symmetry log of sum of exp

Percentage Accurate: 52.8% → 99.0%

Time: 15.7s

Alternatives: 14

Speedup: 2.9×

• could not determine a ground truth

  1. a = 3.882707939318901e+59
  2. b = 7.051952075657564e-94

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: 52.8% 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, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -38:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -38.0) (/ b (+ (exp a) 1.0)) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -38.0) {
		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 (a <= (-38.0d0)) 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 (a <= -38.0) {
		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 a <= -38.0:
		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 (a <= -38.0)
		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 (a <= -38.0)
		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[a, -38.0], 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 a < -38

   1. Initial program 10.0%

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

      1. add-sqr-sqrt 9.9%

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

      2. log-prod 10.0%

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

   3. Applied egg-rr 10.0%

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

      1. log-prod 9.9%

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

      2. rem-square-sqrt 10.0%

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

      3. log1p-expm1 9.8%

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

      4. expm1-def 9.8%

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

      5. rem-exp-log 9.8%

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

      6. associate--l+ 11.4%

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

      7. expm1-def 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around 0 97.8%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 97.8%

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

   if -38 < a

   1. Initial program 69.7%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -38:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p((exp(a) + b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p((Math.exp(a) + b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p((math.exp(a) + b))
	return tmp

Julia

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

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(N[Exp[a], $MachinePrecision] + b), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.6%

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

      1. add-sqr-sqrt 9.6%

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

      2. log-prod 9.7%

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

   3. Applied egg-rr 9.7%

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

      1. log-prod 9.6%

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

      2. rem-square-sqrt 9.6%

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

      3. log1p-expm1 9.5%

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

      4. expm1-def 9.5%

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

      5. rem-exp-log 9.5%

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

      6. associate--l+ 9.5%

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

      7. expm1-def 96.7%

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

   5. Simplified 96.7%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 69.2%

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

      1. add-sqr-sqrt 67.7%

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

      2. log-prod 68.2%

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

   3. Applied egg-rr 68.2%

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

      1. log-prod 67.7%

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

      2. rem-square-sqrt 69.2%

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

      3. log1p-expm1 69.2%

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

      4. expm1-def 69.2%

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

      5. rem-exp-log 69.2%

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

      6. associate--l+ 69.7%

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

      7. expm1-def 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in b around 0 65.9%

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

4. Final simplification 74.0%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-21) (/ b (+ (exp a) 1.0)) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-21) {
		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) <= 2e-21) {
		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) <= 2e-21:
		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) <= 2e-21)
		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], 2e-21], 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) < 1.99999999999999982e-21

   1. Initial program 10.0%

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

      1. add-sqr-sqrt 9.9%

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

      2. log-prod 10.0%

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

   3. Applied egg-rr 10.0%

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

      1. log-prod 9.9%

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

      2. rem-square-sqrt 10.0%

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

      3. log1p-expm1 9.8%

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

      4. expm1-def 9.8%

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

      5. rem-exp-log 9.8%

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

      6. associate--l+ 11.4%

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

      7. expm1-def 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around 0 97.8%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 97.8%

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

   if 1.99999999999999982e-21 < (exp.f64 a)

   1. Initial program 69.7%

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

   2. Taylor expanded in a around 0 66.9%

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

      1. log1p-def 67.0%

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

   4. Simplified 67.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternative 4: 98.1% 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 55.0%

