

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

5. Final simplification 3.8%

   \[\leadsto x \cdot 0.5 \]

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 2023194 
(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)))