Logistic regression 2

Percentage Accurate: 99.2% → 99.2%

Time: 7.0s

Alternatives: 12

Speedup: 1.7×

• could not determine a ground truth

  1. x = 2.058813155757499e+206
  2. y = 5.723603187417002e+192

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: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[((-y) * x), $MachinePrecision] - N[Log[N[Power[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision], -1.0], $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. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip3-+ N/A

      \[\leadsto \log \color{blue}{\left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}\right)} - x \cdot y \]

   4. clear-num N/A

      \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}{{1}^{3} + {\left(e^{x}\right)}^{3}}}\right)} - x \cdot y \]

   5. clear-num N/A

      \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{\frac{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}}\right)} - x \cdot y \]

   6. log-rec N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{\frac{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right)\right)\right)} - x \cdot y \]

   7. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\log \left(\frac{\frac{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right)\right)} - x \cdot y \]

   8. lower-log.f64 N/A

      \[\leadsto \left(-\color{blue}{\log \left(\frac{\frac{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right)}\right) - x \cdot y \]

   9. lower-/.f64 N/A

      \[\leadsto \left(-\log \color{blue}{\left(\frac{\frac{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right)}\right) - x \cdot y \]

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - y \cdot x\\ t_1 := \left(-y\right) \cdot x\\ \mathbf{if}\;t\_0 \leq 0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \log 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* y x))) (t_1 (* (- y) x)))
   (if (<= t_0 0.1) t_1 (if (<= t_0 1.0) (fma 0.5 x (log 2.0)) t_1))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (y * x);
	double t_1 = -y * x;
	double tmp;
	if (t_0 <= 0.1) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = fma(0.5, x, log(2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(y * x))
	t_1 = Float64(Float64(-y) * x)
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = fma(0.5, x, log(2.0));
	else
		tmp = t_1;
	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[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-y) * x), $MachinePrecision]}, If[LessEqual[t$95$0, 0.1], t$95$1, If[LessEqual[t$95$0, 1.0], N[(0.5 * x + N[Log[2.0], $MachinePrecision]), $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)) < 0.10000000000000001 or 1 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 99.2%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if 0.10000000000000001 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \log 2 + \color{blue}{\frac{1}{2} \cdot x} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - y \cdot x \leq 0.1:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;\log \left(1 + e^{x}\right) - y \cdot x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \log 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - y \cdot x\\ t_1 := \left(-y\right) \cdot x\\ \mathbf{if}\;t\_0 \leq 0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{log1p}\left(1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* y x))) (t_1 (* (- y) x)))
   (if (<= t_0 0.1) t_1 (if (<= t_0 1.0) (log1p (+ 1.0 x)) t_1))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (y * x);
	double t_1 = -y * x;
	double tmp;
	if (t_0 <= 0.1) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = log1p((1.0 + x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.log((1.0 + Math.exp(x))) - (y * x);
	double t_1 = -y * x;
	double tmp;
	if (t_0 <= 0.1) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = Math.log1p((1.0 + x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.log((1.0 + math.exp(x))) - (y * x)
	t_1 = -y * x
	tmp = 0
	if t_0 <= 0.1:
		tmp = t_1
	elif t_0 <= 1.0:
		tmp = math.log1p((1.0 + x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(y * x))
	t_1 = Float64(Float64(-y) * x)
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = log1p(Float64(1.0 + x));
	else
		tmp = t_1;
	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[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-y) * x), $MachinePrecision]}, If[LessEqual[t$95$0, 0.1], t$95$1, If[LessEqual[t$95$0, 1.0], N[Log[1 + N[(1.0 + x), $MachinePrecision]], $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)) < 0.10000000000000001 or 1 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 99.2%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 97.9

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

   5. Applied rewrites 97.9%

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

   if 0.10000000000000001 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - y \cdot x \leq 0.1:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;\log \left(1 + e^{x}\right) - y \cdot x \leq 1:\\ \;\;\;\;\mathsf{log1p}\left(1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - y \cdot x\\ t_1 := \left(-y\right) \cdot x\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* y x))) (t_1 (* (- y) x)))
   (if (<= t_0 2e-19) t_1 (if (<= t_0 1.0) (log 2.0) t_1))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (y * x);
	double t_1 = -y * x;
	double tmp;
	if (t_0 <= 2e-19) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		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))) - (y * x)
    t_1 = -y * x
    if (t_0 <= 2d-19) then
        tmp = t_1
    else if (t_0 <= 1.0d0) 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))) - (y * x);
	double t_1 = -y * x;
	double tmp;
	if (t_0 <= 2e-19) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		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))) - (y * x)
	t_1 = -y * x
	tmp = 0
	if t_0 <= 2e-19:
		tmp = t_1
	elif t_0 <= 1.0:
		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(y * x))
	t_1 = Float64(Float64(-y) * x)
	tmp = 0.0
	if (t_0 <= 2e-19)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		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))) - (y * x);
	t_1 = -y * x;
	tmp = 0.0;
	if (t_0 <= 2e-19)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		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[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-y) * x), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-19], t$95$1, If[LessEqual[t$95$0, 1.0], 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)) < 2e-19 or 1 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 99.2%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 2e-19 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 1