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

   1. add-sqr-sqrt 53.9%

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

   2. log-prod 54.3%

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

3. Applied egg-rr 54.3%

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

   1. log-prod 53.9%

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

   2. rem-square-sqrt 55.0%

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

   3. log1p-expm1 55.0%

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

   4. expm1-def 55.0%

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

   5. rem-exp-log 55.0%

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

   6. associate--l+ 55.4%

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

   7. expm1-def 76.4%

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

5. Simplified 76.4%

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

6. Final simplification 76.4%

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

Alternative 5: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := 2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log t_0 + \frac{a}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ b (* 0.5 (* b b))))))
   (if (<= (exp a) 2e-21) (/ b (+ (exp a) 1.0)) (+ (log t_0) (/ a t_0)))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = 2.0 + (b + (0.5 * (b * b)));
	double tmp;
	if (exp(a) <= 2e-21) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(t_0) + (a / t_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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (b + (0.5d0 * (b * b)))
    if (exp(a) <= 2d-21) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(t_0) + (a / t_0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = 2.0 + (b + (0.5 * (b * b)));
	double tmp;
	if (Math.exp(a) <= 2e-21) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(t_0) + (a / t_0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = 2.0 + (b + (0.5 * (b * b)))
	tmp = 0
	if math.exp(a) <= 2e-21:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(t_0) + (a / t_0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(2.0 + Float64(b + Float64(0.5 * Float64(b * b))))
	tmp = 0.0
	if (exp(a) <= 2e-21)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(t_0) + Float64(a / t_0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = 2.0 + (b + (0.5 * (b * b)));
	tmp = 0.0;
	if (exp(a) <= 2e-21)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(t_0) + (a / t_0);
	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[(2.0 + N[(b + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[a], $MachinePrecision], 2e-21], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[t$95$0], $MachinePrecision] + N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1.99999999999999982e-21

   1. Initial program 10.0%

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

      1. add-sqr-sqrt 9.9%

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

      2. log-prod 10.0%

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

   3. Applied egg-rr 10.0%

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

      1. log-prod 9.9%

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

      2. rem-square-sqrt 10.0%

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

      3. log1p-expm1 9.8%

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

      4. expm1-def 9.8%

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

      5. rem-exp-log 9.8%

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

      6. associate--l+ 11.4%

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

      7. expm1-def 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around 0 97.8%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 97.8%

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

   if 1.99999999999999982e-21 < (exp.f64 a)

   1. Initial program 69.7%

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

   2. Taylor expanded in b around 0 66.8%

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

      1. unpow2 66.8%

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

   4. Simplified 66.8%

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

   5. Taylor expanded in a around 0 65.7%

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

      1. unpow2 65.7%

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

      2. unpow2 65.7%

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

   7. Simplified 65.7%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right) + \frac{a}{2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]

Alternative 6: 97.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -45:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + \left(a + 0.5 \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -45.0) (/ b (+ (exp a) 1.0)) (log (+ 2.0 (+ a (* 0.5 (* a a)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -45.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((2.0 + (a + (0.5 * (a * a)))));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-45.0d0)) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((2.0d0 + (a + (0.5d0 * (a * a)))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -45.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((2.0 + (a + (0.5 * (a * a)))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -45.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((2.0 + (a + (0.5 * (a * a)))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -45.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(2.0 + Float64(a + Float64(0.5 * Float64(a * a)))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -45.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((2.0 + (a + (0.5 * (a * a)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -45.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(2.0 + N[(a + N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -45

   1. Initial program 10.0%

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

      1. add-sqr-sqrt 9.9%

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

      2. log-prod 10.0%

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

   3. Applied egg-rr 10.0%

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

      1. log-prod 9.9%

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

      2. rem-square-sqrt 10.0%

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

      3. log1p-expm1 9.8%

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

      4. expm1-def 9.8%

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

      5. rem-exp-log 9.8%

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

      6. associate--l+ 11.4%

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

      7. expm1-def 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around 0 97.8%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 97.8%

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

   if -45 < a

   1. Initial program 69.7%

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

   2. Taylor expanded in b around 0 65.9%

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

   3. Taylor expanded in a around 0 65.0%

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

      1. unpow2 65.0%

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

   5. Simplified 65.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -45:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + \left(a + 0.5 \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 7: 97.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -22:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -22.0) (/ b (+ (exp a) 1.0)) (log (+ 2.0 (+ b (* 0.5 (* b b)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -22.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((2.0 + (b + (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 (a <= (-22.0d0)) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((2.0d0 + (b + (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 (a <= -22.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((2.0 + (b + (0.5 * (b * b)))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -22.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((2.0 + (b + (0.5 * (b * b)))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -22.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(2.0 + Float64(b + 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 (a <= -22.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((2.0 + (b + (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[a, -22.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(2.0 + N[(b + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -22

