Logistic regression 2

Percentage Accurate: 99.2% → 99.6%

Time: 7.4s

Alternatives: 7

Speedup: 1.7×

• could not determine a ground truth

  1. x = 9.839650529755316e+267
  2. y = -5.002394760233901e+252

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: 99.2% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{if}\;t\_0 \leq 3 \cdot 10^{+281}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 - y, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))))
   (if (<= t_0 3e+281) t_0 (fma x (- 0.5 y) (log 2.0)))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double tmp;
	if (t_0 <= 3e+281) {
		tmp = t_0;
	} else {
		tmp = fma(x, (0.5 - y), log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
	tmp = 0.0
	if (t_0 <= 3e+281)
		tmp = t_0;
	else
		tmp = fma(x, Float64(0.5 - y), log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 3e+281], t$95$0, N[(x * N[(0.5 - y), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 3.0000000000000001e281

   1. Initial program 100.0%

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

   if 3.0000000000000001e281 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 69.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\frac{1}{2} - y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} - y, \log 2\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} - y}, \log 2\right) \]

      4. lower-log.f64 94.9

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

   5. Applied rewrites 94.9%

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

Alternative 2: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ t_1 := -x \cdot y\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))) (t_1 (- (* x y))))
   (if (<= t_0 5e-34) t_1 (if (<= t_0 500.0) (log 2.0) t_1))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double t_1 = -(x * y);
	double tmp;
	if (t_0 <= 5e-34) {
		tmp = t_1;
	} else if (t_0 <= 500.0) {
		tmp = log(2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log((1.0d0 + exp(x))) - (x * y)
    t_1 = -(x * y)
    if (t_0 <= 5d-34) then
        tmp = t_1
    else if (t_0 <= 500.0d0) then
        tmp = log(2.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.log((1.0 + Math.exp(x))) - (x * y);
	double t_1 = -(x * y);
	double tmp;
	if (t_0 <= 5e-34) {
		tmp = t_1;
	} else if (t_0 <= 500.0) {
		tmp = Math.log(2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.log((1.0 + math.exp(x))) - (x * y)
	t_1 = -(x * y)
	tmp = 0
	if t_0 <= 5e-34:
		tmp = t_1
	elif t_0 <= 500.0:
		tmp = math.log(2.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
	t_1 = Float64(-Float64(x * y))
	tmp = 0.0
	if (t_0 <= 5e-34)
		tmp = t_1;
	elseif (t_0 <= 500.0)
		tmp = log(2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = log((1.0 + exp(x))) - (x * y);
	t_1 = -(x * y);
	tmp = 0.0;
	if (t_0 <= 5e-34)
		tmp = t_1;
	elseif (t_0 <= 500.0)
		tmp = log(2.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x * y), $MachinePrecision])}, If[LessEqual[t$95$0, 5e-34], t$95$1, If[LessEqual[t$95$0, 500.0], N[Log[2.0], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 5.0000000000000003e-34 or 500 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 96.4%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if 5.0000000000000003e-34 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 500

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-log.f64 96.8

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

   5. Applied rewrites 96.8%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 5 \cdot 10^{-34}:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 500:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;-x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.125, 0.5\right) - y, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -200.0) (- (* x y)) (fma x (- (fma x 0.125 0.5) y) (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -200.0) {
		tmp = -(x * y);
	} else {
		tmp = fma(x, (fma(x, 0.125, 0.5) - y), log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -200.0)
		tmp = Float64(-Float64(x * y));
	else
		tmp = fma(x, Float64(fma(x, 0.125, 0.5) - y), log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -200.0], (-N[(x * y), $MachinePrecision]), N[(x * N[(N[(x * 0.125 + 0.5), $MachinePrecision] - y), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -200

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -200 < x

   1. Initial program 97.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) + \log 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y, \log 2\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y}, \log 2\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{8} \cdot x + \frac{1}{2}\right)} - y, \log 2\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \left(\color{blue}{x \cdot \frac{1}{8}} + \frac{1}{2}\right) - y, \log 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{8}, \frac{1}{2}\right)} - y, \log 2\right) \]

      7. lower-log.f64 98.7

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.125, 0.5\right) - y, \color{blue}{\log 2}\right) \]

   5. Applied rewrites 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.125, 0.5\right) - y, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 - y, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -200.0) (- (* x y)) (fma x (- 0.5 y) (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -200.0) {
		tmp = -(x * y);
	} else {
		tmp = fma(x, (0.5 - y), log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -200.0)
		tmp = Float64(-Float64(x * y));
	else
		tmp = fma(x, Float64(0.5 - y), log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -200.0], (-N[(x * y), $MachinePrecision]), N[(x * N[(0.5 - y), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -200

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -200 < x

   1. Initial program 97.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\frac{1}{2} - y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} - y, \log 2\right)} \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} - y}, \log 2\right) \]

      4. lower-log.f64 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 - y, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -200.0) (- (* x y)) (fma (- y) x (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -200.0) {
		tmp = -(x * y);
	} else {
		tmp = fma(-y, x, log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -200.0)
		tmp = Float64(-Float64(x * y));
	else
		tmp = fma(Float64(-y), x, log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -200.0], (-N[(x * y), $MachinePrecision]), N[((-y) * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -200

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -200 < x

   1. Initial program 97.1%

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

      1. lift-exp.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. metadata-eval N/A

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

      9. lower-expm1.f64 N/A

         \[\leadsto \log \left(\frac{\color{blue}{\mathsf{expm1}\left(x + x\right)}}{e^{x} - 1}\right) - x \cdot y \]

      10. lower-+.f64 N/A

          \[\leadsto \log \left(\frac{\mathsf{expm1}\left(\color{blue}{x + x}\right)}{e^{x} - 1}\right) - x \cdot y \]

      11. lift-exp.f64 N/A

          \[\leadsto \log \left(\frac{\mathsf{expm1}\left(x + x\right)}{\color{blue}{e^{x}} - 1}\right) - x \cdot y \]

      12. lower-expm1.f64 96.5

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

   4. Applied rewrites 96.5%

      \[\leadsto \log \color{blue}{\left(\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}\right)} - x \cdot y \]
   5. Step-by-step derivation

      1. exp-sum N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-exp.f64 N/A

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

      6. flip-+ N/A

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

      7. +-commutative N/A

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

      8. lift-exp.f64 N/A

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

      9. lift-approx N/A

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

      10. lift-log.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-rgt-neg-out N/A

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

      16. lift-neg.f64 N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 98.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -200.0) (- (* x y)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -200.0) {
		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 <= (-200.0d0)) 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 <= -200.0) {
		tmp = -(x * y);
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -200.0:
		tmp = -(x * y)
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -200.0)
		tmp = Float64(-Float64(x * 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 <= -200.0)
		tmp = -(x * y);
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -200.0], (-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 < -200

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -200 < x

   1. Initial program 97.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.9% accurate, 26.5× 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(-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 98.1%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. distribute-rgt-neg-in N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 53.3

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

5. Applied rewrites 53.3%

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

6. Final simplification 53.3%

   \[\leadsto -x \cdot y \]
7. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× 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 2024212 
(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)))