symmetry log of sum of exp

Percentage Accurate: 52.8% → 98.7%

Time: 12.3s

Alternatives: 11

Speedup: 2.5×

• could not determine a ground truth

  1. a = -3.264140369286671e-148
  2. b = 1.359040309795994e+209

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: 52.8% 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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -38:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -38.0) (/ b (+ (exp a) 1.0)) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -38.0) {
		tmp = b / (exp(a) + 1.0);
	} 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 (a <= (-38.0d0)) then
        tmp = b / (exp(a) + 1.0d0)
    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 (a <= -38.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} 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 a <= -38.0:
		tmp = b / (math.exp(a) + 1.0)
	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 (a <= -38.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	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 (a <= -38.0)
		tmp = b / (exp(a) + 1.0);
	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[a, -38.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -38

   1. Initial program 7.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 97.0

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

   5. Simplified 97.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 97.0

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

   8. Simplified 97.0%

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

   if -38 < a

   1. Initial program 66.4%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -38:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

   \[\log \left(e^{a} + e^{b}\right) \]
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. *-rgt-identity N/A

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

   2. associate-*r/ N/A

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

   3. lower-+.f64 N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. associate-*r/ N/A

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

   7. *-rgt-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 70.7

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

5. Simplified 70.7%

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

6. Final simplification 70.7%

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

Alternative 3: 55.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} + e^{b} \leq 1.5:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (+ (exp a) (exp b)) 1.5) (* b 0.5) (fma 0.5 b (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if ((exp(a) + exp(b)) <= 1.5) {
		tmp = b * 0.5;
	} else {
		tmp = fma(0.5, b, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) + exp(b)) <= 1.5)
		tmp = Float64(b * 0.5);
	else
		tmp = fma(0.5, b, log(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[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision], 1.5], N[(b * 0.5), $MachinePrecision], N[(0.5 * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 a) (exp.f64 b)) < 1.5

   1. Initial program 8.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 46.6

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

   5. Simplified 46.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot b + \log 2} \]

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 2.9

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

   8. Simplified 2.9%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{b \cdot \frac{1}{2}} \]

       2. lower-*.f64 9.8

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

   11. Simplified 9.8%

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

   if 1.5 < (+.f64 (exp.f64 a) (exp.f64 b))

   1. Initial program 95.4%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 94.8

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

   5. Simplified 94.8%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot b + \log 2} \]

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 88.5

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

   8. Simplified 88.5%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

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 (+ (exp a) 1.0)) (log1p (exp b))))

C

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

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p(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 / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(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(exp(a) + 1.0));
	else
		tmp = log1p(exp(b));
	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[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 97.0

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

   5. Simplified 97.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 97.0

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

   8. Simplified 97.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 66.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 60.4

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

   5. Simplified 60.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, b, \mathsf{fma}\left(a, \mathsf{fma}\left(0.125, a, 0.5\right), \log 2\right)\right)\\ \end{array} \end{array} \]

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 (+ (exp a) 1.0))
   (fma 0.5 b (fma a (fma 0.125 a 0.5) (log 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = fma(0.5, b, fma(a, fma(0.125, a, 0.5), log(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(exp(a) + 1.0));
	else
		tmp = fma(0.5, b, fma(a, fma(0.125, a, 0.5), log(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[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * b + N[(a * N[(0.125 * a + 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 97.0

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

   5. Simplified 97.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 97.0

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

   8. Simplified 97.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 66.4%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 62.3

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

   5. Simplified 62.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot b + a \cdot \left(\left(\frac{1}{2} + a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right)\right) - \frac{1}{4} \cdot b\right)\right) + \log 2} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot b + \left(a \cdot \left(\left(\frac{1}{2} + a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right)\right) - \frac{1}{4} \cdot b\right) + \log 2\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, a \cdot \left(\left(\frac{1}{2} + a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right)\right) - \frac{1}{4} \cdot b\right) + \log 2\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, b, \color{blue}{\mathsf{fma}\left(a, \left(\frac{1}{2} + a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right)\right) - \frac{1}{4} \cdot b, \log 2\right)}\right) \]

   8. Simplified 60.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, b, \mathsf{fma}\left(a, \mathsf{fma}\left(b, -0.25, \mathsf{fma}\left(a, 0.125, 0.5\right)\right), \log 2\right)\right)} \]

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-log.f64 60.8

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

   11. Simplified 60.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, b, \mathsf{fma}\left(a, \mathsf{fma}\left(0.125, a, 0.5\right), \log 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.125, 0.5\right), \log 2\right)\\ \end{array} \end{array} \]

