symmetry log of sum of exp

Percentage Accurate: 53.5% → 98.9%

Time: 15.1s

Alternatives: 16

Speedup: 1.5×

• could not determine a ground truth

  1. a = 6.556993890603645e+79
  2. b = 6.027587650062667e+178

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: 53.5% 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: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.0%

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

   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 0.0 < (exp.f64 a)

   1. Initial program 72.3%

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

4. Final simplification 78.3%

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

Alternative 2: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{a} + 1\right)\right)} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (+ (log1p (exp a)) (/ b (expm1 (log1p (+ (exp a) 1.0))))))

C

assert(a < b);
double code(double a, double b) {
	return log1p(exp(a)) + (b / expm1(log1p((exp(a) + 1.0))));
}

Java

assert a < b;
public static double code(double a, double b) {
	return Math.log1p(Math.exp(a)) + (b / Math.expm1(Math.log1p((Math.exp(a) + 1.0))));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log1p(math.exp(a)) + (b / math.expm1(math.log1p((math.exp(a) + 1.0))))

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(log1p(exp(a)) + Float64(b / expm1(log1p(Float64(exp(a) + 1.0)))))
end

Wolfram

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

Derivation

1. Initial program 58.2%

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

2. Taylor expanded in b around 0 76.0%

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

   1. log1p-def 76.0%

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

4. Simplified 76.0%

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

   1. expm1-log1p-u 76.0%

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

6. Applied egg-rr 76.0%

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

7. Final simplification 76.0%

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

Alternative 3: 97.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.0%

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

   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 0.0 < (exp.f64 a) < 1

   1. Initial program 72.7%

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

   2. Taylor expanded in b around 0 69.5%

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

      1. log1p-def 69.5%

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

   4. Simplified 69.5%

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

   if 1 < (exp.f64 a)

   1. Initial program 63.0%

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

   2. Taylor expanded in a around 0 38.7%

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

      1. log1p-def 38.7%

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

   4. Simplified 38.7%

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

4. Final simplification 75.2%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

2. Taylor expanded in b around 0 76.0%

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

   1. log1p-def 76.0%

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

4. Simplified 76.0%

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

5. Final simplification 76.0%

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

Alternative 5: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.0%

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

   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 0.0 < (exp.f64 a)

   1. Initial program 72.3%

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

   2. Taylor expanded in b around 0 68.8%

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

      1. log1p-def 68.9%

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

   4. Simplified 68.9%

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

4. Final simplification 75.7%

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

Alternative 6: 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 58.2%

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

   1. add-sqr-sqrt 57.0%

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

   2. log-prod 57.4%

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

3. Applied egg-rr 57.4%

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

   1. log-prod 57.0%

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

   2. rem-square-sqrt 58.2%

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

   3. log1p-expm1 58.2%

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

   4. expm1-def 58.2%

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

   5. rem-exp-log 58.2%

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

   6. associate--l+ 58.3%

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

   7. expm1-def 78.0%

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

5. Simplified 78.0%

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

6. Final simplification 78.0%

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

Alternative 7: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -38:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \left(e^{a} + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -38.0)
   (/ b (+ (exp a) 1.0))
   (log (+ 1.0 (+ (exp a) (+ b (* 0.5 (* b 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((1.0 + (exp(a) + (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 <= (-38.0d0)) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((1.0d0 + (exp(a) + (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 <= -38.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((1.0 + (Math.exp(a) + (b + (0.5 * (b * 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((1.0 + (math.exp(a) + (b + (0.5 * (b * 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(1.0 + Float64(exp(a) + 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 <= -38.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((1.0 + (exp(a) + (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, -38.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(1.0 + N[(N[Exp[a], $MachinePrecision] + N[(b + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -38

   1. Initial program 8.0%

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

   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 -38 < a

   1. Initial program 72.3%

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

   2. Taylor expanded in b around 0 69.4%

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

      1. unpow2 69.4%

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

   4. Simplified 69.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -38:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \left(e^{a} + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.005:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{b + 2} + \log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.005)
   (/ b (+ (exp a) 1.0))
   (+ (/ a (+ b 2.0)) (log (+ b 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.005) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = (a / (b + 2.0)) + 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 (exp(a) <= 0.005d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = (a / (b + 2.0d0)) + 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 (Math.exp(a) <= 0.005) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = (a / (b + 2.0)) + Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.005:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = (a / (b + 2.0)) + math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.005)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(Float64(a / Float64(b + 2.0)) + 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 (exp(a) <= 0.005)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = (a / (b + 2.0)) + 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[N[Exp[a], $MachinePrecision], 0.005], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a / N[(b + 2.0), $MachinePrecision]), $MachinePrecision] + N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0050000000000000001

   1. Initial program 9.6%

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

   2. Taylor expanded in b around 0 99.9%

      \[\leadsto \color{blue}{\log \left(1 + e^{a}\right) + \frac{b}{1 + e^{a}}} \]
   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 98.3%

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

   if 0.0050000000000000001 < (exp.f64 a)

