Logistic regression 2

Percentage Accurate: 99.3% → 99.4%

Time: 10.3s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. x = 6.339478703528047e+268
  2. y = 1.6089043273948158e-46

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))

C

double code(double x, double y) {
	return log((1.0 + exp(x))) - (x * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = log((1.0d0 + exp(x))) - (x * y)
end function

Java

public static double code(double x, double y) {
	return Math.log((1.0 + Math.exp(x))) - (x * y);
}

Python

def code(x, y):
	return math.log((1.0 + math.exp(x))) - (x * y)

Julia

function code(x, y)
	return Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
end

MATLAB

function tmp = code(x, y)
	tmp = log((1.0 + exp(x))) - (x * y);
end

Wolfram

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))

C

double code(double x, double y) {
	return log((1.0 + exp(x))) - (x * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = log((1.0d0 + exp(x))) - (x * y)
end function

Java

public static double code(double x, double y) {
	return Math.log((1.0 + Math.exp(x))) - (x * y);
}

Python

def code(x, y):
	return math.log((1.0 + math.exp(x))) - (x * y)

Julia

function code(x, y)
	return Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
end

MATLAB

function tmp = code(x, y)
	tmp = log((1.0 + exp(x))) - (x * y);
end

Wolfram

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. sub-neg 99.6%

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

   2. +-commutative 99.6%

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

   3. *-commutative 99.6%

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

   4. distribute-lft-neg-in 99.6%

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

   5. fma-define 99.6%

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

   6. log1p-define 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. *-commutative 99.6%

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

   3. log-prod 99.6%

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

   4. log1p-define 100.0%

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

   5. metadata-eval 100.0%

      \[\leadsto \left(\mathsf{log1p}\left(e^{x}\right) + \color{blue}{0}\right) - x \cdot y \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(e^{x}\right) + 0\right)} - x \cdot y \]
5. Step-by-step derivation

   1. +-rgt-identity 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 3: 80.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-61}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-71}:\\ \;\;\;\;\log \left(x + 2\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.75e-61)
   (* (- y) x)
   (if (<= x 5.2e-71)
     (log (+ x 2.0))
     (if (<= x 6.5e-34) (* x (- 0.5 y)) (+ (log 2.0) (* x 0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.75e-61) {
		tmp = -y * x;
	} else if (x <= 5.2e-71) {
		tmp = log((x + 2.0));
	} else if (x <= 6.5e-34) {
		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 <= (-2.75d-61)) then
        tmp = -y * x
    else if (x <= 5.2d-71) then
        tmp = log((x + 2.0d0))
    else if (x <= 6.5d-34) 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 <= -2.75e-61) {
		tmp = -y * x;
	} else if (x <= 5.2e-71) {
		tmp = Math.log((x + 2.0));
	} else if (x <= 6.5e-34) {
		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 <= -2.75e-61:
		tmp = -y * x
	elif x <= 5.2e-71:
		tmp = math.log((x + 2.0))
	elif x <= 6.5e-34:
		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 <= -2.75e-61)
		tmp = Float64(Float64(-y) * x);
	elseif (x <= 5.2e-71)
		tmp = log(Float64(x + 2.0));
	elseif (x <= 6.5e-34)
		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 <= -2.75e-61)
		tmp = -y * x;
	elseif (x <= 5.2e-71)
		tmp = log((x + 2.0));
	elseif (x <= 6.5e-34)
		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, -2.75e-61], N[((-y) * x), $MachinePrecision], If[LessEqual[x, 5.2e-71], N[Log[N[(x + 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 6.5e-34], 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 < -2.7499999999999998e-61

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf 93.2%

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

      1. associate-*r* 93.2%

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

      2. neg-mul-1 93.2%

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

   5. Simplified 93.2%

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

   if -2.7499999999999998e-61 < x < 5.1999999999999997e-71

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   4. Taylor expanded in y around 0 85.0%

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

   if 5.1999999999999997e-71 < x < 6.49999999999999985e-34

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   4. Taylor expanded in x around inf 89.6%

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

   if 6.49999999999999985e-34 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0 89.4%

