symmetry log of sum of exp

Percentage Accurate: 53.1% → 98.6%

Time: 12.0s

Alternatives: 14

Speedup: 1.4×

• could not determine a ground truth

  1. a = -5.739407391704725e+77
  2. b = 5.94406962275951e+110

Specification

Math

\[\begin{array}{l} \\ \log \left(e^{a} + e^{b}\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return log((exp(a) + exp(b)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = log((exp(a) + exp(b)))
end function

Java

public static double code(double a, double b) {
	return Math.log((Math.exp(a) + Math.exp(b)));
}

Python

def code(a, b):
	return math.log((math.exp(a) + math.exp(b)))

Julia

function code(a, b)
	return log(Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = log((exp(a) + exp(b)));
end

Wolfram

code[a_, b_] := N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 53.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(e^{a} + e^{b}\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return log((exp(a) + exp(b)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = log((exp(a) + exp(b)))
end function

Java

public static double code(double a, double b) {
	return Math.log((Math.exp(a) + Math.exp(b)));
}

Python

def code(a, b):
	return math.log((math.exp(a) + math.exp(b)))

Julia

function code(a, b)
	return log(Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = log((exp(a) + exp(b)));
end

Wolfram

code[a_, b_] := N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ b (+ 1.0 (exp a))) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((exp(a) + exp(b)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (1.0d0 + exp(a))
    else
        tmp = log((exp(a) + exp(b)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log((Math.exp(a) + Math.exp(b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log((math.exp(a) + math.exp(b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(exp(a) + exp(b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = b / (1.0 + exp(a));
	else
		tmp = log((exp(a) + exp(b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 11.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b \cdot 1}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{b \cdot 1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{b}{\color{blue}{e^{a}} + 1} + \log \left(1 + e^{a}\right) \]

      11. lower-log1p.f64 N/A

          \[\leadsto \frac{b}{e^{a} + 1} + \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]

      12. lower-exp.f64 100.0

          \[\leadsto \frac{b}{e^{a} + 1} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.3%

      \[\leadsto \left({\left(\frac{b}{1 + e^{a}}\right)}^{2} - {\left(\mathsf{log1p}\left(e^{a}\right)\right)}^{2}\right) \cdot \color{blue}{{\left(\frac{b}{1 + e^{a}} - \mathsf{log1p}\left(e^{a}\right)\right)}^{-1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.3%

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

   10. Taylor expanded in b around inf

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 100.0%

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 71.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ 1.0 (exp a)))
   (log (+ (exp a) (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((exp(a) + fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(exp(a) + fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 11.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b \cdot 1}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{b \cdot 1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{b}{\color{blue}{e^{a}} + 1} + \log \left(1 + e^{a}\right) \]

      11. lower-log1p.f64 N/A

          \[\leadsto \frac{b}{e^{a} + 1} + \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]

      12. lower-exp.f64 100.0

          \[\leadsto \frac{b}{e^{a} + 1} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.3%

      \[\leadsto \left({\left(\frac{b}{1 + e^{a}}\right)}^{2} - {\left(\mathsf{log1p}\left(e^{a}\right)\right)}^{2}\right) \cdot \color{blue}{{\left(\frac{b}{1 + e^{a}} - \mathsf{log1p}\left(e^{a}\right)\right)}^{-1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.3%

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

   10. Taylor expanded in b around inf

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 100.0%

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 71.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \log \left(e^{a} + \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log \left(e^{a} + \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log \left(e^{a} + \left(\color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) \cdot b} + 1\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \log \left(e^{a} + \color{blue}{\mathsf{fma}\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), b, 1\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(\color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, b, 1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right) \cdot b} + 1, b, 1\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot b, b, 1\right)}, b, 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, b, 1\right), b, 1\right)\right) \]

      8. lower-fma.f64 68.2

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, b, 0.5\right)}, b, 1\right), b, 1\right)\right) \]

   5. Applied rewrites 68.2%

      \[\leadsto \log \left(e^{a} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (+ (/ b (+ (exp a) 1.0)) (log1p (exp a))))

C

assert(a < b);
double code(double a, double b) {
	return (b / (exp(a) + 1.0)) + log1p(exp(a));
}

Java

assert a < b;
public static double code(double a, double b) {
	return (b / (Math.exp(a) + 1.0)) + Math.log1p(Math.exp(a));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (b / (math.exp(a) + 1.0)) + math.log1p(math.exp(a))