   1. Initial program 10.0%

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

      1. add-sqr-sqrt 9.9%

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

      2. log-prod 10.0%

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

   3. Applied egg-rr 10.0%

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

      1. log-prod 9.9%

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

      2. rem-square-sqrt 10.0%

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

      3. log1p-expm1 9.8%

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

      4. expm1-def 9.8%

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

      5. rem-exp-log 9.8%

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

      6. associate--l+ 11.4%

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

      7. expm1-def 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around 0 97.8%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 97.8%

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

   if -22 < a

   1. Initial program 69.7%

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

   2. Taylor expanded in b around 0 66.8%

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

      1. unpow2 66.8%

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

   4. Simplified 66.8%

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

   5. Taylor expanded in a around 0 65.2%

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

      1. unpow2 65.2%

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

   7. Simplified 65.2%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -22:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 8: 97.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -28:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -28.0)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (* b (+ 0.5 (* b 0.125))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -28.0) {
		tmp = b / (exp(a) + 1.0);
	} 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 (a <= (-28.0d0)) then
        tmp = b / (exp(a) + 1.0d0)
    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 (a <= -28.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} 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 a <= -28.0:
		tmp = b / (math.exp(a) + 1.0)
	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 (a <= -28.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	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 (a <= -28.0)
		tmp = b / (exp(a) + 1.0);
	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[a, -28.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $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 2 regimes

2. if a < -28

   1. Initial program 10.0%

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

      1. add-sqr-sqrt 9.9%

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

      2. log-prod 10.0%

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

   3. Applied egg-rr 10.0%

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

      1. log-prod 9.9%

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

      2. rem-square-sqrt 10.0%

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

      3. log1p-expm1 9.8%

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

      4. expm1-def 9.8%

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

      5. rem-exp-log 9.8%

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

      6. associate--l+ 11.4%

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

      7. expm1-def 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around 0 97.8%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 97.8%

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

   if -28 < a

   1. Initial program 69.7%

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

   2. Taylor expanded in b around 0 66.8%

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

      1. unpow2 66.8%

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

   4. Simplified 66.8%

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

   5. Taylor expanded in a around 0 65.2%

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

      1. unpow2 65.2%

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

   7. Simplified 65.2%

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

   8. Taylor expanded in b around 0 65.1%

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

      1. +-commutative 65.1%

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

      2. unpow2 65.1%

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

      3. associate-*r* 65.1%

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

      4. distribute-rgt-out 65.1%

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

      5. *-commutative 65.1%

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

   10. Simplified 65.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -28:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \]

Alternative 9: 56.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -48:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -48.0) (* b 0.5) (+ (log 2.0) (* b 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -48.0) {
		tmp = b * 0.5;
	} else {
		tmp = log(2.0) + (b * 0.5);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-48.0d0)) then
        tmp = b * 0.5d0
    else
        tmp = log(2.0d0) + (b * 0.5d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -48.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log(2.0) + (b * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -48.0:
		tmp = b * 0.5
	else:
		tmp = math.log(2.0) + (b * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -48.0)
		tmp = Float64(b * 0.5);
	else
		tmp = Float64(log(2.0) + Float64(b * 0.5));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -48.0)
		tmp = b * 0.5;
	else
		tmp = log(2.0) + (b * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -48.0], N[(b * 0.5), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -48

