Logistic regression 2

Percentage Accurate: 99.3% → 99.4%

Time: 9.7s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. x = 4.057622269280747e+303
  2. y = 1.330876975815016e-198

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-define 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: 82.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log 2 + x \cdot 0.5\\ \mathbf{if}\;x \leq -0.000112:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-93}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{-0.5}{y} - x\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-56}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (log 2.0) (* x 0.5))))
   (if (<= x -0.000112)
     (* y (- x))
     (if (<= x -3.2e-71)
       t_0
       (if (<= x -5.2e-93)
         (* y (- (* x (/ -0.5 y)) x))
         (if (<= x 5.2e-56) t_0 (* x (- 0.5 y))))))))

C

double code(double x, double y) {
	double t_0 = log(2.0) + (x * 0.5);
	double tmp;
	if (x <= -0.000112) {
		tmp = y * -x;
	} else if (x <= -3.2e-71) {
		tmp = t_0;
	} else if (x <= -5.2e-93) {
		tmp = y * ((x * (-0.5 / y)) - x);
	} else if (x <= 5.2e-56) {
		tmp = t_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 = log(2.0d0) + (x * 0.5d0)
    if (x <= (-0.000112d0)) then
        tmp = y * -x
    else if (x <= (-3.2d-71)) then
        tmp = t_0
    else if (x <= (-5.2d-93)) then
        tmp = y * ((x * ((-0.5d0) / y)) - x)
    else if (x <= 5.2d-56) then
        tmp = t_0
    else
        tmp = x * (0.5d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.log(2.0) + (x * 0.5);
	double tmp;
	if (x <= -0.000112) {
		tmp = y * -x;
	} else if (x <= -3.2e-71) {
		tmp = t_0;
	} else if (x <= -5.2e-93) {
		tmp = y * ((x * (-0.5 / y)) - x);
	} else if (x <= 5.2e-56) {
		tmp = t_0;
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.log(2.0) + (x * 0.5)
	tmp = 0
	if x <= -0.000112:
		tmp = y * -x
	elif x <= -3.2e-71:
		tmp = t_0
	elif x <= -5.2e-93:
		tmp = y * ((x * (-0.5 / y)) - x)
	elif x <= 5.2e-56:
		tmp = t_0
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(log(2.0) + Float64(x * 0.5))
	tmp = 0.0
	if (x <= -0.000112)
		tmp = Float64(y * Float64(-x));
	elseif (x <= -3.2e-71)
		tmp = t_0;
	elseif (x <= -5.2e-93)
		tmp = Float64(y * Float64(Float64(x * Float64(-0.5 / y)) - x));
	elseif (x <= 5.2e-56)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(0.5 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = log(2.0) + (x * 0.5);
	tmp = 0.0;
	if (x <= -0.000112)
		tmp = y * -x;
	elseif (x <= -3.2e-71)
		tmp = t_0;
	elseif (x <= -5.2e-93)
		tmp = y * ((x * (-0.5 / y)) - x);
	elseif (x <= 5.2e-56)
		tmp = t_0;
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.000112], N[(y * (-x)), $MachinePrecision], If[LessEqual[x, -3.2e-71], t$95$0, If[LessEqual[x, -5.2e-93], N[(y * N[(N[(x * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e-56], t$95$0, N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.11999999999999998e-4

   1. Initial program 100.0%

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

      1. log1p-define 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 44.4%

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

   6. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   8. Simplified 100.0%

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

   if -1.11999999999999998e-4 < x < -3.1999999999999999e-71 or -5.1999999999999997e-93 < x < 5.19999999999999994e-56

   1. Initial program 100.0%

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

      1. log1p-define 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%

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

   6. Taylor expanded in y around 0 76.4%

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

   if -3.1999999999999999e-71 < x < -5.1999999999999997e-93

   1. Initial program 100.0%

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

      1. log1p-define 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%

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

   6. Taylor expanded in x around inf 78.1%

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

   7. Taylor expanded in y around inf 78.1%

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot x + 0.5 \cdot \frac{x}{y}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 78.1%

         \[\leadsto y \cdot \color{blue}{\left(0.5 \cdot \frac{x}{y} + -1 \cdot x\right)} \]