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(b / Float64(exp(a) + 1.0)) + log1p(exp(a)))
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.9%

   \[\log \left(e^{a} + e^{b}\right) \]
2. Add Preprocessing

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

   2. *-rgt-identity N/A

      \[\leadsto \frac{\color{blue}{b \cdot 1}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

   5. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{b \cdot 1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

   6. *-rgt-identity N/A

      \[\leadsto \frac{\color{blue}{b}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

   8. +-commutative N/A

      \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

   9. lower-+.f64 N/A

      \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

   10. lower-exp.f64 N/A

       \[\leadsto \frac{b}{\color{blue}{e^{a}} + 1} + \log \left(1 + e^{a}\right) \]

   11. lower-log1p.f64 N/A

       \[\leadsto \frac{b}{e^{a} + 1} + \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]

   12. lower-exp.f64 75.2

       \[\leadsto \frac{b}{e^{a} + 1} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

5. Applied rewrites 75.2%

   \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
6. Add Preprocessing

Alternative 4: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot b + \mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ b (+ 1.0 (exp a))) (+ (* 0.5 b) (log1p (exp a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = (0.5 * b) + log1p(exp(a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = (0.5 * b) + Math.log1p(Math.exp(a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = (0.5 * b) + math.log1p(math.exp(a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = Float64(Float64(0.5 * b) + log1p(exp(a)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * b), $MachinePrecision] + N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 11.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b \cdot 1}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{b \cdot 1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{b}{\color{blue}{e^{a}} + 1} + \log \left(1 + e^{a}\right) \]

      11. lower-log1p.f64 N/A

          \[\leadsto \frac{b}{e^{a} + 1} + \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]

      12. lower-exp.f64 100.0

          \[\leadsto \frac{b}{e^{a} + 1} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.3%

      \[\leadsto \left({\left(\frac{b}{1 + e^{a}}\right)}^{2} - {\left(\mathsf{log1p}\left(e^{a}\right)\right)}^{2}\right) \cdot \color{blue}{{\left(\frac{b}{1 + e^{a}} - \mathsf{log1p}\left(e^{a}\right)\right)}^{-1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.3%

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

   10. Taylor expanded in b around inf

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 100.0%

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 71.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b \cdot 1}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{b \cdot 1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{b}{\color{blue}{e^{a}} + 1} + \log \left(1 + e^{a}\right) \]

      11. lower-log1p.f64 N/A

          \[\leadsto \frac{b}{e^{a} + 1} + \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]

      12. lower-exp.f64 69.2

          \[\leadsto \frac{b}{e^{a} + 1} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   5. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} \cdot b + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.1%

      \[\leadsto 0.5 \cdot b + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ b (+ 1.0 (exp a))) (log1p (exp a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log1p(exp(a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log1p(Math.exp(a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log1p(math.exp(a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log1p(exp(a));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 11.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b \cdot 1}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{b \cdot 1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{b}{\color{blue}{e^{a}} + 1} + \log \left(1 + e^{a}\right) \]

      11. lower-log1p.f64 N/A

          \[\leadsto \frac{b}{e^{a} + 1} + \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]

      12. lower-exp.f64 100.0

          \[\leadsto \frac{b}{e^{a} + 1} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.3%

      \[\leadsto \left({\left(\frac{b}{1 + e^{a}}\right)}^{2} - {\left(\mathsf{log1p}\left(e^{a}\right)\right)}^{2}\right) \cdot \color{blue}{{\left(\frac{b}{1 + e^{a}} - \mathsf{log1p}\left(e^{a}\right)\right)}^{-1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.3%

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

   10. Taylor expanded in b around inf

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 100.0%

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 71.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 68.5

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

   5. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, \mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ 1.0 (exp a)))
   (log (fma (fma 0.5 b 1.0) b (fma (fma 0.5 a 1.0) a 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log(fma(fma(0.5, b, 1.0), b, fma(fma(0.5, a, 1.0), a, 2.0)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(fma(fma(0.5, b, 1.0), b, fma(fma(0.5, a, 1.0), a, 2.0)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 11.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b \cdot 1}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{b \cdot 1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{b}{\color{blue}{e^{a}} + 1} + \log \left(1 + e^{a}\right) \]

      11. lower-log1p.f64 N/A

          \[\leadsto \frac{b}{e^{a} + 1} + \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]

      12. lower-exp.f64 100.0

          \[\leadsto \frac{b}{e^{a} + 1} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.3%

      \[\leadsto \left({\left(\frac{b}{1 + e^{a}}\right)}^{2} - {\left(\mathsf{log1p}\left(e^{a}\right)\right)}^{2}\right) \cdot \color{blue}{{\left(\frac{b}{1 + e^{a}} - \mathsf{log1p}\left(e^{a}\right)\right)}^{-1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.3%

