symmetry log of sum of exp

Percentage Accurate: 53.0% → 98.2%

Time: 14.5s

Alternatives: 12

Speedup: 1.5×

• Unsound rule application detected in e-graph. Results may not be sound.

• could not determine a ground truth

  1. a = 3.993279008596225e-227
  2. b = 5.539566819883392e+193

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: 53.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (log1p (+ (exp a) (expm1 b))))

C

assert(a < b);
double code(double a, double b) {
	return log1p((exp(a) + expm1(b)));
}

Java

assert a < b;
public static double code(double a, double b) {
	return Math.log1p((Math.exp(a) + Math.expm1(b)));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log1p((math.exp(a) + math.expm1(b)))

Julia

a, b = sort([a, b])
function code(a, b)
	return log1p(Float64(exp(a) + expm1(b)))
end

Wolfram

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

Derivation

1. Initial program 49.7%

   \[\log \left(e^{a} + e^{b}\right) \]
2. Step-by-step derivation

   1. add-sqr-sqrt 48.6%

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

   2. log-prod 49.0%

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

3. Applied egg-rr 49.0%

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

   1. log-prod 48.6%

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

   2. rem-square-sqrt 49.7%

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

   3. log1p-expm1 49.7%

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

   4. expm1-def 49.7%

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

   5. rem-exp-log 49.7%

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

   6. associate--l+ 50.3%

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

   7. expm1-def 72.3%

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

5. Simplified 72.3%

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

6. Final simplification 72.3%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-78) (/ b (+ (exp a) 1.0)) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-78) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((exp(a) + exp(b)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 5d-78) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((exp(a) + exp(b)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-78) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((Math.exp(a) + Math.exp(b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-78:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((math.exp(a) + math.exp(b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-78)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(exp(a) + exp(b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-78)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((exp(a) + exp(b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-78], 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) < 4.9999999999999996e-78

   1. Initial program 10.4%

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

   2. Taylor expanded in b around 0 98.6%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 98.6%

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

   if 4.9999999999999996e-78 < (exp.f64 a)

   1. Initial program 62.8%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

Alternative 3: 98.1% accurate, 1.0× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-8) (/ b (+ (exp a) 1.0)) (log1p (+ a (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-8) {
		tmp = b / (exp(a) + 1.0);
	} 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-8) {
		tmp = b / (Math.exp(a) + 1.0);
	} 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-8:
		tmp = b / (math.exp(a) + 1.0)
	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-8)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	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-8], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $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) < 2e-8

   1. Initial program 11.2%

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

   2. Taylor expanded in b around 0 98.1%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 97.1%

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

   if 2e-8 < (exp.f64 a)

   1. Initial program 62.8%

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

   2. Taylor expanded in a around 0 61.8%

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

      1. associate-+r+ 61.8%

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

      2. +-commutative 61.8%

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

   4. Simplified 61.8%

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

   5. Taylor expanded in b around inf 61.8%

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

      1. log1p-def 98.3%

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

   7. Simplified 98.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + e^{b}\right)\\ \end{array} \]

Alternative 4: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-297}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-297) (/ b (+ (exp a) 1.0)) (log1p (exp a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-297) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p(exp(a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-297) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p(Math.exp(a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-297:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(math.exp(a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-297)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(exp(a));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-297], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 5e-297

   1. Initial program 10.5%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 100.0%

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

   if 5e-297 < (exp.f64 a)

   1. Initial program 62.5%

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

   2. Taylor expanded in b around 0 59.7%

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

      1. log1p-def 60.4%

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

   4. Simplified 60.4%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-297}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]

Alternative 5: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-78) (/ b (+ (exp a) 1.0)) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-78) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-78) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-78:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-78)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(exp(b));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-78], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.9999999999999996e-78

   1. Initial program 10.4%

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

   2. Taylor expanded in b around 0 98.6%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 98.6%

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

   if 4.9999999999999996e-78 < (exp.f64 a)

   1. Initial program 62.8%

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

   2. Taylor expanded in a around 0 60.2%

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

      1. log1p-def 60.2%

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

   4. Simplified 60.2%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternative 6: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + \left(2 + 0.5 \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-78)
   (/ b (+ (exp a) 1.0))
   (log (+ b (+ 2.0 (* 0.5 (* b b)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-78) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((b + (2.0 + (0.5 * (b * b)))));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 5d-78) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((b + (2.0d0 + (0.5d0 * (b * b)))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-78) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((b + (2.0 + (0.5 * (b * b)))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-78:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((b + (2.0 + (0.5 * (b * b)))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-78)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(b + Float64(2.0 + Float64(0.5 * Float64(b * b)))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-78)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((b + (2.0 + (0.5 * (b * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-78], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(b + N[(2.0 + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.9999999999999996e-78

   1. Initial program 10.4%

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

   2. Taylor expanded in b around 0 98.6%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 98.6%

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

   if 4.9999999999999996e-78 < (exp.f64 a)

