symmetry log of sum of exp

Percentage Accurate: 53.7% → 98.8%

Time: 13.4s

Alternatives: 13

Speedup: 2.9×

• could not determine a ground truth

  1. a = 8.339971248356647e-260
  2. b = 1.2538764209557917e+189

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: 53.7% 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: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-36) (/ b (+ (exp a) 1.0)) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-36) {
		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 (exp(a) <= 5d-36) 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 (Math.exp(a) <= 5e-36) {
		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 math.exp(a) <= 5e-36:
		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 (exp(a) <= 5e-36)
		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 (exp(a) <= 5e-36)
		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[N[Exp[a], $MachinePrecision], 5e-36], 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 (exp.f64 a) < 5.00000000000000004e-36

   1. Initial program 12.0%

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

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 98.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 98.5%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 97.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 97.0%

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

   if 5.00000000000000004e-36 < (exp.f64 a)

   1. Initial program 68.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% 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 54.9%

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

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

   2. associate-*r/ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

   4. log1p-define N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

   5. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

   6. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

   11. exp-lowering-exp.f64 73.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

5. Simplified 73.3%

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

6. Final simplification 73.3%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-251}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-251) (/ b (+ (exp a) 1.0)) (log1p (exp a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-251) {
		tmp = b / (exp(a) + 1.0);
	} 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) <= 1e-251) {
		tmp = b / (Math.exp(a) + 1.0);
	} 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) <= 1e-251:
		tmp = b / (math.exp(a) + 1.0)
	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) <= 1e-251)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	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], 1e-251], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1.00000000000000002e-251

   1. Initial program 12.1%

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

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 98.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 98.5%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 98.5%

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

   if 1.00000000000000002e-251 < (exp.f64 a)

   1. Initial program 68.3%

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

   3. Taylor expanded in b around 0

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

      1. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a}\right)\right) \]

      3. exp-lowering-exp.f64 65.6%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right) \]

   5. Simplified 65.6%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-251}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\log 2 + a \cdot 0.5\right) + a \cdot \left(a \cdot \left(0.125 + -0.005208333333333333 \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.1)
   (/ b (+ (exp a) 1.0))
   (+
    (+ (log 2.0) (* a 0.5))
    (* a (* a (+ 0.125 (* -0.005208333333333333 (* a a))))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.1) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = (log(2.0) + (a * 0.5)) + (a * (a * (0.125 + (-0.005208333333333333 * (a * a)))));
	}
	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) <= 0.1d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = (log(2.0d0) + (a * 0.5d0)) + (a * (a * (0.125d0 + ((-0.005208333333333333d0) * (a * a)))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.1) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = (Math.log(2.0) + (a * 0.5)) + (a * (a * (0.125 + (-0.005208333333333333 * (a * a)))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.1:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = (math.log(2.0) + (a * 0.5)) + (a * (a * (0.125 + (-0.005208333333333333 * (a * a)))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.1)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(Float64(log(2.0) + Float64(a * 0.5)) + Float64(a * Float64(a * Float64(0.125 + Float64(-0.005208333333333333 * Float64(a * a))))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.1)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = (log(2.0) + (a * 0.5)) + (a * (a * (0.125 + (-0.005208333333333333 * (a * a)))));
	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], 0.1], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[2.0], $MachinePrecision] + N[(a * 0.5), $MachinePrecision]), $MachinePrecision] + N[(a * N[(a * N[(0.125 + N[(-0.005208333333333333 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.10000000000000001

   1. Initial program 14.7%

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

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 97.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 97.1%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 94.2%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 94.2%

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

   if 0.10000000000000001 < (exp.f64 a)

   1. Initial program 68.4%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a}\right)\right) \]

      3. exp-lowering-exp.f64 65.6%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right) \]

   5. Simplified 65.6%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log 2, \color{blue}{\left(a \cdot \left(\frac{1}{2} + a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {a}^{2}\right)\right)\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(\color{blue}{a} \cdot \left(\frac{1}{2} + a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {a}^{2}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{2} + a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {a}^{2}\right)\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {a}^{2}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{8} + \frac{-1}{192} \cdot {a}^{2}\right)}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{8}, \color{blue}{\left(\frac{-1}{192} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\frac{-1}{192}, \color{blue}{\left({a}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\frac{-1}{192}, \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 64.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\frac{-1}{192}, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 64.5%

      \[\leadsto \color{blue}{\log 2 + a \cdot \left(0.5 + a \cdot \left(0.125 + -0.005208333333333333 \cdot \left(a \cdot a\right)\right)\right)} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \log \left(1 + 1\right) + a \cdot \left(\frac{1}{2} + a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot \left(a \cdot a\right)\right)\right) \]

