symmetry log of sum of exp

Percentage Accurate: 53.4% → 98.6%

Time: 12.7s

Alternatives: 16

Speedup: 2.9×

• could not determine a ground truth

  1. a = -7.074035622847869e+244
  2. b = -2.990056061100834e+164

• Rival profile vector overflowed, profile may not be complete

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: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 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 52.4%

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

3. Taylor expanded in b around 0 69.6%

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

   1. log1p-define 70.7%

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

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 2: 98.6% accurate, 1.0× speedup

Math

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

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 a) 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(a) + 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(a) + 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(a) + 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(Float64(exp(a) + 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[(N[Exp[a], $MachinePrecision] + b), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.9%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0 60.0%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{a} + b \cdot \left(1 + 0.5 \cdot b\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 60.0%

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

      2. *-commutative 60.0%

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

   5. Simplified 60.0%

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

   6. Taylor expanded in b around 0 58.6%

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

      1. associate-+r+ 58.6%

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

   8. Simplified 58.6%

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

      1. associate-+l+ 58.6%

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

      2. log1p-define 60.1%

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

      3. +-commutative 60.1%

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

   10. Applied egg-rr 60.1%

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

4. Final simplification 69.9%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

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(a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (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) <= 0.0:
		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) <= 0.0)
		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], 0.0], 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) < 0.0

   1. Initial program 9.9%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0 59.9%

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

      1. log1p-define 61.4%

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

   5. Simplified 61.4%

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

4. Final simplification 70.9%

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

Alternative 4: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(b \cdot 0.5 + a \cdot \left(\left(0.5 + a \cdot \left(0.125 - \left(b \cdot -0.125 + b \cdot 0.125\right)\right)\right) - b \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

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))
   (+
    (log 2.0)
    (+
     (* b 0.5)
     (*
      a
      (- (+ 0.5 (* a (- 0.125 (+ (* b -0.125) (* b 0.125))))) (* b 0.25)))))))

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 = log(2.0) + ((b * 0.5) + (a * ((0.5 + (a * (0.125 - ((b * -0.125) + (b * 0.125))))) - (b * 0.25))));
	}
	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 / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + ((b * 0.5d0) + (a * ((0.5d0 + (a * (0.125d0 - ((b * (-0.125d0)) + (b * 0.125d0))))) - (b * 0.25d0))))
    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 / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + ((b * 0.5) + (a * ((0.5 + (a * (0.125 - ((b * -0.125) + (b * 0.125))))) - (b * 0.25))));
	}
	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.log(2.0) + ((b * 0.5) + (a * ((0.5 + (a * (0.125 - ((b * -0.125) + (b * 0.125))))) - (b * 0.25))))
	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 = Float64(log(2.0) + Float64(Float64(b * 0.5) + Float64(a * Float64(Float64(0.5 + Float64(a * Float64(0.125 - Float64(Float64(b * -0.125) + Float64(b * 0.125))))) - Float64(b * 0.25)))));
	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 / (exp(a) + 1.0);
	else
		tmp = log(2.0) + ((b * 0.5) + (a * ((0.5 + (a * (0.125 - ((b * -0.125) + (b * 0.125))))) - (b * 0.25))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(N[(b * 0.5), $MachinePrecision] + N[(a * N[(N[(0.5 + N[(a * N[(0.125 - N[(N[(b * -0.125), $MachinePrecision] + N[(b * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.9%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0 59.6%

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

      1. log1p-define 61.1%

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

   5. Simplified 61.1%

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

   6. Taylor expanded in a around 0 59.5%

      \[\leadsto \color{blue}{\log 2 + \left(0.5 \cdot b + a \cdot \left(\left(0.5 + a \cdot \left(0.125 - \left(-0.125 \cdot b + 0.125 \cdot b\right)\right)\right) - 0.25 \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(b \cdot 0.5 + a \cdot \left(\left(0.5 + a \cdot \left(0.125 - \left(b \cdot -0.125 + b \cdot 0.125\right)\right)\right) - b \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.4× speedup

Math

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

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))
   (+ (log 2.0) (+ (* b 0.5) (* a (+ 0.5 (* a 0.125)))))))

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 = log(2.0) + ((b * 0.5) + (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) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + ((b * 0.5d0) + (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) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + ((b * 0.5) + (a * (0.5 + (a * 0.125))));
	}
	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.log(2.0) + ((b * 0.5) + (a * (0.5 + (a * 0.125))))
	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 = Float64(log(2.0) + Float64(Float64(b * 0.5) + 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) <= 0.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + ((b * 0.5) + (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], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(N[(b * 0.5), $MachinePrecision] + N[(a * N[(0.5 + N[(a * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.9%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0 59.6%

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

      1. log1p-define 61.1%

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

   5. Simplified 61.1%

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

      1. expm1-log1p-u 61.1%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{a}\right)\right)}} \]