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

   10. Taylor expanded in b around inf

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 100.0%

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 71.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

         \[\leadsto \log \color{blue}{\left(\left(1 + e^{a}\right) + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \log \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right) + \left(1 + e^{a}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \log \left(\color{blue}{\left(1 + \frac{1}{2} \cdot b\right) \cdot b} + \left(1 + e^{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(1 + \frac{1}{2} \cdot b, b, 1 + e^{a}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot b + 1}, b, 1 + e^{a}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, 1\right)}, b, 1 + e^{a}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, b, 1\right), b, \color{blue}{e^{a} + 1}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, b, 1\right), b, \color{blue}{e^{a} + 1}\right)\right) \]

      9. lower-exp.f64 69.0

         \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, \color{blue}{e^{a}} + 1\right)\right) \]

   5. Applied rewrites 69.0%

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, e^{a} + 1\right)\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, b, 1\right), b, 2 + a \cdot \left(1 + \frac{1}{2} \cdot a\right)\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 66.9%

      \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, \mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 2\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2 + a\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05)
   (/ b (+ 1.0 (exp a)))
   (log (fma (fma 0.5 b 1.0) b (+ 2.0 a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log(fma(fma(0.5, b, 1.0), b, (2.0 + a)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(fma(fma(0.5, b, 1.0), b, Float64(2.0 + a)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.05], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + N[(2.0 + a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 16.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b \cdot 1}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{b \cdot 1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{b}{\color{blue}{e^{a}} + 1} + \log \left(1 + e^{a}\right) \]

      11. lower-log1p.f64 N/A

          \[\leadsto \frac{b}{e^{a} + 1} + \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]

      12. lower-exp.f64 98.1

          \[\leadsto \frac{b}{e^{a} + 1} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.0%

      \[\leadsto \left({\left(\frac{b}{1 + e^{a}}\right)}^{2} - {\left(\mathsf{log1p}\left(e^{a}\right)\right)}^{2}\right) \cdot \color{blue}{{\left(\frac{b}{1 + e^{a}} - \mathsf{log1p}\left(e^{a}\right)\right)}^{-1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.0%

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

   10. Taylor expanded in b around inf

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 94.5%

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 71.4%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

         \[\leadsto \log \color{blue}{\left(\left(1 + e^{a}\right) + b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \log \color{blue}{\left(b \cdot \left(1 + \frac{1}{2} \cdot b\right) + \left(1 + e^{a}\right)\right)} \]

      3. *-commutative N/A

         \[\leadsto \log \left(\color{blue}{\left(1 + \frac{1}{2} \cdot b\right) \cdot b} + \left(1 + e^{a}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(1 + \frac{1}{2} \cdot b, b, 1 + e^{a}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot b + 1}, b, 1 + e^{a}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, 1\right)}, b, 1 + e^{a}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, b, 1\right), b, \color{blue}{e^{a} + 1}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, b, 1\right), b, \color{blue}{e^{a} + 1}\right)\right) \]

      9. lower-exp.f64 69.3

         \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, \color{blue}{e^{a}} + 1\right)\right) \]

   5. Applied rewrites 69.3%

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, e^{a} + 1\right)\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, b, 1\right), b, 2 + a\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 67.3%

      \[\leadsto \log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2 + a\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(1 + a\right) + \left(1 + b\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05) (/ b (+ 1.0 (exp a))) (log (+ (+ 1.0 a) (+ 1.0 b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log(((1.0 + a) + (1.0 + b)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.05d0) then
        tmp = b / (1.0d0 + exp(a))
    else
        tmp = log(((1.0d0 + a) + (1.0d0 + b)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.05) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log(((1.0 + a) + (1.0 + b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.05:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log(((1.0 + a) + (1.0 + b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(Float64(1.0 + a) + Float64(1.0 + b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.05)
		tmp = b / (1.0 + exp(a));
	else
		tmp = log(((1.0 + a) + (1.0 + b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.05], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(1.0 + a), $MachinePrecision] + N[(1.0 + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 16.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b \cdot 1}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{b \cdot 1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{b}{\color{blue}{e^{a}} + 1} + \log \left(1 + e^{a}\right) \]

      11. lower-log1p.f64 N/A

          \[\leadsto \frac{b}{e^{a} + 1} + \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]

