symmetry log of sum of exp

Percentage Accurate: 53.6% → 99.0%

Time: 17.1s

Alternatives: 18

Speedup: 2.9×

• could not determine a ground truth

  1. a = 7.842827628618423e+78
  2. b = -2.2670212485353172e+135

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: 53.6% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;e^{a} \leq 0.99999995:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (if (<= (exp a) 0.99999995) (log1p (exp a)) (log1p (+ a (exp b))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else if (exp(a) <= 0.99999995) {
		tmp = log1p(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) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else if (Math.exp(a) <= 0.99999995) {
		tmp = Math.log1p(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) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	elif math.exp(a) <= 0.99999995:
		tmp = math.log1p(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) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	elseif (exp(a) <= 0.99999995)
		tmp = log1p(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], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[a], $MachinePrecision], 0.99999995], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision], N[Log[1 + N[(a + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-define 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 b around inf 98.4%

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

   if 0.0 < (exp.f64 a) < 0.999999949999999971

   1. Initial program 85.6%

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

   3. Taylor expanded in b around 0 85.6%

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

      1. log1p-define 99.7%

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

   5. Simplified 99.7%

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

   if 0.999999949999999971 < (exp.f64 a)

   1. Initial program 63.9%

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

   3. Taylor expanded in a around 0 62.6%

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

      1. +-commutative 62.6%

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

   5. Simplified 62.6%

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

      1. log1p-define 97.9%

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

   7. Applied egg-rr 97.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;e^{a} \leq 0.99999995:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;e^{a} \leq 0.99999995:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (if (<= (exp a) 0.99999995) (log1p (exp a)) (log1p (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else if (exp(a) <= 0.99999995) {
		tmp = log1p(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) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else if (Math.exp(a) <= 0.99999995) {
		tmp = Math.log1p(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) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	elif math.exp(a) <= 0.99999995:
		tmp = math.log1p(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) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	elseif (exp(a) <= 0.99999995)
		tmp = log1p(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], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[a], $MachinePrecision], 0.99999995], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-define 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 b around inf 98.4%

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

   if 0.0 < (exp.f64 a) < 0.999999949999999971

   1. Initial program 85.6%

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

   3. Taylor expanded in b around 0 85.6%

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

      1. log1p-define 99.7%

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

   5. Simplified 99.7%

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

   if 0.999999949999999971 < (exp.f64 a)

   1. Initial program 63.9%

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

   3. Taylor expanded in a around 0 60.8%

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

      1. log1p-define 61.3%

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

   5. Simplified 61.3%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;e^{a} \leq 0.99999995:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;e^{a} \leq 0.99999999:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \end{array} \]

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))
   (if (<= (exp a) 0.99999999)
     (log1p (exp a))
     (+ (log 2.0) (* b (+ 0.5 (* b 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 if (exp(a) <= 0.99999999) {
		tmp = log1p(exp(a));
	} else {
		tmp = log(2.0) + (b * (0.5 + (b * 0.125)));
	}
	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 if (Math.exp(a) <= 0.99999999) {
		tmp = Math.log1p(Math.exp(a));
	} else {
		tmp = Math.log(2.0) + (b * (0.5 + (b * 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)
	elif math.exp(a) <= 0.99999999:
		tmp = math.log1p(math.exp(a))
	else:
		tmp = math.log(2.0) + (b * (0.5 + (b * 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));
	elseif (exp(a) <= 0.99999999)
		tmp = log1p(exp(a));
	else
		tmp = Float64(log(2.0) + Float64(b * Float64(0.5 + Float64(b * 0.125))));
	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], If[LessEqual[N[Exp[a], $MachinePrecision], 0.99999999], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * N[(0.5 + N[(b * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-define 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 b around inf 98.4%

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

   if 0.0 < (exp.f64 a) < 0.99999998999999995

   1. Initial program 85.6%

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

   3. Taylor expanded in b around 0 85.6%

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

      1. log1p-define 99.7%

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

   5. Simplified 99.7%

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

   if 0.99999998999999995 < (exp.f64 a)

   1. Initial program 63.9%

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

   3. Taylor expanded in a around 0 60.8%

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

   4. Taylor expanded in b around 0 59.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;e^{a} \leq 0.99999999:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-define 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 b around inf 98.4%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 64.4%

