Logistic regression 2

Percentage Accurate: 99.3% → 99.1%

Time: 11.5s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. x = 1.3040930015610837e+273
  2. y = -9.952456887472764e-284

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: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y):
	tmp = 0
	if x <= -1.4:
		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 <= -1.4)
		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 <= -1.4)
		tmp = x * -y;
	else
		tmp = log(2.0) + (x * (0.5 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4], 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 < -1.3999999999999999

   1. Initial program 98.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 98.9%

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

      1. mul-1-neg 98.9%

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

      2. distribute-rgt-neg-out 98.9%

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

   6. Simplified 98.9%

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

   if -1.3999999999999999 < x

   1. Initial program 97.7%

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

      1. log1p-def 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in x around 0 98.6%

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

4. Final simplification 98.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y) {
	return Math.log1p(Math.exp(x)) - (x * y);
}

Python

def code(x, y):
	return math.log1p(math.exp(x)) - (x * y)

Julia

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

Wolfram

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

Derivation

1. Initial program 98.1%

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

   1. log1p-def 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 3: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -30:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -30.0) (* x (- y)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -30.0) {
		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 <= (-30.0d0)) 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 <= -30.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -30.0:
		tmp = x * -y
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -30.0)
		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 <= -30.0)
		tmp = x * -y;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -30.0], 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 < -30

   1. Initial program 98.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 98.9%

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

      1. mul-1-neg 98.9%

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

      2. distribute-rgt-neg-out 98.9%

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

   6. Simplified 98.9%

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

   if -30 < x

   1. Initial program 97.7%

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

      1. log1p-def 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in x around 0 98.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -30:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 4: 82.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-81}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.5e-23)
   (* x (- y))
   (if (<= x 2.8e-81) (log 2.0) (- 1.0 (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.5e-23) {
		tmp = x * -y;
	} else if (x <= 2.8e-81) {
		tmp = log(2.0);
	} else {
		tmp = 1.0 - (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 <= (-2.5d-23)) then
        tmp = x * -y
    else if (x <= 2.8d-81) then
        tmp = log(2.0d0)
    else
        tmp = 1.0d0 - (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.5e-23) {
		tmp = x * -y;
	} else if (x <= 2.8e-81) {
		tmp = Math.log(2.0);
	} else {
		tmp = 1.0 - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.5e-23:
		tmp = x * -y
	elif x <= 2.8e-81:
		tmp = math.log(2.0)
	else:
		tmp = 1.0 - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.5e-23)
		tmp = Float64(x * Float64(-y));
	elseif (x <= 2.8e-81)
		tmp = log(2.0);
	else
		tmp = Float64(1.0 - Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.5e-23)
		tmp = x * -y;
	elseif (x <= 2.8e-81)
		tmp = log(2.0);
	else
		tmp = 1.0 - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.5e-23], N[(x * (-y)), $MachinePrecision], If[LessEqual[x, 2.8e-81], N[Log[2.0], $MachinePrecision], N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5000000000000001e-23

   1. Initial program 99.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 97.9%

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

      1. mul-1-neg 97.9%

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

      2. distribute-rgt-neg-out 97.9%

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

   6. Simplified 97.9%

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

   if -2.5000000000000001e-23 < x < 2.7999999999999999e-81

   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%

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

   5. Taylor expanded in x around 0 76.5%

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

   if 2.7999999999999999e-81 < x

   1. Initial program 82.0%

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

      1. log1p-def 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in x around 0 87.6%

      \[\leadsto \color{blue}{\log 2} - x \cdot y \]
   5. Step-by-step derivation

      1. expm1-log1p-u 78.0%

         \[\leadsto \log 2 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot y\right)\right)} \]

      2. expm1-def 78.0%

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

      3. associate--r- 78.0%

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

      4. metadata-eval 78.0%

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

      5. log1p-udef 78.0%

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

      6. log1p-udef 78.0%

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

      7. add-exp-log 87.6%

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

      8. +-commutative 87.6%

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

   6. Applied egg-rr 87.6%

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

   7. Taylor expanded in x around inf 63.5%

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

      1. mul-1-neg 63.5%

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

      2. distribute-rgt-neg-in 63.5%

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

   9. Simplified 63.5%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-81}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot y\\ \end{array} \]

Alternative 5: 59.5% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -45:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -45.0) (* x (- y)) (- 1.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -45.0) {
		tmp = x * -y;
	} else {
		tmp = 1.0 - (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 <= (-45.0d0)) then
        tmp = x * -y
    else
        tmp = 1.0d0 - (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -45.0) {
		tmp = x * -y;
	} else {
		tmp = 1.0 - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -45.0:
		tmp = x * -y
	else:
		tmp = 1.0 - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -45.0)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(1.0 - Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -45.0)
		tmp = x * -y;
	else
		tmp = 1.0 - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -45.0], N[(x * (-y)), $MachinePrecision], N[(1.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -45

   1. Initial program 98.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 98.9%

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

      1. mul-1-neg 98.9%

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

      2. distribute-rgt-neg-out 98.9%

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

   6. Simplified 98.9%

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

   if -45 < x

   1. Initial program 97.7%

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

      1. log1p-def 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in x around 0 98.4%

      \[\leadsto \color{blue}{\log 2} - x \cdot y \]
   5. Step-by-step derivation

      1. expm1-log1p-u 84.0%

         \[\leadsto \log 2 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot y\right)\right)} \]

      2. expm1-def 84.0%

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

      3. associate--r- 84.0%

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

      4. metadata-eval 84.0%

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

      5. log1p-udef 84.0%

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

      6. log1p-udef 84.0%

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

      7. add-exp-log 98.4%

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

      8. +-commutative 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in x around inf 42.7%

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

      1. mul-1-neg 42.7%

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

      2. distribute-rgt-neg-in 42.7%

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

   9. Simplified 42.7%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -45:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot y\\ \end{array} \]

Alternative 6: 51.4% 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 98.1%

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

   1. log1p-def 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in x around inf 52.5%

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

   1. mul-1-neg 52.5%

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

   2. distribute-rgt-neg-out 52.5%

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

6. Simplified 52.5%

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

7. Final simplification 52.5%

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

Alternative 7: 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 98.1%

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

   1. log1p-def 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in x around 0 82.5%

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

5. Taylor expanded in x around inf 36.6%

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

6. Taylor expanded in y around 0 3.2%

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

   1. *-commutative 3.2%

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

8. Simplified 3.2%

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

9. Final simplification 3.2%

   \[\leadsto x \cdot 0.5 \]

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0)
   (- (log (+ 1.0 (exp x))) (* x y))
   (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0) {
		tmp = log((1.0 + exp(x))) - (x * y);
	} else {
		tmp = log((1.0 + exp(-x))) - (-x * (1.0 - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.0d0) then
        tmp = log((1.0d0 + exp(x))) - (x * y)
    else
        tmp = log((1.0d0 + exp(-x))) - (-x * (1.0d0 - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.0) {
		tmp = Math.log((1.0 + Math.exp(x))) - (x * y);
	} else {
		tmp = Math.log((1.0 + Math.exp(-x))) - (-x * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.0:
		tmp = math.log((1.0 + math.exp(x))) - (x * y)
	else:
		tmp = math.log((1.0 + math.exp(-x))) - (-x * (1.0 - y))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.0)
		tmp = log((1.0 + exp(x))) - (x * y);
	else
		tmp = log((1.0 + exp(-x))) - (-x * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0], N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[((-x) * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023336 
(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)))