Logistic regression 2

Percentage Accurate: 99.3% → 99.4%

Time: 11.7s

Alternatives: 9

Speedup: 1.8×

• could not determine a ground truth

  1. x = 1.365345049647716e+276
  2. y = 8.041032435427806e-260

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, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -53:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -53.0) (* x (- y)) (+ (log 2.0) (* x (- (+ 0.5 (* x 0.125)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -53.0) {
		tmp = x * -y;
	} else {
		tmp = log(2.0) + (x * ((0.5 + (x * 0.125)) - 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 <= (-53.0d0)) then
        tmp = x * -y
    else
        tmp = log(2.0d0) + (x * ((0.5d0 + (x * 0.125d0)) - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -53.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) + (x * ((0.5 + (x * 0.125)) - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -53.0:
		tmp = x * -y
	else:
		tmp = math.log(2.0) + (x * ((0.5 + (x * 0.125)) - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -53.0)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(log(2.0) + Float64(x * Float64(Float64(0.5 + Float64(x * 0.125)) - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -53.0)
		tmp = x * -y;
	else
		tmp = log(2.0) + (x * ((0.5 + (x * 0.125)) - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -53.0], N[(x * (-y)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(N[(0.5 + N[(x * 0.125), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -53

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 inf 99.4%

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

      1. neg-mul-1 99.4%

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

      2. distribute-lft-neg-in 99.4%

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

      3. *-commutative 99.4%

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

   7. Simplified 99.4%

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

   if -53 < x

   1. Initial program 98.8%

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

      1. log1p-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -53:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 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 99.0%

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

   1. cancel-sign-sub-inv 99.0%

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

   2. +-commutative 99.0%

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

   3. distribute-lft-neg-out 99.0%

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

   4. distribute-rgt-neg-out 99.0%

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

   5. fma-define 99.0%

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

   6. log1p-define 99.2%

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

3. Simplified 99.2%

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

Alternative 3: 99.3% 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 99.0%

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

   1. log1p-define 99.2%

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

3. Simplified 99.2%

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

Alternative 4: 83.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.45 \cdot 10^{-6} \lor \neg \left(x \leq 2.35 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.45e-6) (not (<= x 2.35e-36))) (* x (- y)) (log1p (+ x 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.45e-6) || !(x <= 2.35e-36)) {
		tmp = x * -y;
	} else {
		tmp = log1p((x + 1.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.45e-6) || !(x <= 2.35e-36)) {
		tmp = x * -y;
	} else {
		tmp = Math.log1p((x + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.45e-6) or not (x <= 2.35e-36):
		tmp = x * -y
	else:
		tmp = math.log1p((x + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.45e-6) || !(x <= 2.35e-36))
		tmp = Float64(x * Float64(-y));
	else
		tmp = log1p(Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.45e-6], N[Not[LessEqual[x, 2.35e-36]], $MachinePrecision]], N[(x * (-y)), $MachinePrecision], N[Log[1 + N[(x + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.45e-6 or 2.3500000000000001e-36 < x

   1. Initial program 97.6%

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

      1. log1p-define 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in x around inf 93.9%

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

      1. neg-mul-1 93.9%

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

      2. distribute-lft-neg-in 93.9%

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

      3. *-commutative 93.9%

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

   7. Simplified 93.9%

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

   if -3.45e-6 < x < 2.3500000000000001e-36

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 98.0%

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

      2. log-prod 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. count-2 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in y around 0 78.4%

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

      1. unpow1/2 78.4%

         \[\leadsto 2 \cdot \log \color{blue}{\left({\left(1 + e^{x}\right)}^{0.5}\right)} \]

      2. exp-to-pow 78.4%

         \[\leadsto 2 \cdot \log \color{blue}{\left(e^{\log \left(1 + e^{x}\right) \cdot 0.5}\right)} \]

      3. log1p-define 78.4%

         \[\leadsto 2 \cdot \log \left(e^{\color{blue}{\mathsf{log1p}\left(e^{x}\right)} \cdot 0.5}\right) \]

      4. *-commutative 78.4%

         \[\leadsto 2 \cdot \log \left(e^{\color{blue}{0.5 \cdot \mathsf{log1p}\left(e^{x}\right)}}\right) \]

      5. log1p-expm1 78.4%

         \[\leadsto 2 \cdot \log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right)\right)\right)}}\right) \]

      6. log1p-undefine 78.4%

         \[\leadsto 2 \cdot \log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right)\right)\right)}}\right) \]

      7. rem-exp-log 78.4%

         \[\leadsto 2 \cdot \log \color{blue}{\left(1 + \mathsf{expm1}\left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right)\right)\right)} \]

      8. log1p-undefine 79.6%

         \[\leadsto 2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right)\right)\right)} \]

      9. log1p-expm1 79.6%

         \[\leadsto 2 \cdot \color{blue}{\left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right)\right)} \]

      10. associate-*r* 79.6%

          \[\leadsto \color{blue}{\left(2 \cdot 0.5\right) \cdot \mathsf{log1p}\left(e^{x}\right)} \]

      11. metadata-eval 79.6%

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

      12. *-lft-identity 79.6%

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

   9. Simplified 79.6%

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

   10. Taylor expanded in x around 0 79.7%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{1 + x}\right) \]
   11. Step-by-step derivation

