symmetry log of sum of exp

Percentage Accurate: 53.9% → 99.1%

Time: 18.3s

Alternatives: 11

Speedup: 2.9×

• could not determine a ground truth

  1. a = 4.477272053044382e+69
  2. b = -7.944667550574141e+307

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: 53.9% 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: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-27) (/ b (+ 1.0 (exp a))) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-27) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((exp(a) + exp(b)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 2d-27) then
        tmp = b / (1.0d0 + exp(a))
    else
        tmp = log((exp(a) + exp(b)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 2e-27) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log((Math.exp(a) + Math.exp(b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 2e-27:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log((math.exp(a) + math.exp(b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-27)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(exp(a) + exp(b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 2e-27)
		tmp = b / (1.0 + exp(a));
	else
		tmp = log((exp(a) + exp(b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 2e-27], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2.0000000000000001e-27

   1. Initial program 7.4%

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

   3. Taylor expanded in b around 0 99.7%

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

      1. log1p-def 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 99.7%

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

   if 2.0000000000000001e-27 < (exp.f64 a)

   1. Initial program 73.3%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := 1 + e^{a}\\ \log t_0 + \left(0.5 \cdot \left({b}^{2} \cdot \left(\frac{1}{t_0} - \frac{1}{{t_0}^{2}}\right)\right) + \frac{b}{t_0}\right) \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp a))))
   (+
    (log t_0)
    (+
     (* 0.5 (* (pow b 2.0) (- (/ 1.0 t_0) (/ 1.0 (pow t_0 2.0)))))
     (/ b t_0)))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = 1.0 + exp(a);
	return log(t_0) + ((0.5 * (pow(b, 2.0) * ((1.0 / t_0) - (1.0 / pow(t_0, 2.0))))) + (b / t_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
    real(8) :: t_0
    t_0 = 1.0d0 + exp(a)
    code = log(t_0) + ((0.5d0 * ((b ** 2.0d0) * ((1.0d0 / t_0) - (1.0d0 / (t_0 ** 2.0d0))))) + (b / t_0))
end function

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = 1.0 + Math.exp(a);
	return Math.log(t_0) + ((0.5 * (Math.pow(b, 2.0) * ((1.0 / t_0) - (1.0 / Math.pow(t_0, 2.0))))) + (b / t_0));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = 1.0 + math.exp(a)
	return math.log(t_0) + ((0.5 * (math.pow(b, 2.0) * ((1.0 / t_0) - (1.0 / math.pow(t_0, 2.0))))) + (b / t_0))

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(1.0 + exp(a))
	return Float64(log(t_0) + Float64(Float64(0.5 * Float64((b ^ 2.0) * Float64(Float64(1.0 / t_0) - Float64(1.0 / (t_0 ^ 2.0))))) + Float64(b / t_0)))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	t_0 = 1.0 + exp(a);
	tmp = log(t_0) + ((0.5 * ((b ^ 2.0) * ((1.0 / t_0) - (1.0 / (t_0 ^ 2.0))))) + (b / t_0));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]}, N[(N[Log[t$95$0], $MachinePrecision] + N[(N[(0.5 * N[(N[Power[b, 2.0], $MachinePrecision] * N[(N[(1.0 / t$95$0), $MachinePrecision] - N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 58.9%

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

3. Taylor expanded in b around 0 76.8%

   \[\leadsto \color{blue}{\log \left(1 + e^{a}\right) + \left(0.5 \cdot \left({b}^{2} \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{b}{1 + e^{a}}\right)} \]

4. Final simplification 76.8%

   \[\leadsto \log \left(1 + e^{a}\right) + \left(0.5 \cdot \left({b}^{2} \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{b}{1 + e^{a}}\right) \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-27) (/ b (+ 1.0 (exp a))) (log1p (+ a (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-27) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log1p((a + exp(b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 2e-27) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log1p((a + Math.exp(b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 2e-27:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log1p((a + math.exp(b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-27)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log1p(Float64(a + exp(b)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 2e-27], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(a + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2.0000000000000001e-27