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

4. Final simplification 72.1%

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

Alternative 5: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := e^{a} + 1\\ \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\log \left(t\_0 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (exp a) 1.0)))
   (if (<= (exp a) 0.0)
     (/ b t_0)
     (log (+ t_0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = exp(a) + 1.0;
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / t_0;
	} else {
		tmp = log((t_0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = exp(a) + 1.0d0
    if (exp(a) <= 0.0d0) then
        tmp = b / t_0
    else
        tmp = log((t_0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = Math.exp(a) + 1.0;
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / t_0;
	} else {
		tmp = Math.log((t_0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = math.exp(a) + 1.0
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / t_0
	else:
		tmp = math.log((t_0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(exp(a) + 1.0)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / t_0);
	else
		tmp = log(Float64(t_0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = exp(a) + 1.0;
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = b / t_0;
	else
		tmp = log((t_0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / t$95$0), $MachinePrecision], N[Log[N[(t$95$0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-define 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 b around inf 98.4%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 64.4%

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

   3. Taylor expanded in b around 0 60.9%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{a} + b \cdot \left(1 + b \cdot \left(0.5 + 0.16666666666666666 \cdot b\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 60.9%

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

      2. *-commutative 60.9%

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

   5. Simplified 60.9%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(e^{a} + 1\right) + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;e^{a} \leq 0.99999999:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \end{array} \]

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))
   (if (<= (exp a) 0.99999999)
     (+ (log 2.0) (* a (+ 0.5 (* a 0.125))))
     (+ (log 2.0) (* b (+ 0.5 (* b 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 if (exp(a) <= 0.99999999) {
		tmp = log(2.0) + (a * (0.5 + (a * 0.125)));
	} else {
		tmp = log(2.0) + (b * (0.5 + (b * 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 if (exp(a) <= 0.99999999d0) then
        tmp = log(2.0d0) + (a * (0.5d0 + (a * 0.125d0)))
    else
        tmp = log(2.0d0) + (b * (0.5d0 + (b * 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 if (Math.exp(a) <= 0.99999999) {
		tmp = Math.log(2.0) + (a * (0.5 + (a * 0.125)));
	} else {
		tmp = Math.log(2.0) + (b * (0.5 + (b * 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)
	elif math.exp(a) <= 0.99999999:
		tmp = math.log(2.0) + (a * (0.5 + (a * 0.125)))
	else:
		tmp = math.log(2.0) + (b * (0.5 + (b * 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));
	elseif (exp(a) <= 0.99999999)
		tmp = Float64(log(2.0) + Float64(a * Float64(0.5 + Float64(a * 0.125))));
	else
		tmp = Float64(log(2.0) + Float64(b * Float64(0.5 + Float64(b * 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);
	elseif (exp(a) <= 0.99999999)
		tmp = log(2.0) + (a * (0.5 + (a * 0.125)));
	else
		tmp = log(2.0) + (b * (0.5 + (b * 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], If[LessEqual[N[Exp[a], $MachinePrecision], 0.99999999], N[(N[Log[2.0], $MachinePrecision] + N[(a * N[(0.5 + N[(a * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * N[(0.5 + N[(b * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-define 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 b around inf 98.4%

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

   if 0.0 < (exp.f64 a) < 0.99999998999999995

   1. Initial program 85.6%

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

   3. Taylor expanded in b around 0 85.6%

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

      1. log1p-define 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in a around 0 67.2%

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

      1. *-commutative 67.2%

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

   8. Simplified 67.2%

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

   if 0.99999998999999995 < (exp.f64 a)