      2. +-commutative 61.1%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{e^{a} + 1}\right)\right)} \]

   7. Applied egg-rr 61.1%

      \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a} + 1\right)\right)}} \]

   8. Taylor expanded in a around 0 61.1%

      \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{1} + 1\right)\right)} \]

   9. Taylor expanded in a around 0 59.4%

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

4. Final simplification 69.4%

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

Alternative 6: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

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((b + (1.0 + (a * (1.0 + (a * 0.5))))));
	}
	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((b + (1.0 + (a * (1.0 + (a * 0.5))))));
	}
	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((b + (1.0 + (a * (1.0 + (a * 0.5))))))
	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(Float64(b + Float64(1.0 + Float64(a * Float64(1.0 + Float64(a * 0.5))))));
	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[(b + N[(1.0 + N[(a * N[(1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.9%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0 60.0%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{a} + b \cdot \left(1 + 0.5 \cdot b\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 60.0%

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

      2. *-commutative 60.0%

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

   5. Simplified 60.0%

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

   6. Taylor expanded in b around 0 58.6%

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

      1. associate-+r+ 58.6%

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

   8. Simplified 58.6%

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

      1. associate-+l+ 58.6%

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

      2. log1p-define 60.1%

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

      3. +-commutative 60.1%

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

   10. Applied egg-rr 60.1%

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

   11. Taylor expanded in a around 0 58.2%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(1 + a \cdot \left(1 + 0.5 \cdot a\right)\right)} + b\right) \]
   12. Step-by-step derivation

       1. *-commutative 58.2%

          \[\leadsto \mathsf{log1p}\left(\left(1 + a \cdot \left(1 + \color{blue}{a \cdot 0.5}\right)\right) + b\right) \]

   13. Simplified 58.2%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(1 + a \cdot \left(1 + a \cdot 0.5\right)\right)} + b\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + \left(1 + a \cdot \left(1 + a \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

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 = log((2.0 + (b + (a * (1.0 + (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) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((2.0d0 + (b + (a * (1.0d0 + (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) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((2.0 + (b + (a * (1.0 + (a * 0.5))))));
	}
	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.log((2.0 + (b + (a * (1.0 + (a * 0.5))))))
	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 = log(Float64(2.0 + Float64(b + Float64(a * Float64(1.0 + 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) <= 0.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((2.0 + (b + (a * (1.0 + (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], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(2.0 + N[(b + N[(a * N[(1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.9%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0 60.0%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{a} + b \cdot \left(1 + 0.5 \cdot b\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 60.0%

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

      2. *-commutative 60.0%

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

   5. Simplified 60.0%

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

   6. Taylor expanded in b around 0 58.6%

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

      1. associate-+r+ 58.6%

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

   8. Simplified 58.6%

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

   9. Taylor expanded in a around 0 58.2%

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

4. Final simplification 68.5%

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

Alternative 8: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -350:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right) + b \cdot \left(0.5 + a \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -350.0)
   (/ b (+ (exp a) 1.0))
   (+ (log1p (exp a)) (* b (+ 0.5 (* a -0.25))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -350.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p(exp(a)) + (b * (0.5 + (a * -0.25)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -350.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p(Math.exp(a)) + (b * (0.5 + (a * -0.25)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -350.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(math.exp(a)) + (b * (0.5 + (a * -0.25)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -350.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log1p(exp(a)) + Float64(b * Float64(0.5 + Float64(a * -0.25))));
	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, -350.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision] + N[(b * N[(0.5 + N[(a * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -350

   1. Initial program 9.9%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if -350 < a

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0 59.6%

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

      1. log1p-define 61.1%

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

   5. Simplified 61.1%

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

   6. Taylor expanded in a around 0 61.2%

      \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \color{blue}{\left(-0.25 \cdot \left(a \cdot b\right) + 0.5 \cdot b\right)} \]
   7. Step-by-step derivation

      1. +-commutative 61.2%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \color{blue}{\left(0.5 \cdot b + -0.25 \cdot \left(a \cdot b\right)\right)} \]

      2. associate-*r* 61.2%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \left(0.5 \cdot b + \color{blue}{\left(-0.25 \cdot a\right) \cdot b}\right) \]

      3. distribute-rgt-out 61.2%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \color{blue}{b \cdot \left(0.5 + -0.25 \cdot a\right)} \]

   8. Simplified 61.2%

      \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \color{blue}{b \cdot \left(0.5 + -0.25 \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -350:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right) + b \cdot \left(0.5 + a \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.3% accurate, 1.4× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ 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) <= 0.0) {
		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) <= 0.0d0) 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) <= 0.0) {
		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) <= 0.0:
		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) <= 0.0)
		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) <= 0.0)
		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], 0.0], 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) < 0.0