      2. mul-1-neg 78.1%

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

      3. unsub-neg 78.1%

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

      4. associate-*r/ 78.1%

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

      5. *-commutative 78.1%

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

      6. associate-/l* 78.1%

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

   9. Simplified 78.1%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{0.5}{y} - x\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 78.1%

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

       2. associate-*l/ 78.1%

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

       3. *-commutative 78.1%

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

       4. frac-2neg 78.1%

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

       5. mul-1-neg 78.1%

          \[\leadsto y \cdot \left(0.5 \cdot \frac{\color{blue}{-1 \cdot x}}{-y} - x\right) \]

       6. associate-*r/ 78.1%

          \[\leadsto y \cdot \left(\color{blue}{\frac{0.5 \cdot \left(-1 \cdot x\right)}{-y}} - x\right) \]

       7. add-sqr-sqrt 78.1%

          \[\leadsto y \cdot \left(\frac{0.5 \cdot \color{blue}{\left(\sqrt{-1 \cdot x} \cdot \sqrt{-1 \cdot x}\right)}}{-y} - x\right) \]

       8. sqrt-unprod 78.1%

          \[\leadsto y \cdot \left(\frac{0.5 \cdot \color{blue}{\sqrt{\left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}}}{-y} - x\right) \]

       9. mul-1-neg 78.1%

          \[\leadsto y \cdot \left(\frac{0.5 \cdot \sqrt{\color{blue}{\left(-x\right)} \cdot \left(-1 \cdot x\right)}}{-y} - x\right) \]

       10. mul-1-neg 78.1%

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

       11. sqr-neg 78.1%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 78.7%

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

       14. *-commutative 78.7%

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

   11. Applied egg-rr 78.7%

       \[\leadsto y \cdot \left(\color{blue}{\frac{x \cdot 0.5}{-y}} - x\right) \]
   12. Step-by-step derivation

       1. associate-/l* 78.7%

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

       2. distribute-frac-neg2 78.7%

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

       3. distribute-neg-frac 78.7%

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

       4. metadata-eval 78.7%

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

   13. Simplified 78.7%

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

   if 5.19999999999999994e-56 < x

   1. Initial program 93.8%

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

      1. log1p-define 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in x around 0 86.2%

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

   6. Taylor expanded in x around inf 70.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000112:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-71}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-93}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{-0.5}{y} - x\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-56}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-24}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -3.25 \cdot 10^{-68}:\\ \;\;\;\;\log 2\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-94}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{-0.5}{y} - x\right)\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{-62}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.8e-24)
   (* y (- x))
   (if (<= x -3.25e-68)
     (log 2.0)
     (if (<= x -1.06e-94)
       (* y (- (* x (/ -0.5 y)) x))
       (if (<= x 3.25e-62) (log 2.0) (* x (- 0.5 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.8e-24) {
		tmp = y * -x;
	} else if (x <= -3.25e-68) {
		tmp = log(2.0);
	} else if (x <= -1.06e-94) {
		tmp = y * ((x * (-0.5 / y)) - x);
	} else if (x <= 3.25e-62) {
		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 <= (-5.8d-24)) then
        tmp = y * -x
    else if (x <= (-3.25d-68)) then
        tmp = log(2.0d0)
    else if (x <= (-1.06d-94)) then
        tmp = y * ((x * ((-0.5d0) / y)) - x)
    else if (x <= 3.25d-62) 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 <= -5.8e-24) {
		tmp = y * -x;
	} else if (x <= -3.25e-68) {
		tmp = Math.log(2.0);
	} else if (x <= -1.06e-94) {
		tmp = y * ((x * (-0.5 / y)) - x);
	} else if (x <= 3.25e-62) {
		tmp = Math.log(2.0);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.8e-24:
		tmp = y * -x
	elif x <= -3.25e-68:
		tmp = math.log(2.0)
	elif x <= -1.06e-94:
		tmp = y * ((x * (-0.5 / y)) - x)
	elif x <= 3.25e-62:
		tmp = math.log(2.0)
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.8e-24)
		tmp = Float64(y * Float64(-x));
	elseif (x <= -3.25e-68)
		tmp = log(2.0);
	elseif (x <= -1.06e-94)
		tmp = Float64(y * Float64(Float64(x * Float64(-0.5 / y)) - x));
	elseif (x <= 3.25e-62)
		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 <= -5.8e-24)
		tmp = y * -x;
	elseif (x <= -3.25e-68)
		tmp = log(2.0);
	elseif (x <= -1.06e-94)
		tmp = y * ((x * (-0.5 / y)) - x);
	elseif (x <= 3.25e-62)
		tmp = log(2.0);
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.8e-24], N[(y * (-x)), $MachinePrecision], If[LessEqual[x, -3.25e-68], N[Log[2.0], $MachinePrecision], If[LessEqual[x, -1.06e-94], N[(y * N[(N[(x * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.25e-62], N[Log[2.0], $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.7999999999999997e-24