   1. Initial program 99.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - y \cdot x \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;\log \left(1 + e^{x}\right) - y \cdot x \leq 1:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $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. Final simplification 99.6%

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

Alternative 6: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-y\right) \cdot x\\ \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0020833333333333333, x \cdot x, 0.020833333333333332\right), x \cdot x, -0.25\right), x, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- y) x)))
   (if (<= x -2.5)
     t_0
     (-
      t_0
      (log
       (fma
        (fma
         (fma -0.0020833333333333333 (* x x) 0.020833333333333332)
         (* x x)
         -0.25)
        x
        0.5))))))

C

double code(double x, double y) {
	double t_0 = -y * x;
	double tmp;
	if (x <= -2.5) {
		tmp = t_0;
	} else {
		tmp = t_0 - log(fma(fma(fma(-0.0020833333333333333, (x * x), 0.020833333333333332), (x * x), -0.25), x, 0.5));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(-y) * x)
	tmp = 0.0
	if (x <= -2.5)
		tmp = t_0;
	else
		tmp = Float64(t_0 - log(fma(fma(fma(-0.0020833333333333333, Float64(x * x), 0.020833333333333332), Float64(x * x), -0.25), x, 0.5)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[((-y) * x), $MachinePrecision]}, If[LessEqual[x, -2.5], t$95$0, N[(t$95$0 - N[Log[N[(N[(N[(-0.0020833333333333333 * N[(x * x), $MachinePrecision] + 0.020833333333333332), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * x + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5

   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. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.5 < x

   1. Initial program 99.4%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

         \[\leadsto \log \color{blue}{\left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}\right)} - x \cdot y \]

      4. clear-num N/A

         \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}{{1}^{3} + {\left(e^{x}\right)}^{3}}}\right)} - x \cdot y \]

      5. clear-num N/A

         \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{\frac{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}}\right)} - x \cdot y \]

      6. log-rec N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{\frac{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right)\right)\right)} - x \cdot y \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\log \left(\frac{\frac{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right)\right)} - x \cdot y \]

      8. lower-log.f64 N/A

         \[\leadsto \left(-\color{blue}{\log \left(\frac{\frac{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right)}\right) - x \cdot y \]

      9. lower-/.f64 N/A

         \[\leadsto \left(-\log \color{blue}{\left(\frac{\frac{1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)}{{1}^{3} + {\left(e^{x}\right)}^{3}}}{1}\right)}\right) - x \cdot y \]

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(-\log \color{blue}{\left(\frac{1}{2} + x \cdot \left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) - \frac{1}{4}\right)\right)}\right) - x \cdot y \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-\log \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)}\right) - x \cdot y \]

      2. *-commutative N/A

         \[\leadsto \left(-\log \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x} + \frac{1}{2}\right)\right) - x \cdot y \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(-\log \color{blue}{\left(\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) - \frac{1}{4}, x, \frac{1}{2}\right)\right)}\right) - x \cdot y \]

      4. sub-neg N/A

         \[\leadsto \left(-\log \left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, x, \frac{1}{2}\right)\right)\right) - x \cdot y \]

      5. *-commutative N/A

         \[\leadsto \left(-\log \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), x, \frac{1}{2}\right)\right)\right) - x \cdot y \]

      6. metadata-eval N/A

         \[\leadsto \left(-\log \left(\mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{4}}, x, \frac{1}{2}\right)\right)\right) - x \cdot y \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-\log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}, {x}^{2}, \frac{-1}{4}\right)}, x, \frac{1}{2}\right)\right)\right) - x \cdot y \]

      8. +-commutative N/A

         \[\leadsto \left(-\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{480} \cdot {x}^{2} + \frac{1}{48}}, {x}^{2}, \frac{-1}{4}\right), x, \frac{1}{2}\right)\right)\right) - x \cdot y \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(-\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{480}, {x}^{2}, \frac{1}{48}\right)}, {x}^{2}, \frac{-1}{4}\right), x, \frac{1}{2}\right)\right)\right) - x \cdot y \]

      10. unpow2 N/A

          \[\leadsto \left(-\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, \color{blue}{x \cdot x}, \frac{1}{48}\right), {x}^{2}, \frac{-1}{4}\right), x, \frac{1}{2}\right)\right)\right) - x \cdot y \]

      11. lower-*.f64 N/A

          \[\leadsto \left(-\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, \color{blue}{x \cdot x}, \frac{1}{48}\right), {x}^{2}, \frac{-1}{4}\right), x, \frac{1}{2}\right)\right)\right) - x \cdot y \]

      12. unpow2 N/A

          \[\leadsto \left(-\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, x \cdot x, \frac{1}{48}\right), \color{blue}{x \cdot x}, \frac{-1}{4}\right), x, \frac{1}{2}\right)\right)\right) - x \cdot y \]