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

   4. Taylor expanded in y around 0 69.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-61}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-71}:\\ \;\;\;\;\log \left(x + 2\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-62}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-71} \lor \neg \left(x \leq 1.9 \cdot 10^{-27}\right):\\ \;\;\;\;\log \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.4e-62)
   (* (- y) x)
   (if (or (<= x 5.6e-71) (not (<= x 1.9e-27)))
     (log (+ x 2.0))
     (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.4e-62) {
		tmp = -y * x;
	} else if ((x <= 5.6e-71) || !(x <= 1.9e-27)) {
		tmp = log((x + 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 <= (-5.4d-62)) then
        tmp = -y * x
    else if ((x <= 5.6d-71) .or. (.not. (x <= 1.9d-27))) then
        tmp = log((x + 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 <= -5.4e-62) {
		tmp = -y * x;
	} else if ((x <= 5.6e-71) || !(x <= 1.9e-27)) {
		tmp = Math.log((x + 2.0));
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.4e-62:
		tmp = -y * x
	elif (x <= 5.6e-71) or not (x <= 1.9e-27):
		tmp = math.log((x + 2.0))
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.4e-62)
		tmp = Float64(Float64(-y) * x);
	elseif ((x <= 5.6e-71) || !(x <= 1.9e-27))
		tmp = log(Float64(x + 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 <= -5.4e-62)
		tmp = -y * x;
	elseif ((x <= 5.6e-71) || ~((x <= 1.9e-27)))
		tmp = log((x + 2.0));
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.4e-62], N[((-y) * x), $MachinePrecision], If[Or[LessEqual[x, 5.6e-71], N[Not[LessEqual[x, 1.9e-27]], $MachinePrecision]], N[Log[N[(x + 2.0), $MachinePrecision]], $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.40000000000000039e-62

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf 93.2%

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

      1. associate-*r* 93.2%

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

      2. neg-mul-1 93.2%

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

   5. Simplified 93.2%

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

   if -5.40000000000000039e-62 < x < 5.60000000000000001e-71 or 1.9e-27 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.2%

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

   4. Taylor expanded in y around 0 83.9%

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

   if 5.60000000000000001e-71 < x < 1.9e-27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   4. Taylor expanded in x around inf 89.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-62}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-71} \lor \neg \left(x \leq 1.9 \cdot 10^{-27}\right):\\ \;\;\;\;\log \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-61}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-70}:\\ \;\;\;\;\log 2\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{-35}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.6e-61)
   (* (- y) x)
   (if (<= x 2.65e-70)
     (log 2.0)
     (if (<= x 1.08e-35) (* x (- 0.5 y)) (log 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-61) {
		tmp = -y * x;
	} else if (x <= 2.65e-70) {
		tmp = log(2.0);
	} else if (x <= 1.08e-35) {
		tmp = x * (0.5 - 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 <= (-2.6d-61)) then
        tmp = -y * x
    else if (x <= 2.65d-70) then
        tmp = log(2.0d0)
    else if (x <= 1.08d-35) then
        tmp = x * (0.5d0 - 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 <= -2.6e-61) {
		tmp = -y * x;
	} else if (x <= 2.65e-70) {
		tmp = Math.log(2.0);
	} else if (x <= 1.08e-35) {
		tmp = x * (0.5 - y);
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.6e-61:
		tmp = -y * x
	elif x <= 2.65e-70:
		tmp = math.log(2.0)
	elif x <= 1.08e-35:
		tmp = x * (0.5 - y)
	else:
		tmp = math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.6e-61)
		tmp = Float64(Float64(-y) * x);
	elseif (x <= 2.65e-70)
		tmp = log(2.0);
	elseif (x <= 1.08e-35)
		tmp = Float64(x * Float64(0.5 - y));
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.6e-61)
		tmp = -y * x;
	elseif (x <= 2.65e-70)
		tmp = log(2.0);
	elseif (x <= 1.08e-35)
		tmp = x * (0.5 - y);
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.6e-61], N[((-y) * x), $MachinePrecision], If[LessEqual[x, 2.65e-70], N[Log[2.0], $MachinePrecision], If[LessEqual[x, 1.08e-35], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision], N[Log[2.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6000000000000001e-61