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. metadata-eval N/A

         \[\leadsto \left(\log 2 + \frac{1}{2} \cdot a\right) + a \cdot \left(a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot \left(a \cdot a\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\log 2 + \frac{1}{2} \cdot a\right), \color{blue}{\left(a \cdot \left(a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot \left(a \cdot a\right)\right)\right)\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\log \left(1 + 1\right) + \frac{1}{2} \cdot a\right), \left(a \cdot \left(a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot \left(a \cdot a\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\log \left(1 + 1\right), \left(\frac{1}{2} \cdot a\right)\right), \left(\color{blue}{a} \cdot \left(a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot \left(a \cdot a\right)\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(\left(1 + 1\right)\right), \left(\frac{1}{2} \cdot a\right)\right), \left(a \cdot \left(a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot \left(a \cdot a\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(\frac{1}{2} \cdot a\right)\right), \left(a \cdot \left(a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot \left(a \cdot a\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(a \cdot \frac{1}{2}\right)\right), \left(a \cdot \left(a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot \left(a \cdot a\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \frac{1}{2}\right)\right), \left(a \cdot \left(a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot \left(a \cdot a\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(a, \color{blue}{\left(a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot \left(a \cdot a\right)\right)\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{8} + \frac{-1}{192} \cdot \left(a \cdot a\right)\right)}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{8}, \color{blue}{\left(\frac{-1}{192} \cdot \left(a \cdot a\right)\right)}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\frac{-1}{192}, \color{blue}{\left(a \cdot a\right)}\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 64.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\frac{-1}{192}, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{\left(\log 2 + a \cdot 0.5\right) + a \cdot \left(a \cdot \left(0.125 + -0.005208333333333333 \cdot \left(a \cdot a\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\log 2 + a \cdot 0.5\right) + a \cdot \left(a \cdot \left(0.125 + -0.005208333333333333 \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot \left(0.125 + -0.005208333333333333 \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.1)
   (/ b (+ (exp a) 1.0))
   (+
    (log 2.0)
    (* a (+ 0.5 (* a (+ 0.125 (* -0.005208333333333333 (* a a)))))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.1) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (a * (0.5 + (a * (0.125 + (-0.005208333333333333 * (a * a))))));
	}
	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) <= 0.1d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (a * (0.5d0 + (a * (0.125d0 + ((-0.005208333333333333d0) * (a * a))))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.1) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (a * (0.5 + (a * (0.125 + (-0.005208333333333333 * (a * a))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.1:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (a * (0.5 + (a * (0.125 + (-0.005208333333333333 * (a * a))))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.1)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(a * Float64(0.5 + Float64(a * Float64(0.125 + Float64(-0.005208333333333333 * Float64(a * a)))))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.1)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (a * (0.5 + (a * (0.125 + (-0.005208333333333333 * (a * a))))));
	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], 0.1], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * N[(0.5 + N[(a * N[(0.125 + N[(-0.005208333333333333 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.10000000000000001

   1. Initial program 14.7%

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

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 97.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 97.1%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 94.2%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 94.2%

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

   if 0.10000000000000001 < (exp.f64 a)

   1. Initial program 68.4%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a}\right)\right) \]

      3. exp-lowering-exp.f64 65.6%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right) \]

   5. Simplified 65.6%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log 2, \color{blue}{\left(a \cdot \left(\frac{1}{2} + a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {a}^{2}\right)\right)\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(\color{blue}{a} \cdot \left(\frac{1}{2} + a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {a}^{2}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{2} + a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {a}^{2}\right)\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {a}^{2}\right)\right)}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{8} + \frac{-1}{192} \cdot {a}^{2}\right)}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{8}, \color{blue}{\left(\frac{-1}{192} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\frac{-1}{192}, \color{blue}{\left({a}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\frac{-1}{192}, \left(a \cdot \color{blue}{a}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 64.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\frac{-1}{192}, \mathsf{*.f64}\left(a, \color{blue}{a}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 64.5%

      \[\leadsto \color{blue}{\log 2 + a \cdot \left(0.5 + a \cdot \left(0.125 + -0.005208333333333333 \cdot \left(a \cdot a\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot \left(0.125 + -0.005208333333333333 \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-36)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (* a (+ 0.5 (* a 0.125))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-36) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (a * (0.5 + (a * 0.125)));
	}
	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) <= 5d-36) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (a * (0.5d0 + (a * 0.125d0)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-36) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (a * (0.5 + (a * 0.125)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-36:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (a * (0.5 + (a * 0.125)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-36)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(a * Float64(0.5 + Float64(a * 0.125))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-36)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (a * (0.5 + (a * 0.125)));
	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], 5e-36], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * N[(0.5 + N[(a * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 5.00000000000000004e-36