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 0.5) (fma b (fma b 0.125 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b * 0.5;
	} else {
		tmp = fma(b, fma(b, 0.125, 0.5), log(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 * 0.5);
	else
		tmp = fma(b, fma(b, 0.125, 0.5), log(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 * 0.5), $MachinePrecision], N[(b * N[(b * 0.125 + 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 97.0

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

   5. Simplified 97.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot b + \log 2} \]

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 4.1

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

   8. Simplified 4.1%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{b \cdot \frac{1}{2}} \]

       2. lower-*.f64 18.3

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

   11. Simplified 18.3%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 66.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-exp.f64 60.4

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

   5. Simplified 60.4%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 59.2

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

   8. Simplified 59.2%

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

Alternative 7: 55.8% accurate, 1.4× speedup

Math

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

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 0.5) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b * 0.5;
	} else {
		tmp = log((b + 2.0));
	}
	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 * 0.5d0
    else
        tmp = log((b + 2.0d0))
    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 * 0.5;
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b * 0.5
	else:
		tmp = math.log((b + 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 * 0.5);
	else
		tmp = log(Float64(b + 2.0));
	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 * 0.5;
	else
		tmp = log((b + 2.0));
	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 * 0.5), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 97.0

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

   5. Simplified 97.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot b + \log 2} \]

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 4.1

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

   8. Simplified 4.1%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{b \cdot \frac{1}{2}} \]

       2. lower-*.f64 18.3

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

   11. Simplified 18.3%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 66.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-exp.f64 60.4

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

   5. Simplified 60.4%

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

   6. Taylor expanded in b around 0

      \[\leadsto \log \color{blue}{\left(2 + b\right)} \]
   7. Step-by-step derivation

      1. lower-+.f64 58.1

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

   8. Simplified 58.1%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.4% accurate, 1.5× speedup

Math

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

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 0.5) (log1p 1.0)))

C

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b * 0.5
	else:
		tmp = math.log1p(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 * 0.5);
	else
		tmp = log1p(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 * 0.5), $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 97.0

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

   5. Simplified 97.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot b + \log 2} \]

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 4.1

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

   8. Simplified 4.1%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{b \cdot \frac{1}{2}} \]

       2. lower-*.f64 18.3

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

   11. Simplified 18.3%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 66.4%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 62.6

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

   5. Simplified 62.6%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 59.4%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 97.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(2.0 + fma(b, fma(0.5, b, 1.0), 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[a, -1.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(2.0 + N[(b * N[(0.5 * b + 1.0), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 7.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 97.0

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

   5. Simplified 97.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 97.0

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

   8. Simplified 97.0%

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

   if -1 < a

   1. Initial program 66.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 62.7

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

   5. Simplified 62.7%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 60.8

         \[\leadsto \log \left(2 + \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(0.5, b, 1\right)}, a\right)\right) \]

   8. Simplified 60.8%

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

4. Final simplification 69.6%

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

Alternative 10: 97.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -38:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.125, 0.5\right), \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -38.0) (/ b (+ (exp a) 1.0)) (fma b (fma b 0.125 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -38.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = fma(b, fma(b, 0.125, 0.5), log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -38.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = fma(b, fma(b, 0.125, 0.5), log(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[a, -38.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * 0.125 + 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -38

   1. Initial program 7.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 97.0

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

   5. Simplified 97.0%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 97.0

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

   8. Simplified 97.0%

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

   if -38 < a

   1. Initial program 66.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-exp.f64 60.4

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

   5. Simplified 60.4%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 59.2

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

   8. Simplified 59.2%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -38:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.125, 0.5\right), \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 12.1% accurate, 50.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ b \cdot 0.5 \end{array} \]

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return b * 0.5;
}

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 = b * 0.5d0
end function

Java

assert a < b;
public static double code(double a, double b) {
	return b * 0.5;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return b * 0.5

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(b * 0.5)
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = b * 0.5;
end

Wolfram

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

Derivation

1. Initial program 52.1%

   \[\log \left(e^{a} + e^{b}\right) \]
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. *-rgt-identity N/A

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

   2. associate-*r/ N/A

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

   3. lower-+.f64 N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. associate-*r/ N/A

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

   7. *-rgt-identity N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 70.7

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

5. Simplified 70.7%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot b + \log 2} \]

   2. lower-fma.f64 N/A

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

   3. lower-log.f64 45.7

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

8. Simplified 45.7%

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

9. Taylor expanded in b around inf

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

    1. *-commutative N/A

       \[\leadsto \color{blue}{b \cdot \frac{1}{2}} \]

    2. lower-*.f64 7.0

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

11. Simplified 7.0%

    \[\leadsto \color{blue}{b \cdot 0.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))