   1. Initial program 63.9%

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

   3. Taylor expanded in a around 0 60.8%

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

   4. Taylor expanded in b around 0 59.0%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;e^{a} \leq 0.99999999:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := e^{a} + 1\\ \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\log \left(t\_0 + b \cdot \left(1 + b \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (exp a) 1.0)))
   (if (<= (exp a) 0.0) (/ b t_0) (log (+ t_0 (* b (+ 1.0 (* b 0.5))))))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = exp(a) + 1.0;
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / t_0;
	} else {
		tmp = log((t_0 + (b * (1.0 + (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) :: t_0
    real(8) :: tmp
    t_0 = exp(a) + 1.0d0
    if (exp(a) <= 0.0d0) then
        tmp = b / t_0
    else
        tmp = log((t_0 + (b * (1.0d0 + (b * 0.5d0)))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = Math.exp(a) + 1.0;
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / t_0;
	} else {
		tmp = Math.log((t_0 + (b * (1.0 + (b * 0.5)))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = math.exp(a) + 1.0
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / t_0
	else:
		tmp = math.log((t_0 + (b * (1.0 + (b * 0.5)))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(exp(a) + 1.0)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / t_0);
	else
		tmp = log(Float64(t_0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = exp(a) + 1.0;
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = b / t_0;
	else
		tmp = log((t_0 + (b * (1.0 + (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_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / t$95$0), $MachinePrecision], N[Log[N[(t$95$0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-define 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 b around inf 98.4%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 64.4%

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

   3. Taylor expanded in b around 0 61.6%

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

      1. associate-+r+ 61.6%

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

      2. *-commutative 61.6%

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

   5. Simplified 61.6%

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

4. Final simplification 69.9%

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

Alternative 8: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.6%

   \[\log \left(e^{a} + e^{b}\right) \]
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 69.9%

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

5. Simplified 69.9%

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

6. Final simplification 69.9%

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

Alternative 9: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(b + 2\right) + a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

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 (+ (+ b 2.0) (* a (+ 1.0 (* a (+ 0.5 (* a 0.16666666666666666)))))))))

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(((b + 2.0) + (a * (1.0 + (a * (0.5 + (a * 0.16666666666666666)))))));
	}
	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(((b + 2.0d0) + (a * (1.0d0 + (a * (0.5d0 + (a * 0.16666666666666666d0)))))))
    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(((b + 2.0) + (a * (1.0 + (a * (0.5 + (a * 0.16666666666666666)))))));
	}
	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(((b + 2.0) + (a * (1.0 + (a * (0.5 + (a * 0.16666666666666666)))))))
	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(Float64(b + 2.0) + Float64(a * Float64(1.0 + Float64(a * Float64(0.5 + Float64(a * 0.16666666666666666)))))));
	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(((b + 2.0) + (a * (1.0 + (a * (0.5 + (a * 0.16666666666666666)))))));
	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[(N[(b + 2.0), $MachinePrecision] + N[(a * N[(1.0 + N[(a * N[(0.5 + N[(a * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-define 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 b around inf 98.4%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 64.4%

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

   3. Taylor expanded in b around 0 60.2%

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

      1. associate-+r+ 60.2%

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

      2. +-commutative 60.2%

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

   5. Simplified 60.2%

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

   6. Taylor expanded in a around 0 59.3%

      \[\leadsto \log \color{blue}{\left(2 + \left(b + a \cdot \left(1 + a \cdot \left(0.5 + 0.16666666666666666 \cdot a\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 59.3%

         \[\leadsto \log \color{blue}{\left(\left(2 + b\right) + a \cdot \left(1 + a \cdot \left(0.5 + 0.16666666666666666 \cdot a\right)\right)\right)} \]

      2. *-commutative 59.3%

         \[\leadsto \log \left(\left(2 + b\right) + a \cdot \left(1 + a \cdot \left(0.5 + \color{blue}{a \cdot 0.16666666666666666}\right)\right)\right) \]

   8. Simplified 59.3%

      \[\leadsto \log \color{blue}{\left(\left(2 + b\right) + a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(b + 2\right) + a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.1% accurate, 1.4× speedup

Math

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

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 7.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-define 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 b around inf 98.4%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 64.4%

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

   3. Taylor expanded in b around 0 60.3%

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

      1. log1p-define 60.7%

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

   5. Simplified 60.7%

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

   6. Taylor expanded in a around 0 59.7%

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

      1. *-commutative 59.7%

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

   8. Simplified 59.7%

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

4. Final simplification 68.5%

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

Alternative 11: 97.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = (b * 0.5) + 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 (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = (b * 0.5d0) + log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = (b * 0.5) + Math.log(2.0);
	}
	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 = (b * 0.5) + math.log(2.0)
	return tmp

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.7%

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

   3. Taylor expanded in b around 0 98.4%

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

      1. log1p-define 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 b around inf 98.4%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 64.4%