   1. Initial program 7.4%

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

   3. Taylor expanded in b around 0 99.7%

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

      1. log1p-def 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 99.7%

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

   if 2.0000000000000001e-27 < (exp.f64 a)

   1. Initial program 73.3%

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

   3. Taylor expanded in a around 0 72.0%

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

      1. +-commutative 72.0%

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

   5. Simplified 72.0%

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

   6. Taylor expanded in b around inf 72.0%

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

      1. log1p-def 97.3%

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

   8. Simplified 97.3%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.9%

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

3. Taylor expanded in b around 0 76.7%

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

   1. log1p-def 76.8%

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

5. Simplified 76.8%

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

6. Final simplification 76.8%

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

Alternative 5: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-27) (/ b (+ 1.0 (exp a))) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-27) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 2e-27) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 2e-27:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-27)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log1p(exp(b));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 2e-27], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2.0000000000000001e-27

   1. Initial program 7.4%

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

   3. Taylor expanded in b around 0 99.7%

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

      1. log1p-def 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 99.7%

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

   if 2.0000000000000001e-27 < (exp.f64 a)

   1. Initial program 73.3%

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

   3. Taylor expanded in a around 0 70.9%

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

      1. log1p-def 70.9%

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

   5. Simplified 70.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + 0.5 \cdot \left(a + b\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-27) (/ b (+ 1.0 (exp a))) (+ (log 2.0) (* 0.5 (+ a b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-27) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log(2.0) + (0.5 * (a + 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) <= 2d-27) then
        tmp = b / (1.0d0 + exp(a))
    else
        tmp = log(2.0d0) + (0.5d0 * (a + 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) <= 2e-27) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log(2.0) + (0.5 * (a + b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 2e-27:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log(2.0) + (0.5 * (a + b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-27)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = Float64(log(2.0) + Float64(0.5 * Float64(a + b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 2e-27)
		tmp = b / (1.0 + exp(a));
	else
		tmp = log(2.0) + (0.5 * (a + 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], 2e-27], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(0.5 * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2.0000000000000001e-27

   1. Initial program 7.4%

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

   3. Taylor expanded in b around 0 99.7%

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

      1. log1p-def 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 99.7%

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

   if 2.0000000000000001e-27 < (exp.f64 a)

   1. Initial program 73.3%

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

   3. Taylor expanded in b around 0 70.2%

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

      1. log1p-def 70.2%

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

   5. Simplified 70.2%

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

   6. Taylor expanded in a around 0 70.3%

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

   7. Taylor expanded in a around 0 69.8%

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

      1. distribute-lft-out 69.8%

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

   9. Simplified 69.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + 0.5 \cdot \left(a + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(a + \left(b + 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-27) (/ b (+ 1.0 (exp a))) (log (+ a (+ b 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-27) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((a + (b + 2.0)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 2d-27) then
        tmp = b / (1.0d0 + exp(a))
    else
        tmp = log((a + (b + 2.0d0)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 2e-27) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log((a + (b + 2.0)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 2e-27:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log((a + (b + 2.0)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-27)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(a + Float64(b + 2.0)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 2e-27)
		tmp = b / (1.0 + exp(a));
	else
		tmp = log((a + (b + 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 2e-27], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(a + N[(b + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2.0000000000000001e-27

   1. Initial program 7.4%

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

   3. Taylor expanded in b around 0 99.7%

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

      1. log1p-def 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 99.7%

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

   if 2.0000000000000001e-27 < (exp.f64 a)

   1. Initial program 73.3%

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

   3. Taylor expanded in a around 0 72.0%

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

      1. +-commutative 72.0%

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

   5. Simplified 72.0%

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

   6. Taylor expanded in b around 0 69.0%

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

      1. associate-+r+ 69.0%

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

      2. +-commutative 69.0%

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

      3. associate-+l+ 69.0%

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

   8. Simplified 69.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(a + \left(b + 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = 0.5 * b;
	} else {
		tmp = log((a + (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 <= (-1.0d0)) then
        tmp = 0.5d0 * b
    else
        tmp = log((a + (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 <= -1.0) {
		tmp = 0.5 * b;
	} else {
		tmp = Math.log((a + (b + 2.0)));
	}
	return tmp;
}