   1. Initial program 100.0%

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

      1. log1p-define 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 45.6%

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

   6. Taylor expanded in y around inf 99.0%

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

      1. mul-1-neg 99.0%

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

      2. *-commutative 99.0%

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

      3. distribute-rgt-neg-in 99.0%

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

   8. Simplified 99.0%

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

   if -5.7999999999999997e-24 < x < -3.2499999999999999e-68 or -1.06e-94 < x < 3.25000000000000013e-62

   1. Initial program 100.0%

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

      1. log1p-define 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%

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

   6. Taylor expanded in x around 0 76.7%

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

   if -3.2499999999999999e-68 < x < -1.06e-94

   1. Initial program 100.0%

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

      1. log1p-define 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%

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

   6. Taylor expanded in x around inf 78.1%

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

   7. Taylor expanded in y around inf 78.1%

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot x + 0.5 \cdot \frac{x}{y}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 78.1%

         \[\leadsto y \cdot \color{blue}{\left(0.5 \cdot \frac{x}{y} + -1 \cdot x\right)} \]

      2. mul-1-neg 78.1%

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

      3. unsub-neg 78.1%

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

      4. associate-*r/ 78.1%

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

      5. *-commutative 78.1%

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

      6. associate-/l* 78.1%

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

   9. Simplified 78.1%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{0.5}{y} - x\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 78.1%

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

       2. associate-*l/ 78.1%

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

       3. *-commutative 78.1%

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

       4. frac-2neg 78.1%

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

       5. mul-1-neg 78.1%

          \[\leadsto y \cdot \left(0.5 \cdot \frac{\color{blue}{-1 \cdot x}}{-y} - x\right) \]

       6. associate-*r/ 78.1%

          \[\leadsto y \cdot \left(\color{blue}{\frac{0.5 \cdot \left(-1 \cdot x\right)}{-y}} - x\right) \]

       7. add-sqr-sqrt 78.1%

          \[\leadsto y \cdot \left(\frac{0.5 \cdot \color{blue}{\left(\sqrt{-1 \cdot x} \cdot \sqrt{-1 \cdot x}\right)}}{-y} - x\right) \]

       8. sqrt-unprod 78.1%

          \[\leadsto y \cdot \left(\frac{0.5 \cdot \color{blue}{\sqrt{\left(-1 \cdot x\right) \cdot \left(-1 \cdot x\right)}}}{-y} - x\right) \]

       9. mul-1-neg 78.1%

          \[\leadsto y \cdot \left(\frac{0.5 \cdot \sqrt{\color{blue}{\left(-x\right)} \cdot \left(-1 \cdot x\right)}}{-y} - x\right) \]

       10. mul-1-neg 78.1%

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

       11. sqr-neg 78.1%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 78.7%

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

       14. *-commutative 78.7%

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

   11. Applied egg-rr 78.7%

       \[\leadsto y \cdot \left(\color{blue}{\frac{x \cdot 0.5}{-y}} - x\right) \]
   12. Step-by-step derivation

       1. associate-/l* 78.7%

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

       2. distribute-frac-neg2 78.7%

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

       3. distribute-neg-frac 78.7%

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

       4. metadata-eval 78.7%

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

   13. Simplified 78.7%

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

   if 3.25000000000000013e-62 < x

   1. Initial program 93.8%

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

      1. log1p-define 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in x around 0 86.2%