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

   3. Taylor expanded in b around 0 61.2%

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

      1. log1p-define 61.6%

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

   5. Simplified 61.6%

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

   6. Taylor expanded in a around 0 57.9%

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

4. Final simplification 67.1%

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

Alternative 12: 56.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+14}:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2e+14)
   (* b 0.5)
   (if (<= a -9e-8) (+ (log 2.0) (* a 0.5)) (+ (* b 0.5) (log 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2e+14) {
		tmp = b * 0.5;
	} else if (a <= -9e-8) {
		tmp = log(2.0) + (a * 0.5);
	} else {
		tmp = (b * 0.5) + 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 <= (-2d+14)) then
        tmp = b * 0.5d0
    else if (a <= (-9d-8)) then
        tmp = log(2.0d0) + (a * 0.5d0)
    else
        tmp = (b * 0.5d0) + 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 <= -2e+14) {
		tmp = b * 0.5;
	} else if (a <= -9e-8) {
		tmp = Math.log(2.0) + (a * 0.5);
	} else {
		tmp = (b * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2e+14:
		tmp = b * 0.5
	elif a <= -9e-8:
		tmp = math.log(2.0) + (a * 0.5)
	else:
		tmp = (b * 0.5) + math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2e+14)
		tmp = Float64(b * 0.5);
	elseif (a <= -9e-8)
		tmp = Float64(log(2.0) + Float64(a * 0.5));
	else
		tmp = Float64(Float64(b * 0.5) + 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 <= -2e+14)
		tmp = b * 0.5;
	elseif (a <= -9e-8)
		tmp = log(2.0) + (a * 0.5);
	else
		tmp = (b * 0.5) + 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, -2e+14], N[(b * 0.5), $MachinePrecision], If[LessEqual[a, -9e-8], N[(N[Log[2.0], $MachinePrecision] + N[(a * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(b * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2e14

   1. Initial program 8.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 a around 0 4.2%

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

   7. Taylor expanded in b around inf 18.8%

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

      1. *-commutative 18.8%

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

   9. Simplified 18.8%

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

   if -2e14 < a < -8.99999999999999986e-8

   1. Initial program 43.3%

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

   3. Taylor expanded in b around 0 43.3%

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

      1. log1p-define 51.9%

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

   5. Simplified 51.9%

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

   6. Taylor expanded in a around 0 22.3%

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

      1. +-commutative 22.3%

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

      2. *-commutative 22.3%

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

   8. Simplified 22.3%

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

   if -8.99999999999999986e-8 < a

   1. Initial program 64.1%

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

   3. Taylor expanded in b around 0 60.7%

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

      1. log1p-define 60.8%

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

   5. Simplified 60.8%

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

   6. Taylor expanded in a around 0 58.5%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+14}:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3700000000:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\log \left(a + 2\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3700000000.0)
   (* b 0.5)
   (if (<= a -9e-8) (log (+ a 2.0)) (+ (* b 0.5) (log 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3700000000.0) {
		tmp = b * 0.5;
	} else if (a <= -9e-8) {
		tmp = log((a + 2.0));
	} else {
		tmp = (b * 0.5) + 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 <= (-3700000000.0d0)) then
        tmp = b * 0.5d0
    else if (a <= (-9d-8)) then
        tmp = log((a + 2.0d0))
    else
        tmp = (b * 0.5d0) + 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 <= -3700000000.0) {
		tmp = b * 0.5;
	} else if (a <= -9e-8) {
		tmp = Math.log((a + 2.0));
	} else {
		tmp = (b * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3700000000.0:
		tmp = b * 0.5
	elif a <= -9e-8:
		tmp = math.log((a + 2.0))
	else:
		tmp = (b * 0.5) + math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3700000000.0)
		tmp = Float64(b * 0.5);
	elseif (a <= -9e-8)
		tmp = log(Float64(a + 2.0));
	else
		tmp = Float64(Float64(b * 0.5) + 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 <= -3700000000.0)
		tmp = b * 0.5;
	elseif (a <= -9e-8)
		tmp = log((a + 2.0));
	else
		tmp = (b * 0.5) + 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, -3700000000.0], N[(b * 0.5), $MachinePrecision], If[LessEqual[a, -9e-8], N[Log[N[(a + 2.0), $MachinePrecision]], $MachinePrecision], N[(N[(b * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.7e9