   1. Initial program 9.9%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0 59.9%

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

      1. log1p-define 61.4%

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

   5. Simplified 61.4%

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

   6. Taylor expanded in a around 0 59.4%

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

      1. metadata-eval 59.4%

         \[\leadsto \log 2 + a \cdot \left(0.5 + \color{blue}{\left(0.125 - 0\right)} \cdot a\right) \]

      2. mul0-rgt 59.4%

         \[\leadsto \log 2 + a \cdot \left(0.5 + \left(0.125 - \color{blue}{b \cdot 0}\right) \cdot a\right) \]

      3. metadata-eval 59.4%

         \[\leadsto \log 2 + a \cdot \left(0.5 + \left(0.125 - b \cdot \color{blue}{\left(-0.125 + 0.125\right)}\right) \cdot a\right) \]

      4. distribute-rgt-out 59.4%

         \[\leadsto \log 2 + a \cdot \left(0.5 + \left(0.125 - \color{blue}{\left(-0.125 \cdot b + 0.125 \cdot b\right)}\right) \cdot a\right) \]

      5. *-commutative 59.4%

         \[\leadsto \log 2 + a \cdot \left(0.5 + \color{blue}{a \cdot \left(0.125 - \left(-0.125 \cdot b + 0.125 \cdot b\right)\right)}\right) \]

      6. distribute-rgt-out 59.4%

         \[\leadsto \log 2 + a \cdot \left(0.5 + a \cdot \left(0.125 - \color{blue}{b \cdot \left(-0.125 + 0.125\right)}\right)\right) \]

      7. metadata-eval 59.4%

         \[\leadsto \log 2 + a \cdot \left(0.5 + a \cdot \left(0.125 - b \cdot \color{blue}{0}\right)\right) \]

      8. mul0-rgt 59.4%

         \[\leadsto \log 2 + a \cdot \left(0.5 + a \cdot \left(0.125 - \color{blue}{0}\right)\right) \]

      9. metadata-eval 59.4%

         \[\leadsto \log 2 + a \cdot \left(0.5 + a \cdot \color{blue}{0.125}\right) \]

   8. Simplified 59.4%

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

4. Final simplification 69.4%

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

Alternative 10: 97.1% accurate, 1.4× speedup

Math

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

FPCore

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

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 = log(2.0) + (b * 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) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (b * 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) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (b * 0.5);
	}
	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.log(2.0) + (b * 0.5)
	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 = Float64(log(2.0) + Float64(b * 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) <= 0.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (b * 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], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.9%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0 59.6%

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

      1. log1p-define 61.1%

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

   5. Simplified 61.1%

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

   6. Taylor expanded in a around 0 58.7%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -140:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -140.0) (log1p b) (+ (log 2.0) (* b 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -140.0) {
		tmp = log1p(b);
	} else {
		tmp = log(2.0) + (b * 0.5);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -140.0) {
		tmp = Math.log1p(b);
	} else {
		tmp = Math.log(2.0) + (b * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -140.0:
		tmp = math.log1p(b)
	else:
		tmp = math.log(2.0) + (b * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -140.0)
		tmp = log1p(b);
	else
		tmp = Float64(log(2.0) + Float64(b * 0.5));
	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, -140.0], N[Log[1 + b], $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -140

   1. Initial program 9.9%

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

   3. Taylor expanded in b around 0 9.2%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{a} + b \cdot \left(1 + 0.5 \cdot b\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 9.2%

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

      2. *-commutative 9.2%

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

   5. Simplified 9.2%

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

   6. Taylor expanded in b around 0 8.4%

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

      1. associate-+r+ 8.4%

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

   8. Simplified 8.4%

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

      1. associate-+l+ 8.4%

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

      2. log1p-define 96.8%

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

      3. +-commutative 96.8%

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

   10. Applied egg-rr 96.8%

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

   11. Taylor expanded in b around inf 96.8%

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

   if -140 < a

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0 59.6%

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

      1. log1p-define 61.1%

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

   5. Simplified 61.1%

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

   6. Taylor expanded in a around 0 58.7%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -140:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -140:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -140.0) (log1p b) (log1p (+ b 1.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -140.0) {
		tmp = log1p(b);
	} else {
		tmp = log1p((b + 1.0));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -140.0) {
		tmp = Math.log1p(b);
	} else {
		tmp = Math.log1p((b + 1.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -140.0:
		tmp = math.log1p(b)
	else:
		tmp = math.log1p((b + 1.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -140.0)
		tmp = log1p(b);
	else
		tmp = log1p(Float64(b + 1.0));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -140.0], N[Log[1 + b], $MachinePrecision], N[Log[1 + N[(b + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -140

   1. Initial program 9.9%

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

   3. Taylor expanded in b around 0 9.2%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{a} + b \cdot \left(1 + 0.5 \cdot b\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 9.2%