      13. lower-*.f64 98.7

          \[\leadsto \left(-\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0020833333333333333, x \cdot x, 0.020833333333333332\right), \color{blue}{x \cdot x}, -0.25\right), x, 0.5\right)\right)\right) - x \cdot y \]

   7. Applied rewrites 98.7%

      \[\leadsto \left(-\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0020833333333333333, x \cdot x, 0.020833333333333332\right), x \cdot x, -0.25\right), x, 0.5\right)\right)}\right) - x \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot x - \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0020833333333333333, x \cdot x, 0.020833333333333332\right), x \cdot x, -0.25\right), x, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.005208333333333333, 0.125\right), x, 0.5 - y\right), x, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.6)
   (* (- y) x)
   (fma
    (fma (fma (* x x) -0.005208333333333333 0.125) x (- 0.5 y))
    x
    (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.6) {
		tmp = -y * x;
	} else {
		tmp = fma(fma(fma((x * x), -0.005208333333333333, 0.125), x, (0.5 - y)), x, log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.6)
		tmp = Float64(Float64(-y) * x);
	else
		tmp = fma(fma(fma(Float64(x * x), -0.005208333333333333, 0.125), x, Float64(0.5 - y)), x, log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.6], N[((-y) * x), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.005208333333333333 + 0.125), $MachinePrecision] * x + N[(0.5 - y), $MachinePrecision]), $MachinePrecision] * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000009

   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. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.60000000000000009 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y\right) + \log 2} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y\right) \cdot x} + \log 2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y, x, \log 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right) + \frac{1}{2}\right)} - y, x, \log 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right) + \left(\frac{1}{2} - y\right)}, x, \log 2\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right) \cdot x} + \left(\frac{1}{2} - y\right), x, \log 2\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}, x, \frac{1}{2} - y\right)}, x, \log 2\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{192} \cdot {x}^{2} + \frac{1}{8}}, x, \frac{1}{2} - y\right), x, \log 2\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \frac{-1}{192}} + \frac{1}{8}, x, \frac{1}{2} - y\right), x, \log 2\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{192}, \frac{1}{8}\right)}, x, \frac{1}{2} - y\right), x, \log 2\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{192}, \frac{1}{8}\right), x, \frac{1}{2} - y\right), x, \log 2\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{192}, \frac{1}{8}\right), x, \frac{1}{2} - y\right), x, \log 2\right) \]

      13. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{192}, \frac{1}{8}\right), x, \color{blue}{\frac{1}{2} - y}\right), x, \log 2\right) \]

      14. lower-log.f64 98.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.005208333333333333, 0.125\right), x, 0.5 - y\right), x, \color{blue}{\log 2}\right) \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.005208333333333333, 0.125\right), x, 0.5 - y\right), x, \log 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -70:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, x, 0.5 - y\right), x, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -70.0) (* (- y) x) (fma (fma 0.125 x (- 0.5 y)) x (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -70.0) {
		tmp = -y * x;
	} else {
		tmp = fma(fma(0.125, x, (0.5 - y)), x, log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -70.0)
		tmp = Float64(Float64(-y) * x);
	else
		tmp = fma(fma(0.125, x, Float64(0.5 - y)), x, log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -70.0], N[((-y) * x), $MachinePrecision], N[(N[(0.125 * x + N[(0.5 - y), $MachinePrecision]), $MachinePrecision] * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -70

   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. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -70 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right)} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 9: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.36:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - y, x, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.36) (* (- y) x) (fma (- 0.5 y) x (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.36) {
		tmp = -y * x;
	} else {
		tmp = fma((0.5 - y), x, log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.36)
		tmp = Float64(Float64(-y) * x);
	else
		tmp = fma(Float64(0.5 - y), x, log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.36], N[((-y) * x), $MachinePrecision], N[(N[(0.5 - y), $MachinePrecision] * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3600000000000001

   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. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -1.3600000000000001 < x

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 10: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -70:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -70.0) (* (- y) x) (fma (- y) x (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -70.0) {
		tmp = -y * x;
	} else {
		tmp = fma(-y, x, log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -70.0)
		tmp = Float64(Float64(-y) * x);
	else
		tmp = fma(Float64(-y), x, log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -70.0], N[((-y) * x), $MachinePrecision], N[((-y) * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -70

   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. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -70 < x

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 97.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-out N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-fma.f64 97.3

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

   7. Applied rewrites 97.3%

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

Alternative 11: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -70:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -70.0) (* (- y) x) (- (log 2.0) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -70.0) {
		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 <= (-70.0d0)) 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 <= -70.0) {
		tmp = -y * x;
	} else {
		tmp = Math.log(2.0) - (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -70.0:
		tmp = -y * x
	else:
		tmp = math.log(2.0) - (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -70.0)
		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 <= -70.0)
		tmp = -y * x;
	else
		tmp = log(2.0) - (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -70.0], 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 < -70

   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. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -70 < x

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 97.3%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -70:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 - y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.5% accurate, 26.5× 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

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 50.4

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

5. Applied rewrites 50.4%

   \[\leadsto \color{blue}{\left(-y\right) \cdot x} \]
6. 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 2024283 
(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)))