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 9.0%

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

   3. Taylor expanded in b around 0 98.0%

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

      1. log1p-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in a around 0 3.9%

      \[\leadsto \color{blue}{\log 2 + 0.5 \cdot b} \]
   7. Step-by-step derivation

      1. *-commutative 3.9%

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

   8. Simplified 3.9%

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

   9. Taylor expanded in b around inf 18.5%

      \[\leadsto \color{blue}{0.5 \cdot b} \]
   10. Step-by-step derivation

       1. *-commutative 18.5%

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

   11. Simplified 18.5%

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

   if -1 < a

   1. Initial program 73.2%

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

   3. Taylor expanded in a around 0 72.3%

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

      1. +-commutative 72.3%

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

   5. Simplified 72.3%

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

   6. Taylor expanded in b around 0 69.3%

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

      1. associate-+r+ 69.3%

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

      2. +-commutative 69.3%

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

      3. associate-+l+ 69.3%

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

   8. Simplified 69.3%

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

4. Final simplification 58.0%

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

Alternative 9: 56.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 9.0%

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

   3. Taylor expanded in b around 0 98.0%

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

      1. log1p-def 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in a around 0 3.9%

      \[\leadsto \color{blue}{\log 2 + 0.5 \cdot b} \]
   7. Step-by-step derivation

      1. *-commutative 3.9%

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

   8. Simplified 3.9%

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

   9. Taylor expanded in b around inf 18.5%

      \[\leadsto \color{blue}{0.5 \cdot b} \]
   10. Step-by-step derivation

       1. *-commutative 18.5%

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

   11. Simplified 18.5%

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

   if -1 < a

   1. Initial program 73.2%

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

   3. Taylor expanded in b around 0 70.1%

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

   4. Taylor expanded in a around 0 69.6%

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

      1. +-commutative 69.6%

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

   6. Simplified 69.6%

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

4. Final simplification 58.2%

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

Alternative 10: 56.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -62:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -62.0) (* 0.5 b) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -62.0) {
		tmp = 0.5 * b;
	} else {
		tmp = log(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 <= (-62.0d0)) then
        tmp = 0.5d0 * b
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -62.0) {
		tmp = 0.5 * b;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -62.0:
		tmp = 0.5 * b
	else:
		tmp = math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -62.0)
		tmp = Float64(0.5 * b);
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -62.0)
		tmp = 0.5 * b;
	else
		tmp = log(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, -62.0], N[(0.5 * b), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -62

   1. Initial program 7.4%

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

   3. Taylor expanded in b around 0 99.7%

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

      1. log1p-def 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 a around 0 3.9%

      \[\leadsto \color{blue}{\log 2 + 0.5 \cdot b} \]
   7. Step-by-step derivation

      1. *-commutative 3.9%

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

   8. Simplified 3.9%

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

   9. Taylor expanded in b around inf 18.8%

      \[\leadsto \color{blue}{0.5 \cdot b} \]
   10. Step-by-step derivation

       1. *-commutative 18.8%

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

   11. Simplified 18.8%

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

   if -62 < a

   1. Initial program 73.3%

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

   3. Taylor expanded in b around 0 69.7%

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

   4. Taylor expanded in a around 0 68.9%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -62:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 12.0% accurate, 101.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.9%

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

3. Taylor expanded in b around 0 76.7%

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

   1. log1p-def 76.8%

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

5. Simplified 76.8%

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

6. Taylor expanded in a around 0 55.1%

   \[\leadsto \color{blue}{\log 2 + 0.5 \cdot b} \]
7. Step-by-step derivation

   1. *-commutative 55.1%

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

8. Simplified 55.1%

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

9. Taylor expanded in b around inf 6.9%

   \[\leadsto \color{blue}{0.5 \cdot b} \]
10. Step-by-step derivation

    1. *-commutative 6.9%

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

11. Simplified 6.9%

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

12. Final simplification 6.9%

    \[\leadsto 0.5 \cdot b \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024023 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))