   1. Initial program 12.0%

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

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 98.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 98.5%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 97.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 97.0%

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

   if 5.00000000000000004e-36 < (exp.f64 a)

   1. Initial program 68.7%

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

   3. Taylor expanded in b around 0

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

      1. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a}\right)\right) \]

      3. exp-lowering-exp.f64 65.4%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right) \]

   5. Simplified 65.4%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log 2, \color{blue}{\left(a \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot a\right)\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(\color{blue}{a} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot a\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - 0\right) \cdot a\right)\right)\right) \]

      4. mul0-rgt N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - b \cdot 0\right) \cdot a\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - b \cdot \left(\frac{-1}{8} + \frac{1}{8}\right)\right) \cdot a\right)\right)\right) \]

      6. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right) \cdot a\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(a \cdot \left(\frac{1}{2} + a \cdot \color{blue}{\left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{2} + a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right)\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \left(\frac{1}{2} + \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right) \cdot \color{blue}{a}\right)\right)\right) \]

      10. distribute-rgt-out N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \left(\frac{1}{2} + \left(\frac{1}{8} - b \cdot \left(\frac{-1}{8} + \frac{1}{8}\right)\right) \cdot a\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \left(\frac{1}{2} + \left(\frac{1}{8} - b \cdot 0\right) \cdot a\right)\right)\right) \]

      12. mul0-rgt N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \left(\frac{1}{2} + \left(\frac{1}{8} - 0\right) \cdot a\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \left(\frac{1}{2} + \frac{1}{8} \cdot a\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{8} \cdot a\right)}\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{\frac{1}{8}}\right)\right)\right)\right) \]

      16. *-lowering-*.f64 64.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{8}}\right)\right)\right)\right) \]

   8. Simplified 64.0%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.1) (/ b (+ (exp a) 1.0)) (+ (log 2.0) (* a 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.1) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (a * 0.5);
	}
	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) <= 0.1d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (a * 0.5d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.1) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (a * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.1:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (a * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.1)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(a * 0.5));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.1)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (a * 0.5);
	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], 0.1], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.10000000000000001

   1. Initial program 14.7%

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

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 97.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 97.1%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 94.2%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 94.2%

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

   if 0.10000000000000001 < (exp.f64 a)

   1. Initial program 68.4%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a}\right)\right) \]

      3. exp-lowering-exp.f64 65.6%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right) \]

   5. Simplified 65.6%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log 2, \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(\color{blue}{\frac{1}{2}} \cdot a\right)\right) \]

      3. *-lowering-*.f64 64.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right) \]

   8. Simplified 64.1%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.36:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.36) (/ b 2.0) (+ (log 2.0) (* a 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.36) {
		tmp = b / 2.0;
	} else {
		tmp = log(2.0) + (a * 0.5);
	}
	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 <= (-1.36d0)) then
        tmp = b / 2.0d0
    else
        tmp = log(2.0d0) + (a * 0.5d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.36) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log(2.0) + (a * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.36:
		tmp = b / 2.0
	else:
		tmp = math.log(2.0) + (a * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.36)
		tmp = Float64(b / 2.0);
	else
		tmp = Float64(log(2.0) + Float64(a * 0.5));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.36)
		tmp = b / 2.0;
	else
		tmp = log(2.0) + (a * 0.5);
	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, -1.36], N[(b / 2.0), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3600000000000001

   1. Initial program 14.7%

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

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 97.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 97.1%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 94.2%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 94.2%

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

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{2}\right) \]
   10. Step-by-step derivation

   11. Simplified 18.1%

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

   if -1.3600000000000001 < a

   1. Initial program 68.4%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a}\right)\right) \]

      3. exp-lowering-exp.f64 65.6%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right) \]

   5. Simplified 65.6%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log 2, \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(\color{blue}{\frac{1}{2}} \cdot a\right)\right) \]

      3. *-lowering-*.f64 64.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right) \]

   8. Simplified 64.1%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.36:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -1.0) (/ b 2.0) (log1p (+ a 1.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = b / 2.0;
	} else {
		tmp = log1p((a + 1.0));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log1p((a + 1.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.0:
		tmp = b / 2.0
	else:
		tmp = math.log1p((a + 1.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.0)
		tmp = Float64(b / 2.0);
	else
		tmp = log1p(Float64(a + 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[a, -1.0], N[(b / 2.0), $MachinePrecision], N[Log[1 + N[(a + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 14.7%

