Logistic regression 2

Percentage Accurate: 99.3% → 99.4%

Time: 6.3s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. x = 1.2862368951532726e+209
  2. y = 1.9543453286039857e-101

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 100.0%

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

   1. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. distribute-rgt-neg-in 100.0%

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

   4. fma-def 100.0%

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

   5. log1p-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

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 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. Final simplification 100.0%

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

Alternative 3: 83.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-27}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e-28)
   (* x (- y))
   (if (<= x 1.7e-27) (+ (log 2.0) (* x 0.5)) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.7e-28) {
		tmp = x * -y;
	} else if (x <= 1.7e-27) {
		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) :: tmp
    if (x <= (-2.7d-28)) then
        tmp = x * -y
    else if (x <= 1.7d-27) 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 tmp;
	if (x <= -2.7e-28) {
		tmp = x * -y;
	} else if (x <= 1.7e-27) {
		tmp = Math.log(2.0) + (x * 0.5);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.7e-28:
		tmp = x * -y
	elif x <= 1.7e-27:
		tmp = math.log(2.0) + (x * 0.5)
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.7e-28)
		tmp = Float64(x * Float64(-y));
	elseif (x <= 1.7e-27)
		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)
	tmp = 0.0;
	if (x <= -2.7e-28)
		tmp = x * -y;
	elseif (x <= 1.7e-27)
		tmp = log(2.0) + (x * 0.5);
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.7e-28], N[(x * (-y)), $MachinePrecision], If[LessEqual[x, 1.7e-27], 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 3 regimes

2. if x < -2.6999999999999999e-28

   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 97.3%

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

      1. mul-1-neg 97.3%

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

      2. distribute-rgt-neg-out 97.3%

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

   6. Simplified 97.3%

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

   if -2.6999999999999999e-28 < x < 1.69999999999999985e-27

   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 0 100.0%

      \[\leadsto \color{blue}{\left(0.5 - y\right) \cdot x + \log 2} \]
   5. Step-by-step derivation

      1. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - y, x, \log 2\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - y, x, \log 2\right)} \]

   7. Taylor expanded in y around 0 79.7%

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

   if 1.69999999999999985e-27 < x

   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 0 100.0%

      \[\leadsto \color{blue}{\left(0.5 - y\right) \cdot x + \log 2} \]
   5. Step-by-step derivation

      1. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - y, x, \log 2\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - y, x, \log 2\right)} \]

   7. Taylor expanded in x around inf 94.3%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-27}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4) {
		tmp = x * -y;
	} 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 <= (-1.4d0)) then
        tmp = x * -y
    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 <= -1.4) {
		tmp = x * -y;
	} else {
		tmp = (x * (0.5 - y)) + Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4:
		tmp = x * -y
	else:
		tmp = (x * (0.5 - y)) + math.log(2.0)
	return tmp

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4], N[(x * (-y)), $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 < -1.3999999999999999

   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. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

   6. Simplified 100.0%

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

   if -1.3999999999999999 < x

   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 0 100.0%

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

4. Final simplification 100.0%

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

Alternative 5: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -86:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -86.0) (* x (- y)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -86.0) {
		tmp = x * -y;
	} 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 <= (-86.0d0)) then
        tmp = x * -y
    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 <= -86.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -86.0:
		tmp = x * -y
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -86.0)
		tmp = Float64(x * Float64(-y));
	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 <= -86.0)
		tmp = x * -y;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -86.0], N[(x * (-y)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -86

   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. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

   6. Simplified 100.0%

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

   if -86 < x

   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 0 100.0%

      \[\leadsto \color{blue}{\left(0.5 - y\right) \cdot x + \log 2} \]
   5. Step-by-step derivation

      1. flip-- 81.6%

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

      2. associate-*l/ 81.6%

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

      3. metadata-eval 81.6%

         \[\leadsto \frac{\left(\color{blue}{0.25} - y \cdot y\right) \cdot x}{0.5 + y} + \log 2 \]

   6. Applied egg-rr 81.6%

      \[\leadsto \color{blue}{\frac{\left(0.25 - y \cdot y\right) \cdot x}{0.5 + y}} + \log 2 \]

   7. Taylor expanded in y around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in y around 0 100.0%

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

       1. associate-*r* 100.0%

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

       2. mul-1-neg 100.0%

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

       3. cancel-sign-sub-inv 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -86:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 6: 83.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-26}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e-27)
   (* x (- y))
   (if (<= x 2.3e-26) (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.5e-27) {
		tmp = x * -y;
	} else if (x <= 2.3e-26) {
		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 <= (-7.5d-27)) then
        tmp = x * -y
    else if (x <= 2.3d-26) 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 <= -7.5e-27) {
		tmp = x * -y;
	} else if (x <= 2.3e-26) {
		tmp = Math.log(2.0);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.5e-27:
		tmp = x * -y
	elif x <= 2.3e-26:
		tmp = math.log(2.0)
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.5e-27)
		tmp = Float64(x * Float64(-y));
	elseif (x <= 2.3e-26)
		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 <= -7.5e-27)
		tmp = x * -y;
	elseif (x <= 2.3e-26)
		tmp = log(2.0);
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.5e-27], N[(x * (-y)), $MachinePrecision], If[LessEqual[x, 2.3e-26], N[Log[2.0], $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.50000000000000029e-27

   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 97.3%

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

      1. mul-1-neg 97.3%

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

      2. distribute-rgt-neg-out 97.3%

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

   6. Simplified 97.3%

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

   if -7.50000000000000029e-27 < x < 2.30000000000000009e-26

   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 0 79.7%

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

   if 2.30000000000000009e-26 < x

   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 0 100.0%

      \[\leadsto \color{blue}{\left(0.5 - y\right) \cdot x + \log 2} \]
   5. Step-by-step derivation

      1. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - y, x, \log 2\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - y, x, \log 2\right)} \]

   7. Taylor expanded in x around inf 94.3%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-26}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 7: 53.5% accurate, 51.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 47.6%

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

   1. mul-1-neg 47.6%

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

   2. distribute-rgt-neg-out 47.6%

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

6. Simplified 47.6%

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

7. Final simplification 47.6%

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

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 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 0 88.2%

   \[\leadsto \color{blue}{\left(0.5 - y\right) \cdot x + \log 2} \]
5. Step-by-step derivation

   1. fma-def 88.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - y, x, \log 2\right)} \]

6. Simplified 88.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - y, x, \log 2\right)} \]

7. Taylor expanded in y around 0 55.4%

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

8. Taylor expanded in x around inf 3.7%

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

9. Final simplification 3.7%

   \[\leadsto x \cdot 0.5 \]

Developer target: 100.0% 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 2023175 
(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)))