   1. Initial program 10.0%

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

      1. add-sqr-sqrt 9.9%

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

      2. log-prod 10.0%

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

   3. Applied egg-rr 10.0%

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

      1. log-prod 9.9%

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

      2. rem-square-sqrt 10.0%

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

      3. log1p-expm1 9.8%

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

      4. expm1-def 9.8%

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

      5. rem-exp-log 9.8%

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

      6. associate--l+ 11.4%

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

      7. expm1-def 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around 0 97.8%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in a around 0 4.7%

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

       1. *-commutative 4.7%

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

   11. Simplified 4.7%

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

   12. Taylor expanded in b around inf 18.5%

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

   if -48 < a

   1. Initial program 69.7%

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

      1. add-sqr-sqrt 68.2%

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

      2. log-prod 68.7%

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

   3. Applied egg-rr 68.7%

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

      1. log-prod 68.2%

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

      2. rem-square-sqrt 69.7%

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

      3. log1p-expm1 69.7%

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

      4. expm1-def 69.7%

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

      5. rem-exp-log 69.7%

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

      6. associate--l+ 69.7%

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

      7. expm1-def 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in b around 0 66.5%

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

      1. log1p-def 66.5%

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

      2. +-commutative 66.5%

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

   8. Simplified 66.5%

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

   9. Taylor expanded in a around 0 64.8%

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

       1. *-commutative 64.8%

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

   11. Simplified 64.8%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -48:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \]

Alternative 10: 97.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -20:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -20.0) (/ b (+ (exp a) 1.0)) (+ (log 2.0) (* b 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -20.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (b * 0.5);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-20.0d0)) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (b * 0.5d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -20.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (b * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -20.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (b * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -20.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(b * 0.5));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -20.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (b * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -20.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -20

   1. Initial program 10.0%

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

      1. add-sqr-sqrt 9.9%

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

      2. log-prod 10.0%

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

   3. Applied egg-rr 10.0%

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

      1. log-prod 9.9%

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

      2. rem-square-sqrt 10.0%

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

      3. log1p-expm1 9.8%

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

      4. expm1-def 9.8%

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

      5. rem-exp-log 9.8%

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

      6. associate--l+ 11.4%

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

      7. expm1-def 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around 0 97.8%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 97.8%

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

   if -20 < a

   1. Initial program 69.7%

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

      1. add-sqr-sqrt 68.2%

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

      2. log-prod 68.7%

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

   3. Applied egg-rr 68.7%

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

      1. log-prod 68.2%

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

      2. rem-square-sqrt 69.7%

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

      3. log1p-expm1 69.7%

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

      4. expm1-def 69.7%

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

      5. rem-exp-log 69.7%

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

      6. associate--l+ 69.7%

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

      7. expm1-def 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in b around 0 66.5%

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

      1. log1p-def 66.5%

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

      2. +-commutative 66.5%

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

   8. Simplified 66.5%

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

   9. Taylor expanded in a around 0 64.8%

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

       1. *-commutative 64.8%

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

   11. Simplified 64.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -20:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \]

Alternative 11: 56.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		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 <= (-1.0d0)) 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 <= -1.0) {
		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 <= -1.0:
		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 <= -1.0)
		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 <= -1.0)
		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, -1.0], N[(b * 0.5), $MachinePrecision], N[Log[N[(a + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 10.0%

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

      1. add-sqr-sqrt 9.9%

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

      2. log-prod 10.0%

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

   3. Applied egg-rr 10.0%

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

      1. log-prod 9.9%

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

      2. rem-square-sqrt 10.0%

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

      3. log1p-expm1 9.8%

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

      4. expm1-def 9.8%

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

      5. rem-exp-log 9.8%

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

      6. associate--l+ 11.4%

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

      7. expm1-def 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around 0 97.8%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in a around 0 4.7%