   1. Initial program 7.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 a around 0 4.2%

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

   7. Taylor expanded in b around inf 18.8%

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

      1. *-commutative 18.8%

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

   9. Simplified 18.8%

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

   if -3.7e9 < a < -8.99999999999999986e-8

   1. Initial program 48.8%

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

   3. Taylor expanded in a around 0 20.7%

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

      1. +-commutative 20.7%

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

   5. Simplified 20.7%

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

   6. Taylor expanded in b around 0 20.7%

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

   if -8.99999999999999986e-8 < a

   1. Initial program 64.1%

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

   3. Taylor expanded in b around 0 60.7%

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

      1. log1p-define 60.8%

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

   5. Simplified 60.8%

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

   6. Taylor expanded in a around 0 58.5%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3700000000:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\log \left(a + 2\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3700000000:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\log \left(a + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3700000000.0)
   (* b 0.5)
   (if (<= a -9e-8) (log (+ a 2.0)) (log1p (+ b 1.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3700000000.0) {
		tmp = b * 0.5;
	} else if (a <= -9e-8) {
		tmp = log((a + 2.0));
	} else {
		tmp = log1p((b + 1.0));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3700000000.0) {
		tmp = b * 0.5;
	} else if (a <= -9e-8) {
		tmp = Math.log((a + 2.0));
	} else {
		tmp = Math.log1p((b + 1.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3700000000.0:
		tmp = b * 0.5
	elif a <= -9e-8:
		tmp = math.log((a + 2.0))
	else:
		tmp = math.log1p((b + 1.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3700000000.0)
		tmp = Float64(b * 0.5);
	elseif (a <= -9e-8)
		tmp = log(Float64(a + 2.0));
	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, -3700000000.0], N[(b * 0.5), $MachinePrecision], If[LessEqual[a, -9e-8], N[Log[N[(a + 2.0), $MachinePrecision]], $MachinePrecision], N[Log[1 + N[(b + 1.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.7e9

   1. Initial program 7.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 a around 0 4.2%

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

   7. Taylor expanded in b around inf 18.8%

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

      1. *-commutative 18.8%

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

   9. Simplified 18.8%

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

   if -3.7e9 < a < -8.99999999999999986e-8

   1. Initial program 48.8%

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

   3. Taylor expanded in a around 0 20.7%

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

      1. +-commutative 20.7%

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

   5. Simplified 20.7%

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

   6. Taylor expanded in b around 0 20.7%

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

   if -8.99999999999999986e-8 < a

   1. Initial program 64.1%

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

   3. Taylor expanded in b around 0 59.7%

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

      1. associate-+r+ 59.7%

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

      2. +-commutative 59.7%

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

   5. Simplified 59.7%

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

   6. Taylor expanded in a around 0 59.3%

      \[\leadsto \log \color{blue}{\left(2 + \left(b + a \cdot \left(1 + a \cdot \left(0.5 + 0.16666666666666666 \cdot a\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 59.3%

         \[\leadsto \log \color{blue}{\left(\left(2 + b\right) + a \cdot \left(1 + a \cdot \left(0.5 + 0.16666666666666666 \cdot a\right)\right)\right)} \]

      2. *-commutative 59.3%

         \[\leadsto \log \left(\left(2 + b\right) + a \cdot \left(1 + a \cdot \left(0.5 + \color{blue}{a \cdot 0.16666666666666666}\right)\right)\right) \]

   8. Simplified 59.3%

      \[\leadsto \log \color{blue}{\left(\left(2 + b\right) + a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right)\right)} \]

   9. Taylor expanded in a around 0 57.5%

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

       1. +-commutative 57.5%

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

   11. Simplified 57.5%

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

       1. log1p-expm1-u 57.5%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(b + 2\right)\right)\right)} \]

       2. expm1-undefine 57.5%

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

       3. add-exp-log 57.5%

          \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(b + 2\right)} - 1\right) \]

   13. Applied egg-rr 57.5%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(b + 2\right) - 1\right)} \]
   14. Step-by-step derivation

       1. associate--l+ 57.5%

          \[\leadsto \mathsf{log1p}\left(\color{blue}{b + \left(2 - 1\right)}\right) \]