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

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 97.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 97.1%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 94.2%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 94.2%

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

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{2}\right) \]
   10. Step-by-step derivation

   11. Simplified 18.1%

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

   if -1 < a

   1. Initial program 68.4%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a}\right)\right) \]

      3. exp-lowering-exp.f64 65.6%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right) \]

   5. Simplified 65.6%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{log1p.f64}\left(\color{blue}{\left(1 + a\right)}\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(a + 1\right)\right) \]

      2. +-lowering-+.f64 64.0%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(a, 1\right)\right) \]

   8. Simplified 64.0%

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

Alternative 10: 56.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -70:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -70.0) (/ b 2.0) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -70.0) {
		tmp = b / 2.0;
	} 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 (a <= (-70.0d0)) then
        tmp = b / 2.0d0
    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 (a <= -70.0) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -70.0:
		tmp = b / 2.0
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -70.0)
		tmp = Float64(b / 2.0);
	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 (a <= -70.0)
		tmp = b / 2.0;
	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[a, -70.0], N[(b / 2.0), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -70

   1. Initial program 12.0%

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

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 98.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 98.5%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 97.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 97.0%

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

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{2}\right) \]
   10. Step-by-step derivation

   11. Simplified 18.4%

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

   if -70 < a

   1. Initial program 68.7%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(1 + e^{b}\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right) \]

      2. exp-lowering-exp.f64 64.1%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 64.1%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(2 + b\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 61.9%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(2, b\right)\right) \]

   8. Simplified 61.9%

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

4. Final simplification 51.3%

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

Alternative 11: 56.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -80:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -80.0) (/ b 2.0) (log1p 1.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -80.0) {
		tmp = b / 2.0;
	} else {
		tmp = log1p(1.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -80.0) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log1p(1.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -80.0:
		tmp = b / 2.0
	else:
		tmp = math.log1p(1.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -80.0)
		tmp = Float64(b / 2.0);
	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[a, -80.0], N[(b / 2.0), $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -80

   1. Initial program 12.0%

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

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

      2. associate-*r/ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 98.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 98.5%

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

   6. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 97.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 97.0%

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

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{2}\right) \]
   10. Step-by-step derivation

   11. Simplified 18.4%

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

   if -80 < a

   1. Initial program 68.7%

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

   3. Taylor expanded in b around 0

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

      1. log1p-define N/A

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

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a}\right)\right) \]

      3. exp-lowering-exp.f64 65.4%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right) \]

   5. Simplified 65.4%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{log1p.f64}\left(\color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 62.9%

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

Alternative 12: 12.0% accurate, 101.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{b}{2} \end{array} \]

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return b / 2.0;
}

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 / 2.0d0
end function

Java

assert a < b;
public static double code(double a, double b) {
	return b / 2.0;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return b / 2.0

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(b / 2.0)
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = b / 2.0;
end

Wolfram

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

Derivation

1. Initial program 54.9%

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

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

   2. associate-*r/ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\log \left(1 + e^{a}\right), \color{blue}{\left(b \cdot \frac{1}{1 + e^{a}}\right)}\right) \]

   4. log1p-define N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

   5. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

   6. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

   11. exp-lowering-exp.f64 73.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

5. Simplified 73.3%

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

6. Taylor expanded in b around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

   3. exp-lowering-exp.f64 26.0%

      \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

8. Simplified 26.0%

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

9. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{2}\right) \]
10. Step-by-step derivation

11. Simplified 7.0%

    \[\leadsto \frac{b}{\color{blue}{2}} \]
12. Add Preprocessing

Alternative 13: 2.6% accurate, 101.0× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return a * 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 = a * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

   \[\log \left(e^{a} + e^{b}\right) \]
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. log1p-define N/A

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

   2. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{log1p.f64}\left(\left(e^{a}\right)\right) \]

   3. exp-lowering-exp.f64 51.1%

      \[\leadsto \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right) \]

5. Simplified 51.1%

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

6. Taylor expanded in a around 0

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

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\log 2, \color{blue}{\left(\frac{1}{2} \cdot a\right)}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(\color{blue}{\frac{1}{2}} \cdot a\right)\right) \]

   3. *-lowering-*.f64 48.8%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{a}\right)\right) \]

8. Simplified 48.8%

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

9. Taylor expanded in a around inf

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

    1. *-commutative N/A

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

    2. *-lowering-*.f64 7.5%

       \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{2}}\right) \]

11. Simplified 7.5%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))