Logistic regression 2

Percentage Accurate: 99.3% → 99.3%

Time: 7.7s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. x = 1.1054993351237636e+60
  2. y = -1.5223185850326296e+85

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: 99.3% 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]

Derivation

1. Initial program 99.2%

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

2. Final simplification 99.2%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - y, x, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.0) (* x (- y)) (fma (- 0.5 y) x (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.0) {
		tmp = x * -y;
	} else {
		tmp = fma((0.5 - y), x, log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.0)
		tmp = Float64(x * Float64(-y));
	else
		tmp = fma(Float64(0.5 - y), x, log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.0], N[(x * (-y)), $MachinePrecision], N[(N[(0.5 - y), $MachinePrecision] * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5

   1. Initial program 98.7%

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

   2. Taylor expanded in x around inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-out 98.7%

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

   4. Simplified 98.7%

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

   if -5 < x

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 99.1%

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

      1. fma-def 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - y, x, \log 2\right)\\ \end{array} \]

Alternative 3: 82.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-64}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-104}:\\ \;\;\;\;\log 2\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-71} \lor \neg \left(x \leq 3.45 \cdot 10^{-15}\right):\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.1e-64)
   (* x (- y))
   (if (<= x 1.9e-104)
     (log 2.0)
     (if (or (<= x 9.2e-71) (not (<= x 3.45e-15)))
       (* x (- 0.5 y))
       (+ (log 2.0) (* x 0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.1e-64) {
		tmp = x * -y;
	} else if (x <= 1.9e-104) {
		tmp = log(2.0);
	} else if ((x <= 9.2e-71) || !(x <= 3.45e-15)) {
		tmp = x * (0.5 - y);
	} else {
		tmp = log(2.0) + (x * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.1d-64)) then
        tmp = x * -y
    else if (x <= 1.9d-104) then
        tmp = log(2.0d0)
    else if ((x <= 9.2d-71) .or. (.not. (x <= 3.45d-15))) then
        tmp = x * (0.5d0 - y)
    else
        tmp = log(2.0d0) + (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.1e-64) {
		tmp = x * -y;
	} else if (x <= 1.9e-104) {
		tmp = Math.log(2.0);
	} else if ((x <= 9.2e-71) || !(x <= 3.45e-15)) {
		tmp = x * (0.5 - y);
	} else {
		tmp = Math.log(2.0) + (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.1e-64:
		tmp = x * -y
	elif x <= 1.9e-104:
		tmp = math.log(2.0)
	elif (x <= 9.2e-71) or not (x <= 3.45e-15):
		tmp = x * (0.5 - y)
	else:
		tmp = math.log(2.0) + (x * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.1e-64)
		tmp = Float64(x * Float64(-y));
	elseif (x <= 1.9e-104)
		tmp = log(2.0);
	elseif ((x <= 9.2e-71) || !(x <= 3.45e-15))
		tmp = Float64(x * Float64(0.5 - y));
	else
		tmp = Float64(log(2.0) + Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.1e-64)
		tmp = x * -y;
	elseif (x <= 1.9e-104)
		tmp = log(2.0);
	elseif ((x <= 9.2e-71) || ~((x <= 3.45e-15)))
		tmp = x * (0.5 - y);
	else
		tmp = log(2.0) + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.1e-64], N[(x * (-y)), $MachinePrecision], If[LessEqual[x, 1.9e-104], N[Log[2.0], $MachinePrecision], If[Or[LessEqual[x, 9.2e-71], N[Not[LessEqual[x, 3.45e-15]], $MachinePrecision]], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.1e-64

   1. Initial program 98.9%

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

   2. Taylor expanded in x around inf 94.6%

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

      1. mul-1-neg 94.6%

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

      2. distribute-rgt-neg-out 94.6%

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

   4. Simplified 94.6%

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

   if -4.1e-64 < x < 1.9e-104

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   3. Taylor expanded in x around 0 87.3%

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

   if 1.9e-104 < x < 9.1999999999999994e-71 or 3.45000000000000005e-15 < x

   1. Initial program 96.5%

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

   2. Taylor expanded in x around 0 94.3%

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

   3. Taylor expanded in x around inf 78.0%

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

   if 9.1999999999999994e-71 < x < 3.45000000000000005e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   3. Taylor expanded in y around 0 76.5%

      \[\leadsto \color{blue}{0.5 \cdot x} + \log 2 \]
   4. Step-by-step derivation

      1. *-commutative 6.4%

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

   5. Simplified 76.5%

      \[\leadsto \color{blue}{x \cdot 0.5} + \log 2 \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-64}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-104}:\\ \;\;\;\;\log 2\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-71} \lor \neg \left(x \leq 3.45 \cdot 10^{-15}\right):\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \]