       2. metadata-eval 57.5%

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

   15. Simplified 57.5%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3700000000:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\log \left(a + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -3700000000:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\log \left(a + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -3700000000.0)
   (* b 0.5)
   (if (<= a -9e-8) (log (+ a 2.0)) (log (+ b 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -3700000000.0) {
		tmp = b * 0.5;
	} else if (a <= -9e-8) {
		tmp = log((a + 2.0));
	} else {
		tmp = log((b + 2.0));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-3700000000.0d0)) then
        tmp = b * 0.5d0
    else if (a <= (-9d-8)) then
        tmp = log((a + 2.0d0))
    else
        tmp = log((b + 2.0d0))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -3700000000.0) {
		tmp = b * 0.5;
	} else if (a <= -9e-8) {
		tmp = Math.log((a + 2.0));
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -3700000000.0:
		tmp = b * 0.5
	elif a <= -9e-8:
		tmp = math.log((a + 2.0))
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -3700000000.0)
		tmp = Float64(b * 0.5);
	elseif (a <= -9e-8)
		tmp = log(Float64(a + 2.0));
	else
		tmp = log(Float64(b + 2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3700000000.0)
		tmp = b * 0.5;
	elseif (a <= -9e-8)
		tmp = log((a + 2.0));
	else
		tmp = log((b + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -3700000000.0], N[(b * 0.5), $MachinePrecision], If[LessEqual[a, -9e-8], N[Log[N[(a + 2.0), $MachinePrecision]], $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.7e9

   1. Initial program 7.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 a around 0 4.2%

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

   7. Taylor expanded in b around inf 18.8%

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

      1. *-commutative 18.8%

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

   9. Simplified 18.8%

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

   if -3.7e9 < a < -8.99999999999999986e-8

   1. Initial program 48.8%

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

   3. Taylor expanded in a around 0 20.7%

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

      1. +-commutative 20.7%

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

   5. Simplified 20.7%

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

   6. Taylor expanded in b around 0 20.7%

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

   if -8.99999999999999986e-8 < a

   1. Initial program 64.1%

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

   3. Taylor expanded in a around 0 60.8%

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

   4. Taylor expanded in b around 0 57.5%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3700000000:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\log \left(a + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+30}:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(a + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.9e+30) (* b 0.5) (log (+ a 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.9e+30) {
		tmp = b * 0.5;
	} 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 <= (-2.9d+30)) then
        tmp = b * 0.5d0
    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 <= -2.9e+30) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log((a + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.9e+30:
		tmp = b * 0.5
	else:
		tmp = math.log((a + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.9e+30)
		tmp = Float64(b * 0.5);
	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 <= -2.9e+30)
		tmp = b * 0.5;
	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, -2.9e+30], N[(b * 0.5), $MachinePrecision], N[Log[N[(a + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.8999999999999998e30

   1. Initial program 8.1%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 a around 0 4.3%

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

   7. Taylor expanded in b around inf 18.8%

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

      1. *-commutative 18.8%

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

   9. Simplified 18.8%

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

   if -2.8999999999999998e30 < a

   1. Initial program 62.4%

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

   3. Taylor expanded in a around 0 60.1%

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

      1. +-commutative 60.1%

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

   5. Simplified 60.1%

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

   6. Taylor expanded in b around 0 56.6%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+30}:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(a + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 54.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{+66}:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -1.8e+66) (* b 0.5) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.8e+66) {
		tmp = b * 0.5;
	} 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 <= (-1.8d+66)) then
        tmp = b * 0.5d0
    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 <= -1.8e+66) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.8e+66:
		tmp = b * 0.5
	else:
		tmp = math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.8e+66)
		tmp = Float64(b * 0.5);
	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 <= -1.8e+66)
		tmp = b * 0.5;
	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, -1.8e+66], N[(b * 0.5), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.8e66

   1. Initial program 8.5%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 a around 0 4.1%

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

   7. Taylor expanded in b around inf 18.8%

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

      1. *-commutative 18.8%

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

   9. Simplified 18.8%

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

   if -1.8e66 < a

   1. Initial program 61.3%

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

   3. Taylor expanded in b around 0 57.4%

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

      1. log1p-define 57.7%

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

   5. Simplified 57.7%

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

   6. Taylor expanded in a around 0 54.4%

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

Alternative 18: 12.0% accurate, 101.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

   \[\log \left(e^{a} + e^{b}\right) \]
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 69.9%

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

5. Simplified 69.9%

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

6. Taylor expanded in a around 0 45.8%

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

7. Taylor expanded in b around inf 6.6%

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

   1. *-commutative 6.6%

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

9. Simplified 6.6%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))