   1. Initial program 62.8%

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

   2. Taylor expanded in b around 0 60.3%

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

      1. +-commutative 60.3%

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

      2. associate-+r+ 60.3%

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

      3. associate-+l+ 60.3%

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

      4. +-commutative 60.3%

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

      5. associate-+l+ 60.3%

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

      6. fma-def 60.3%

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

      7. unpow2 60.3%

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

   4. Simplified 60.3%

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

   5. Taylor expanded in a around 0 58.2%

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

      1. unpow2 58.2%

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

   7. Simplified 58.2%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + \left(2 + 0.5 \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 7: 97.4% accurate, 1.5× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-8) (/ b (+ (exp a) 1.0)) (log (+ b (+ a 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-8) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((b + (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 (exp(a) <= 2d-8) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((b + (a + 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-8) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((b + (a + 2.0)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 2e-8:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((b + (a + 2.0)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-8)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(b + 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 (exp(a) <= 2e-8)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((b + (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[N[Exp[a], $MachinePrecision], 2e-8], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(b + N[(a + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2e-8

   1. Initial program 11.2%

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

   2. Taylor expanded in b around 0 98.1%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 97.1%

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

   if 2e-8 < (exp.f64 a)

   1. Initial program 62.8%

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

   2. Taylor expanded in b around 0 58.9%

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

      1. associate-+r+ 58.9%

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

      2. +-commutative 58.9%

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

      3. associate-+l+ 58.9%

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

   4. Simplified 58.9%

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

   5. Taylor expanded in a around 0 57.9%

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

      1. +-commutative 57.9%

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

      2. +-commutative 57.9%

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

      3. associate-+l+ 57.9%

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

   7. Simplified 57.9%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + \left(a + 2\right)\right)\\ \end{array} \]

Alternative 8: 96.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

2. Taylor expanded in b around 0 46.3%

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

   1. associate-+r+ 46.3%

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

   2. +-commutative 46.3%

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

   3. associate-+l+ 46.3%

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

4. Simplified 46.3%

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

5. Taylor expanded in a around inf 46.3%

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

   1. log1p-def 68.5%

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

   2. +-commutative 68.5%

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

7. Simplified 68.5%

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

8. Final simplification 68.5%

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

Alternative 9: 48.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ b \cdot 0.5 + \log 2 \end{array} \]

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return (b * 0.5) + log(2.0);
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (b * 0.5d0) + log(2.0d0)
end function

Java

assert a < b;
public static double code(double a, double b) {
	return (b * 0.5) + Math.log(2.0);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (b * 0.5) + math.log(2.0)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(b * 0.5) + log(2.0))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (b * 0.5) + log(2.0);
end

Wolfram

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

Derivation

1. Initial program 49.7%

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

2. Taylor expanded in b around 0 47.4%

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

   1. +-commutative 47.4%

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

   2. associate-+r+ 47.4%

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

   3. associate-+l+ 47.4%

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

   4. +-commutative 47.4%

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

   5. associate-+l+ 47.4%

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

   6. fma-def 47.4%

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

   7. unpow2 47.4%

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

4. Simplified 47.4%

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

5. Taylor expanded in a around 0 44.6%

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

   1. unpow2 44.6%

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

7. Simplified 44.6%

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

8. Taylor expanded in b around 0 44.4%

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

9. Final simplification 44.4%

   \[\leadsto b \cdot 0.5 + \log 2 \]

Alternative 10: 48.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \log \left(b + 2\right) \end{array} \]

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = log((b + 2.0d0))
end function

Java

assert a < b;
public static double code(double a, double b) {
	return Math.log((b + 2.0));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log((b + 2.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

2. Taylor expanded in b around 0 46.3%

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

   1. associate-+r+ 46.3%

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

   2. +-commutative 46.3%

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

   3. associate-+l+ 46.3%

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

4. Simplified 46.3%

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

5. Taylor expanded in a around 0 43.5%

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

   1. +-commutative 43.5%

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

7. Simplified 43.5%

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

8. Final simplification 43.5%

   \[\leadsto \log \left(b + 2\right) \]

Alternative 11: 48.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = log(2.0d0)
end function

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log(2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

2. Taylor expanded in b around 0 47.4%

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

   1. +-commutative 47.4%

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

   2. associate-+r+ 47.4%

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

   3. associate-+l+ 47.4%

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

   4. +-commutative 47.4%

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

   5. associate-+l+ 47.4%

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

   6. fma-def 47.4%

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

   7. unpow2 47.4%

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

4. Simplified 47.4%

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

5. Taylor expanded in a around 0 44.6%

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

   1. unpow2 44.6%

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

7. Simplified 44.6%

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

8. Taylor expanded in b around 0 44.4%

   \[\leadsto \color{blue}{\log 2} \]

9. Final simplification 44.4%

   \[\leadsto \log 2 \]

Alternative 12: 2.6% accurate, 60.6× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a / (b + 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

2. Taylor expanded in b around 0 46.3%

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

   1. associate-+r+ 46.3%

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

   2. +-commutative 46.3%

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

   3. associate-+l+ 46.3%

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

4. Simplified 46.3%

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

5. Taylor expanded in a around 0 43.9%

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

6. Taylor expanded in a around inf 3.9%

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

   1. +-commutative 3.9%

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

8. Simplified 3.9%

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

9. Final simplification 3.9%

   \[\leadsto \frac{a}{b + 2} \]

Reproduce

herbie shell --seed 2023174 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))