      12. lower-exp.f64 98.1

          \[\leadsto \frac{b}{e^{a} + 1} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.0%

      \[\leadsto \left({\left(\frac{b}{1 + e^{a}}\right)}^{2} - {\left(\mathsf{log1p}\left(e^{a}\right)\right)}^{2}\right) \cdot \color{blue}{{\left(\frac{b}{1 + e^{a}} - \mathsf{log1p}\left(e^{a}\right)\right)}^{-1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.0%

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

   10. Taylor expanded in b around inf

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 94.5%

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 71.4%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 69.2

         \[\leadsto \log \left(\color{blue}{\left(1 + a\right)} + e^{b}\right) \]

   5. Applied rewrites 69.2%

      \[\leadsto \log \left(\color{blue}{\left(1 + a\right)} + e^{b}\right) \]

   6. Taylor expanded in b around 0

      \[\leadsto \log \left(\left(1 + a\right) + \color{blue}{\left(1 + b\right)}\right) \]
   7. Step-by-step derivation

      1. lower-+.f64 66.4

         \[\leadsto \log \left(\left(1 + a\right) + \color{blue}{\left(1 + b\right)}\right) \]

   8. Applied rewrites 66.4%

      \[\leadsto \log \left(\left(1 + a\right) + \color{blue}{\left(1 + b\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 98.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -36:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \left(1 + b\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -36.0) (/ b (+ 1.0 (exp a))) (log (+ (exp a) (+ 1.0 b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -36.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((exp(a) + (1.0 + b)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-36.0d0)) then
        tmp = b / (1.0d0 + exp(a))
    else
        tmp = log((exp(a) + (1.0d0 + b)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -36.0) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log((Math.exp(a) + (1.0 + b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -36.0:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log((math.exp(a) + (1.0 + b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -36.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(exp(a) + Float64(1.0 + b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -36.0)
		tmp = b / (1.0 + exp(a));
	else
		tmp = log((exp(a) + (1.0 + b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -36.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[(1.0 + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -36

   1. Initial program 13.5%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b \cdot 1}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{b \cdot 1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{b}{\color{blue}{e^{a}} + 1} + \log \left(1 + e^{a}\right) \]

      11. lower-log1p.f64 N/A

          \[\leadsto \frac{b}{e^{a} + 1} + \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]

      12. lower-exp.f64 98.1

          \[\leadsto \frac{b}{e^{a} + 1} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.3%

      \[\leadsto \left({\left(\frac{b}{1 + e^{a}}\right)}^{2} - {\left(\mathsf{log1p}\left(e^{a}\right)\right)}^{2}\right) \cdot \color{blue}{{\left(\frac{b}{1 + e^{a}} - \mathsf{log1p}\left(e^{a}\right)\right)}^{-1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 53.3%

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

   10. Taylor expanded in b around inf

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 98.1%

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]

   if -36 < a

   1. Initial program 71.5%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 68.5

         \[\leadsto \log \left(e^{a} + \color{blue}{\left(1 + b\right)}\right) \]

   5. Applied rewrites 68.5%

      \[\leadsto \log \left(e^{a} + \color{blue}{\left(1 + b\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.05:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(1 + a\right) + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.05) (/ b (+ 1.0 (exp a))) (log (+ (+ 1.0 a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.05) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log(((1.0 + a) + exp(b)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.05d0)) then
        tmp = b / (1.0d0 + exp(a))
    else
        tmp = log(((1.0d0 + a) + exp(b)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.05) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log(((1.0 + a) + Math.exp(b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.05:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log(((1.0 + a) + math.exp(b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.05)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(Float64(1.0 + a) + exp(b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.05)
		tmp = b / (1.0 + exp(a));
	else
		tmp = log(((1.0 + a) + exp(b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.05], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(1.0 + a), $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.05000000000000004

   1. Initial program 16.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b \cdot 1}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{b \cdot 1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

      10. lower-exp.f64 N/A

          \[\leadsto \frac{b}{\color{blue}{e^{a}} + 1} + \log \left(1 + e^{a}\right) \]

      11. lower-log1p.f64 N/A

          \[\leadsto \frac{b}{e^{a} + 1} + \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]

      12. lower-exp.f64 98.1

          \[\leadsto \frac{b}{e^{a} + 1} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 55.0%

      \[\leadsto \left({\left(\frac{b}{1 + e^{a}}\right)}^{2} - {\left(\mathsf{log1p}\left(e^{a}\right)\right)}^{2}\right) \cdot \color{blue}{{\left(\frac{b}{1 + e^{a}} - \mathsf{log1p}\left(e^{a}\right)\right)}^{-1}} \]
   8. Step-by-step derivation