Alternative 4: 82.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-105} \lor \neg \left(x \leq 1.28 \cdot 10^{-70}\right) \land x \leq 10^{-14}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e-61)
   (* x (- y))
   (if (or (<= x 8e-105) (and (not (<= x 1.28e-70)) (<= x 1e-14)))
     (log 2.0)
     (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e-61) {
		tmp = x * -y;
	} else if ((x <= 8e-105) || (!(x <= 1.28e-70) && (x <= 1e-14))) {
		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 <= (-1.25d-61)) then
        tmp = x * -y
    else if ((x <= 8d-105) .or. (.not. (x <= 1.28d-70)) .and. (x <= 1d-14)) 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 <= -1.25e-61) {
		tmp = x * -y;
	} else if ((x <= 8e-105) || (!(x <= 1.28e-70) && (x <= 1e-14))) {
		tmp = Math.log(2.0);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.25e-61:
		tmp = x * -y
	elif (x <= 8e-105) or (not (x <= 1.28e-70) and (x <= 1e-14)):
		tmp = math.log(2.0)
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e-61)
		tmp = Float64(x * Float64(-y));
	elseif ((x <= 8e-105) || (!(x <= 1.28e-70) && (x <= 1e-14)))
		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 <= -1.25e-61)
		tmp = x * -y;
	elseif ((x <= 8e-105) || (~((x <= 1.28e-70)) && (x <= 1e-14)))
		tmp = log(2.0);
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e-61], N[(x * (-y)), $MachinePrecision], If[Or[LessEqual[x, 8e-105], And[N[Not[LessEqual[x, 1.28e-70]], $MachinePrecision], LessEqual[x, 1e-14]]], N[Log[2.0], $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25e-61

   1. Initial program 98.9%

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

   2. Taylor expanded in x around inf 94.6%

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

      1. mul-1-neg 94.6%

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

      2. distribute-rgt-neg-out 94.6%

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

   4. Simplified 94.6%

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

   if -1.25e-61 < x < 7.99999999999999972e-105 or 1.28e-70 < x < 9.99999999999999999e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

   3. Taylor expanded in x around 0 85.8%

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

   if 7.99999999999999972e-105 < x < 1.28e-70 or 9.99999999999999999e-15 < x

   1. Initial program 96.5%

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

   2. Taylor expanded in x around 0 94.3%

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

   3. Taylor expanded in x around inf 78.0%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-105} \lor \neg \left(x \leq 1.28 \cdot 10^{-70}\right) \land x \leq 10^{-14}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 5: 99.1% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.0) (* x (- y)) (+ (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.0) {
		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 <= (-5.0d0)) 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 <= -5.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) + (x * (0.5 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.0:
		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 <= -5.0)
		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 <= -5.0)
		tmp = x * -y;
	else
		tmp = log(2.0) + (x * (0.5 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.0], 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 < -5

   1. Initial program 98.7%

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

   2. Taylor expanded in x around inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. distribute-rgt-neg-out 98.7%

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

   4. Simplified 98.7%

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

   if -5 < x

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 99.1%

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

4. Final simplification 99.0%

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

Alternative 6: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -235:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -235.0) (* x (- y)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -235.0) {
		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 <= (-235.0d0)) 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 <= -235.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -235.0:
		tmp = x * -y
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -235.0)
		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 <= -235.0)
		tmp = x * -y;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -235.0], 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 < -235

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

   4. Simplified 100.0%

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

   if -235 < x

   1. Initial program 98.9%

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

   2. 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 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -235:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 7: 54.2% 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 99.2%

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

2. Taylor expanded in x around inf 50.7%

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

   1. mul-1-neg 50.7%

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

   2. distribute-rgt-neg-out 50.7%

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

4. Simplified 50.7%

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

5. Final simplification 50.7%

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

Alternative 8: 3.6% 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 99.2%

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

2. Taylor expanded in x around 0 82.4%

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

3. Taylor expanded in x around inf 34.4%

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

4. Taylor expanded in y around 0 3.9%

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

   1. *-commutative 3.9%

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

6. Simplified 3.9%

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

7. Final simplification 3.9%

   \[\leadsto x \cdot 0.5 \]

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 2023187 
(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)))