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

   6. Taylor expanded in x around inf 70.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-24}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq -3.25 \cdot 10^{-68}:\\ \;\;\;\;\log 2\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{-94}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{-0.5}{y} - x\right)\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{-62}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4800000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log 2 + x \cdot \left(0.5 + x \cdot 0.125\right)\right) - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4800000.0)
   (* y (- x))
   (- (+ (log 2.0) (* x (+ 0.5 (* x 0.125)))) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4800000.0) {
		tmp = y * -x;
	} else {
		tmp = (log(2.0) + (x * (0.5 + (x * 0.125)))) - (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 <= (-4800000.0d0)) then
        tmp = y * -x
    else
        tmp = (log(2.0d0) + (x * (0.5d0 + (x * 0.125d0)))) - (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4800000.0) {
		tmp = y * -x;
	} else {
		tmp = (Math.log(2.0) + (x * (0.5 + (x * 0.125)))) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4800000.0:
		tmp = y * -x
	else:
		tmp = (math.log(2.0) + (x * (0.5 + (x * 0.125)))) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4800000.0)
		tmp = Float64(y * Float64(-x));
	else
		tmp = Float64(Float64(log(2.0) + Float64(x * Float64(0.5 + Float64(x * 0.125)))) - Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4800000.0)
		tmp = y * -x;
	else
		tmp = (log(2.0) + (x * (0.5 + (x * 0.125)))) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4800000.0], N[(y * (-x)), $MachinePrecision], N[(N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(0.5 + N[(x * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8e6

   1. Initial program 100.0%

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

      1. log1p-define 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 43.2%

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

   6. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   8. Simplified 100.0%

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

   if -4.8e6 < x

   1. Initial program 99.4%

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

      1. log1p-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{\left(\log 2 + x \cdot \left(0.5 + 0.125 \cdot x\right)\right)} - x \cdot y \]
   6. Step-by-step derivation

      1. *-commutative 98.8%

         \[\leadsto \left(\log 2 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.125}\right)\right) - x \cdot y \]

   7. Simplified 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4800000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log 2 + x \cdot \left(0.5 + x \cdot 0.125\right)\right) - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4800000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log 2 + x \cdot 0.5\right) - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4800000.0) (* y (- x)) (- (+ (log 2.0) (* x 0.5)) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4800000.0) {
		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 <= (-4800000.0d0)) 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 <= -4800000.0) {
		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 <= -4800000.0:
		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 <= -4800000.0)
		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 <= -4800000.0)
		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, -4800000.0], 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 < -4.8e6

   1. Initial program 100.0%

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

      1. log1p-define 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 43.2%

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

   6. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   8. Simplified 100.0%

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

   if -4.8e6 < x

   1. Initial program 99.4%

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

      1. log1p-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4800000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log 2 + x \cdot 0.5\right) - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4800000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4800000.0) (* y (- x)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4800000.0) {
		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 <= (-4800000.0d0)) 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 <= -4800000.0) {
		tmp = y * -x;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4800000.0:
		tmp = y * -x
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4800000.0)
		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 <= -4800000.0)
		tmp = y * -x;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4800000.0], 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 < -4.8e6

   1. Initial program 100.0%

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

      1. log1p-define 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 43.2%

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

   6. Taylor expanded in y around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   8. Simplified 100.0%

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

   if -4.8e6 < x

   1. Initial program 99.4%

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

      1. log1p-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4800000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.9% 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-define 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 79.4%

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

6. Taylor expanded in y around inf 56.4%

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

   1. mul-1-neg 56.4%

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

   2. *-commutative 56.4%

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

   3. distribute-rgt-neg-in 56.4%

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

8. Simplified 56.4%

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

9. Final simplification 56.4%

   \[\leadsto y \cdot \left(-x\right) \]
10. 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-define 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 79.4%

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

6. Taylor expanded in x around inf 37.0%

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

7. Taylor expanded in y around 0 3.5%

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

   1. *-commutative 3.5%

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

9. Simplified 3.5%

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

10. Final simplification 3.5%

    \[\leadsto x \cdot 0.5 \]
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 2024076 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :alt
  (if (<= x 0.0) (- (log (+ 1.0 (exp x))) (* x y)) (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y))))

  (- (log (+ 1.0 (exp x))) (* x y)))