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

      2. *-commutative 9.2%

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

   5. Simplified 9.2%

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

   6. Taylor expanded in b around 0 8.4%

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

      1. associate-+r+ 8.4%

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

   8. Simplified 8.4%

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

      1. associate-+l+ 8.4%

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

      2. log1p-define 96.8%

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

      3. +-commutative 96.8%

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

   10. Applied egg-rr 96.8%

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

   11. Taylor expanded in b around inf 96.8%

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

   if -140 < a

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0 60.0%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{a} + b \cdot \left(1 + 0.5 \cdot b\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 60.0%

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

      2. *-commutative 60.0%

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

   5. Simplified 60.0%

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

   6. Taylor expanded in b around 0 58.6%

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

      1. associate-+r+ 58.6%

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

   8. Simplified 58.6%

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

      1. associate-+l+ 58.6%

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

      2. log1p-define 60.1%

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

      3. +-commutative 60.1%

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

   10. Applied egg-rr 60.1%

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

   11. Taylor expanded in a around 0 57.7%

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

       1. +-commutative 57.7%

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

   13. Simplified 57.7%

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

Alternative 13: 95.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -140:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -140.0) (log1p b) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -140.0) {
		tmp = log1p(b);
	} else {
		tmp = log((b + 2.0));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -140.0) {
		tmp = Math.log1p(b);
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -140.0:
		tmp = math.log1p(b)
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -140.0)
		tmp = log1p(b);
	else
		tmp = log(Float64(b + 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, -140.0], N[Log[1 + b], $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -140

   1. Initial program 9.9%

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

   3. Taylor expanded in b around 0 9.2%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{a} + b \cdot \left(1 + 0.5 \cdot b\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 9.2%

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

      2. *-commutative 9.2%

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

   5. Simplified 9.2%

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

   6. Taylor expanded in b around 0 8.4%

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

      1. associate-+r+ 8.4%

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

   8. Simplified 8.4%

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

      1. associate-+l+ 8.4%

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

      2. log1p-define 96.8%

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

      3. +-commutative 96.8%

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

   10. Applied egg-rr 96.8%

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

   11. Taylor expanded in b around inf 96.8%

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

   if -140 < a

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0 60.0%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{a} + b \cdot \left(1 + 0.5 \cdot b\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 60.0%

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

      2. *-commutative 60.0%

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

   5. Simplified 60.0%

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

   6. Taylor expanded in b around 0 58.6%

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

      1. associate-+r+ 58.6%

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

   8. Simplified 58.6%

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

   9. Taylor expanded in a around 0 57.7%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -140:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 94.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -140:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -140.0) (log1p b) (log1p 1.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -140.0) {
		tmp = log1p(b);
	} else {
		tmp = log1p(1.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -140.0) {
		tmp = Math.log1p(b);
	} else {
		tmp = Math.log1p(1.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -140.0:
		tmp = math.log1p(b)
	else:
		tmp = math.log1p(1.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -140.0)
		tmp = log1p(b);
	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, -140.0], N[Log[1 + b], $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -140

   1. Initial program 9.9%

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

   3. Taylor expanded in b around 0 9.2%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{a} + b \cdot \left(1 + 0.5 \cdot b\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 9.2%

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

      2. *-commutative 9.2%

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

   5. Simplified 9.2%

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

   6. Taylor expanded in b around 0 8.4%

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

      1. associate-+r+ 8.4%

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

   8. Simplified 8.4%

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

      1. associate-+l+ 8.4%

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

      2. log1p-define 96.8%

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

      3. +-commutative 96.8%

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

   10. Applied egg-rr 96.8%

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

   11. Taylor expanded in b around inf 96.8%

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

   if -140 < a

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0 59.9%

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

      1. log1p-define 61.4%

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

   5. Simplified 61.4%

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

   6. Taylor expanded in a around 0 58.8%

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

Alternative 15: 48.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in b around 0 46.5%

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

   1. log1p-define 47.6%

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

5. Simplified 47.6%

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

6. Taylor expanded in a around 0 45.2%

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

Alternative 16: 1.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

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 = log(b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in b around 0 47.5%

   \[\leadsto \log \color{blue}{\left(1 + \left(e^{a} + b \cdot \left(1 + 0.5 \cdot b\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-+r+ 47.5%

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

   2. *-commutative 47.5%

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

5. Simplified 47.5%

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

6. Taylor expanded in b around 0 46.3%

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

   1. associate-+r+ 46.3%

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

8. Simplified 46.3%

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

9. Taylor expanded in b around inf 0.9%

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

    1. mul-1-neg 0.9%

       \[\leadsto \color{blue}{-\log \left(\frac{1}{b}\right)} \]

    2. log-rec 0.9%

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

    3. remove-double-neg 0.9%

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

11. Simplified 0.9%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))