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

       1. *-commutative 4.7%

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

   11. Simplified 4.7%

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

   12. Taylor expanded in b around inf 18.5%

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

   if -1 < a

   1. Initial program 69.7%

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

   2. Taylor expanded in b around 0 65.9%

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

   3. Taylor expanded in a around 0 64.7%

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

      1. +-commutative 64.7%

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

   5. Simplified 64.7%

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

4. Final simplification 53.3%

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

Alternative 12: 56.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -48:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -48.0) (* b 0.5) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -48.0) {
		tmp = b * 0.5;
	} 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 <= (-48.0d0)) then
        tmp = b * 0.5d0
    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 <= -48.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -48.0:
		tmp = b * 0.5
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -48.0)
		tmp = Float64(b * 0.5);
	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 <= -48.0)
		tmp = b * 0.5;
	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, -48.0], N[(b * 0.5), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -48

   1. Initial program 10.0%

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

      1. add-sqr-sqrt 9.9%

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

      2. log-prod 10.0%

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

   3. Applied egg-rr 10.0%

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

      1. log-prod 9.9%

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

      2. rem-square-sqrt 10.0%

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

      3. log1p-expm1 9.8%

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

      4. expm1-def 9.8%

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

      5. rem-exp-log 9.8%

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

      6. associate--l+ 11.4%

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

      7. expm1-def 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around 0 97.8%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in a around 0 4.7%

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

       1. *-commutative 4.7%

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

   11. Simplified 4.7%

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

   12. Taylor expanded in b around inf 18.5%

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

   if -48 < a

   1. Initial program 69.7%

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

   2. Taylor expanded in b around 0 66.8%

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

      1. unpow2 66.8%

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

   4. Simplified 66.8%

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

   5. Taylor expanded in a around 0 65.2%

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

      1. unpow2 65.2%

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

   7. Simplified 65.2%

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

   8. Taylor expanded in b around 0 63.9%

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

      1. +-commutative 63.9%

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

   10. Simplified 63.9%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -48:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]

Alternative 13: 55.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -48:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -48.0) (* b 0.5) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -48.0) {
		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 <= (-48.0d0)) 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 <= -48.0) {
		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 <= -48.0:
		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 <= -48.0)
		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 <= -48.0)
		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, -48.0], N[(b * 0.5), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -48

   1. Initial program 10.0%

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

      1. add-sqr-sqrt 9.9%

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

      2. log-prod 10.0%

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

   3. Applied egg-rr 10.0%

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

      1. log-prod 9.9%

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

      2. rem-square-sqrt 10.0%

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

      3. log1p-expm1 9.8%

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

      4. expm1-def 9.8%

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

      5. rem-exp-log 9.8%

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

      6. associate--l+ 11.4%

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

      7. expm1-def 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in b around 0 97.8%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in a around 0 4.7%

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

       1. *-commutative 4.7%

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

   11. Simplified 4.7%

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

   12. Taylor expanded in b around inf 18.5%

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

   if -48 < a

   1. Initial program 69.7%

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

   2. Taylor expanded in b around 0 65.9%

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

      1. log1p-def 65.9%

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

   4. Simplified 65.9%

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

   5. Taylor expanded in a around 0 64.2%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -48:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 14: 12.1% 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 55.0%

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

   1. add-sqr-sqrt 53.9%

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

   2. log-prod 54.3%

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

3. Applied egg-rr 54.3%

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

   1. log-prod 53.9%

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

   2. rem-square-sqrt 55.0%

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

   3. log1p-expm1 55.0%

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

   4. expm1-def 55.0%

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

   5. rem-exp-log 55.0%

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

   6. associate--l+ 55.4%

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

   7. expm1-def 76.4%

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

5. Simplified 76.4%

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

6. Taylor expanded in b around 0 74.2%

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

   1. log1p-def 74.7%

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

   2. +-commutative 74.7%

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

8. Simplified 74.7%

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

9. Taylor expanded in a around 0 50.0%

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

    1. *-commutative 50.0%

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

11. Simplified 50.0%

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

12. Taylor expanded in b around inf 7.5%

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

13. Final simplification 7.5%

    \[\leadsto b \cdot 0.5 \]

Reproduce

herbie shell --seed 2023292 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))