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf 93.2%

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

      1. associate-*r* 93.2%

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

      2. neg-mul-1 93.2%

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

   5. Simplified 93.2%

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

   if -2.6000000000000001e-61 < x < 2.64999999999999992e-70 or 1.08000000000000003e-35 < x

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. distribute-lft-neg-in 99.9%

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

      5. fma-define 99.9%

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

      6. log1p-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.7%

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

   6. Taylor expanded in y around 0 83.4%

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

   if 2.64999999999999992e-70 < x < 1.08000000000000003e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

   4. Taylor expanded in x around inf 89.6%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-61}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-70}:\\ \;\;\;\;\log 2\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{-35}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -370:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(x \cdot 0.125 + \left(0.5 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -370.0) (* (- y) x) (+ (log 2.0) (* x (+ (* x 0.125) (- 0.5 y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -370.0) {
		tmp = -y * x;
	} else {
		tmp = log(2.0) + (x * ((x * 0.125) + (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 <= (-370.0d0)) then
        tmp = -y * x
    else
        tmp = log(2.0d0) + (x * ((x * 0.125d0) + (0.5d0 - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -370.0) {
		tmp = -y * x;
	} else {
		tmp = Math.log(2.0) + (x * ((x * 0.125) + (0.5 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -370.0:
		tmp = -y * x
	else:
		tmp = math.log(2.0) + (x * ((x * 0.125) + (0.5 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -370.0)
		tmp = Float64(Float64(-y) * x);
	else
		tmp = Float64(log(2.0) + Float64(x * Float64(Float64(x * 0.125) + Float64(0.5 - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -370.0)
		tmp = -y * x;
	else
		tmp = log(2.0) + (x * ((x * 0.125) + (0.5 - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -370.0], N[((-y) * x), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(N[(x * 0.125), $MachinePrecision] + N[(0.5 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -370

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   5. Simplified 100.0%

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

   if -370 < x

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

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

      2. unpow2 99.0%

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

      3. associate-*l* 99.0%

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

      4. distribute-lft-out 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -370:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(x \cdot 0.125 + \left(0.5 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y):
	tmp = 0
	if x <= -1.4:
		tmp = -y * x
	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(Float64(-y) * x);
	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 = -y * x;
	else
		tmp = log(2.0) + (x * (0.5 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4], N[((-y) * x), $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 99.0%

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

   3. Taylor expanded in x around inf 99.0%

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

      1. associate-*r* 99.0%

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

      2. neg-mul-1 99.0%

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

   5. Simplified 99.0%

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

   if -1.3999999999999999 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.1%

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

4. Final simplification 99.0%

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

Alternative 8: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -370:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -370.0) (* (- y) x) (- (log 2.0) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -370.0) {
		tmp = -y * x;
	} else {
		tmp = log(2.0) - (y * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-370.0d0)) then
        tmp = -y * x
    else
        tmp = log(2.0d0) - (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -370.0) {
		tmp = -y * x;
	} else {
		tmp = Math.log(2.0) - (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -370.0:
		tmp = -y * x
	else:
		tmp = math.log(2.0) - (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -370.0)
		tmp = Float64(Float64(-y) * x);
	else
		tmp = Float64(log(2.0) - Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -370.0)
		tmp = -y * x;
	else
		tmp = log(2.0) - (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -370.0], N[((-y) * x), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -370

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   5. Simplified 100.0%

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

   if -370 < x

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 97.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -370:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 - y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.8% accurate, 51.8× speedup

Math

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

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(Float64(-y) * 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. Add Preprocessing

3. Taylor expanded in x around inf 53.6%

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

   1. associate-*r* 53.6%

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

   2. neg-mul-1 53.6%

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

5. Simplified 53.6%

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

6. Final simplification 53.6%

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

Alternative 10: 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. Add Preprocessing

3. Taylor expanded in x around 0 80.1%

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

4. Taylor expanded in y around 0 48.1%

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

5. Taylor expanded in x around inf 3.6%

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

   1. *-commutative 3.6%

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

7. Simplified 3.6%

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

8. Final simplification 3.6%

   \[\leadsto x \cdot 0.5 \]
9. 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 2024048 
(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)))