       1. +-commutative 79.7%

          \[\leadsto \mathsf{log1p}\left(\color{blue}{x + 1}\right) \]

   12. Simplified 79.7%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{x + 1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.45 \cdot 10^{-6} \lor \neg \left(x \leq 2.35 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.38:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.38) (* x (- y)) (+ (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.38) {
		tmp = x * -y;
	} else {
		tmp = log(2.0) + (x * (0.5 - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.38d0)) then
        tmp = x * -y
    else
        tmp = log(2.0d0) + (x * (0.5d0 - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.38) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) + (x * (0.5 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.38:
		tmp = x * -y
	else:
		tmp = math.log(2.0) + (x * (0.5 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.38)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(log(2.0) + Float64(x * Float64(0.5 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.38)
		tmp = x * -y;
	else
		tmp = log(2.0) + (x * (0.5 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.38], N[(x * (-y)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3799999999999999

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 inf 99.4%

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

      1. neg-mul-1 99.4%

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

      2. distribute-lft-neg-in 99.4%

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

      3. *-commutative 99.4%

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

   7. Simplified 99.4%

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

   if -1.3799999999999999 < x

   1. Initial program 98.8%

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

      1. log1p-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.38:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-76} \lor \neg \left(x \leq 3.1 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.6e-76) (not (<= x 3.1e-36))) (* x (- y)) (log 2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.6e-76) || !(x <= 3.1e-36)) {
		tmp = x * -y;
	} else {
		tmp = 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 <= (-3.6d-76)) .or. (.not. (x <= 3.1d-36))) then
        tmp = x * -y
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.6e-76) || !(x <= 3.1e-36)) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.6e-76) or not (x <= 3.1e-36):
		tmp = x * -y
	else:
		tmp = math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.6e-76) || !(x <= 3.1e-36))
		tmp = Float64(x * Float64(-y));
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.6e-76) || ~((x <= 3.1e-36)))
		tmp = x * -y;
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.6e-76], N[Not[LessEqual[x, 3.1e-36]], $MachinePrecision]], N[(x * (-y)), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6e-76 or 3.0999999999999999e-36 < x

   1. Initial program 97.9%

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

      1. log1p-define 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in x around inf 87.9%

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

      1. neg-mul-1 87.9%

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

      2. distribute-lft-neg-in 87.9%

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

      3. *-commutative 87.9%

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

   7. Simplified 87.9%

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

   if -3.6e-76 < x < 3.0999999999999999e-36

   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%

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

   6. Taylor expanded in x around 0 83.1%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-76} \lor \neg \left(x \leq 3.1 \cdot 10^{-36}\right):\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -54:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -54.0) (* x (- y)) (- (log1p 1.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -54.0) {
		tmp = x * -y;
	} else {
		tmp = log1p(1.0) - (x * y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -54.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log1p(1.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -54.0:
		tmp = x * -y
	else:
		tmp = math.log1p(1.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -54.0)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(log1p(1.0) - Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -54.0], N[(x * (-y)), $MachinePrecision], N[(N[Log[1 + 1.0], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -54

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 inf 99.4%

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

      1. neg-mul-1 99.4%

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

      2. distribute-lft-neg-in 99.4%

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

      3. *-commutative 99.4%

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

   7. Simplified 99.4%

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

   if -54 < x

   1. Initial program 98.8%

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

      1. log1p-define 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. Add Preprocessing

   5. Taylor expanded in x around 0 98.2%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{1}\right) - x \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -54:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.8% 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 99.0%

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

   1. log1p-define 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around inf 50.3%

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

   1. neg-mul-1 50.3%

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

   2. distribute-lft-neg-in 50.3%

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

   3. *-commutative 50.3%

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

7. Simplified 50.3%

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

8. Final simplification 50.3%

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

Alternative 9: 2.3% accurate, 69.0× speedup

Math

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

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 * y)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. log1p-define 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around inf 50.3%

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

   1. neg-mul-1 50.3%

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

   2. distribute-lft-neg-in 50.3%

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

   3. *-commutative 50.3%

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

7. Simplified 50.3%

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

   1. neg-sub0 50.3%

      \[\leadsto y \cdot \color{blue}{\left(0 - x\right)} \]

   2. sub-neg 50.3%

      \[\leadsto y \cdot \color{blue}{\left(0 + \left(-x\right)\right)} \]

   3. add-sqr-sqrt 41.3%

      \[\leadsto y \cdot \left(0 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right) \]

   4. sqrt-unprod 32.0%

      \[\leadsto y \cdot \left(0 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right) \]

   5. sqr-neg 32.0%

      \[\leadsto y \cdot \left(0 + \sqrt{\color{blue}{x \cdot x}}\right) \]

   6. sqrt-unprod 1.0%

      \[\leadsto y \cdot \left(0 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) \]

   7. add-sqr-sqrt 2.2%

      \[\leadsto y \cdot \left(0 + \color{blue}{x}\right) \]

9. Applied egg-rr 2.2%

   \[\leadsto y \cdot \color{blue}{\left(0 + x\right)} \]
10. Step-by-step derivation

    1. +-lft-identity 2.2%

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

11. Simplified 2.2%

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

12. Final simplification 2.2%

    \[\leadsto x \cdot y \]
13. Add Preprocessing

Developer Target 1: 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 2024150 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (<= x 0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y)))))

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