   9. Applied rewrites 55.0%

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

   10. Taylor expanded in b around inf

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 94.5%

       \[\leadsto \frac{b}{\color{blue}{1 + e^{a}}} \]

   if -1.05000000000000004 < a

   1. Initial program 71.4%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 69.2

         \[\leadsto \log \left(\color{blue}{\left(1 + a\right)} + e^{b}\right) \]

   5. Applied rewrites 69.2%

      \[\leadsto \log \left(\color{blue}{\left(1 + a\right)} + e^{b}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 49.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{fma}\left(0.5, b, \log 2\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (fma 0.5 b (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	return fma(0.5, b, log(2.0));
}

Julia

a, b = sort([a, b])
function code(a, b)
	return fma(0.5, b, log(2.0))
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(0.5 * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.9%

   \[\log \left(e^{a} + e^{b}\right) \]
2. Add Preprocessing

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

   2. *-rgt-identity N/A

      \[\leadsto \frac{\color{blue}{b \cdot 1}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{b \cdot \frac{1}{1 + e^{a}} + \log \left(1 + e^{a}\right)} \]

   5. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{b \cdot 1}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

   6. *-rgt-identity N/A

      \[\leadsto \frac{\color{blue}{b}}{1 + e^{a}} + \log \left(1 + e^{a}\right) \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} + \log \left(1 + e^{a}\right) \]

   8. +-commutative N/A

      \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

   9. lower-+.f64 N/A

      \[\leadsto \frac{b}{\color{blue}{e^{a} + 1}} + \log \left(1 + e^{a}\right) \]

   10. lower-exp.f64 N/A

       \[\leadsto \frac{b}{\color{blue}{e^{a}} + 1} + \log \left(1 + e^{a}\right) \]

   11. lower-log1p.f64 N/A

       \[\leadsto \frac{b}{e^{a} + 1} + \color{blue}{\mathsf{log1p}\left(e^{a}\right)} \]

   12. lower-exp.f64 75.2

       \[\leadsto \frac{b}{e^{a} + 1} + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]

5. Applied rewrites 75.2%

   \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]

6. Taylor expanded in a around 0

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

8. Applied rewrites 53.6%

   \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{b}, \log 2\right) \]
9. Add Preprocessing

Alternative 12: 47.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{fma}\left(0.5, a, \log 2\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (fma 0.5 a (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	return fma(0.5, a, log(2.0));
}

Julia

a, b = sort([a, b])
function code(a, b)
	return fma(0.5, a, log(2.0))
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(0.5 * a + N[Log[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.9%

   \[\log \left(e^{a} + e^{b}\right) \]
2. Add Preprocessing

3. Taylor expanded in b around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 56.2

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

5. Applied rewrites 56.2%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 53.6%

   \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{a}, \log 2\right) \]
9. Add Preprocessing

Alternative 13: 48.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{log1p}\left(1\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (log1p 1.0))

C

assert(a < b);
double code(double a, double b) {
	return log1p(1.0);
}

Java

assert a < b;
public static double code(double a, double b) {
	return Math.log1p(1.0);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log1p(1.0)

Julia

a, b = sort([a, b])
function code(a, b)
	return log1p(1.0)
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[Log[1 + 1.0], $MachinePrecision]

Derivation

1. Initial program 59.9%

   \[\log \left(e^{a} + e^{b}\right) \]
2. Add Preprocessing

3. Taylor expanded in b around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 56.2

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

5. Applied rewrites 56.2%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 53.2%

   \[\leadsto \mathsf{log1p}\left(1\right) \]
9. Add Preprocessing

Alternative 14: 3.2% accurate, 27.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \left(a \cdot a\right) \cdot 0.125 \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (* a a) 0.125))

C

assert(a < b);
double code(double a, double b) {
	return (a * a) * 0.125;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * a) * 0.125d0
end function

Java

assert a < b;
public static double code(double a, double b) {
	return (a * a) * 0.125;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (a * a) * 0.125

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(a * a) * 0.125)
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (a * a) * 0.125;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(a * a), $MachinePrecision] * 0.125), $MachinePrecision]

Derivation

1. Initial program 59.9%

   \[\log \left(e^{a} + e^{b}\right) \]
2. Add Preprocessing

3. Taylor expanded in b around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 56.2

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

5. Applied rewrites 56.2%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 53.9%

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

9. Taylor expanded in a around inf

   \[\leadsto \frac{1}{8} \cdot {a}^{\color{blue}{2}} \]
10. Step-by-step derivation

11. Applied rewrites 4.6%

    \[\leadsto \left(a \cdot a\right) \cdot 0.125 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024322 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))