   1. Initial program 72.2%

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

   2. Taylor expanded in b around 0 68.3%

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

      1. associate-+r+ 68.3%

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

      2. +-commutative 68.3%

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

      3. associate-+l+ 68.3%

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

   4. Simplified 68.3%

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

   5. Taylor expanded in a around 0 67.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.005:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{b + 2} + \log \left(b + 2\right)\\ \end{array} \]

Alternative 9: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.005:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + \left(a + 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.005) (/ 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) <= 0.005) {
		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) <= 0.005d0) 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) <= 0.005) {
		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) <= 0.005:
		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) <= 0.005)
		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) <= 0.005)
		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], 0.005], 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) < 0.0050000000000000001

   1. Initial program 9.6%

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

   2. Taylor expanded in b around 0 99.9%

      \[\leadsto \color{blue}{\log \left(1 + e^{a}\right) + \frac{b}{1 + e^{a}}} \]
   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 98.3%

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

   if 0.0050000000000000001 < (exp.f64 a)

   1. Initial program 72.2%

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

   2. Taylor expanded in b around 0 68.3%

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

      1. associate-+r+ 68.3%

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

      2. +-commutative 68.3%

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

      3. associate-+l+ 68.3%

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

   4. Simplified 68.3%

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

   5. Taylor expanded in a around 0 67.1%

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

      1. associate-+r+ 67.1%

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

      2. +-commutative 67.1%

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

   7. Simplified 67.1%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.005:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + \left(a + 2\right)\right)\\ \end{array} \]

Alternative 10: 98.1% accurate, 1.5× speedup

Math

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

FPCore

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

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) + (b + 1.0)));
	}
	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 <= (-38.0d0)) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((exp(a) + (b + 1.0d0)))
    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) + (b + 1.0)));
	}
	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) + (b + 1.0)))
	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) + Float64(b + 1.0)));
	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) + (b + 1.0)));
	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, -38.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[(b + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -38

   1. Initial program 8.0%

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

   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 -38 < a

   1. Initial program 72.3%

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

   2. Taylor expanded in b around 0 68.5%

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

      1. associate-+r+ 68.5%

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

      2. +-commutative 68.5%

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

      3. associate-+l+ 68.5%

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

   4. Simplified 68.5%

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

4. Final simplification 75.4%

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

Alternative 11: 49.2% 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 58.2%

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

2. Taylor expanded in b around 0 76.0%

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

   1. log1p-def 76.0%

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

4. Simplified 76.0%

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

5. Taylor expanded in a around 0 53.2%

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

6. Final simplification 53.2%

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

Alternative 12: 49.0% 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 58.2%

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

2. Taylor expanded in b around 0 54.9%

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

   1. associate-+r+ 54.9%

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

   2. +-commutative 54.9%

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

   3. associate-+l+ 54.9%

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

4. Simplified 54.9%

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

5. Taylor expanded in a around 0 52.5%

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

   1. +-commutative 52.5%

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

7. Simplified 52.5%

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

8. Final simplification 52.5%

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

Alternative 13: 49.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

2. Taylor expanded in b around 0 54.9%

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

   1. associate-+r+ 54.9%

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

   2. +-commutative 54.9%

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

   3. associate-+l+ 54.9%

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

4. Simplified 54.9%

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

5. Taylor expanded in a around 0 52.5%

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

   1. +-commutative 52.5%

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

7. Simplified 52.5%

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

   1. +-commutative 52.5%

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

   2. log1p-expm1-u 52.5%

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

   3. expm1-udef 52.5%

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

   4. add-exp-log 52.5%

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

   5. +-commutative 52.5%

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

9. Applied egg-rr 52.5%

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

    1. associate--l+ 52.5%

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

    2. metadata-eval 52.5%

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

11. Simplified 52.5%

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

12. Final simplification 52.5%

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

Alternative 14: 48.4% 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 58.2%

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

2. Taylor expanded in a around 0 54.9%

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

   1. log1p-def 54.9%

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

4. Simplified 54.9%

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

5. Taylor expanded in b around 0 52.9%

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

6. Final simplification 52.9%

   \[\leadsto \log 2 \]

Alternative 15: 2.6% accurate, 101.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

2. Taylor expanded in b around 0 54.9%

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

   1. associate-+r+ 54.9%

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

   2. +-commutative 54.9%

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

   3. associate-+l+ 54.9%

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

4. Simplified 54.9%

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

5. Taylor expanded in a around 0 52.8%

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

6. Taylor expanded in b around 0 53.2%

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

7. Taylor expanded in a around inf 7.0%

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

   1. *-commutative 7.0%

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

9. Simplified 7.0%

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

10. Final simplification 7.0%

    \[\leadsto a \cdot 0.5 \]

Alternative 16: 2.6% accurate, 101.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

2. Taylor expanded in b around 0 54.9%

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

   1. associate-+r+ 54.9%

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

   2. +-commutative 54.9%

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

   3. associate-+l+ 54.9%

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

4. Simplified 54.9%

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

5. Taylor expanded in a around 0 52.8%

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

6. Taylor expanded in a around inf 3.9%

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

7. Taylor expanded in b around inf 3.6%

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

8. Final simplification 3.6%

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

Reproduce

herbie shell --seed 2023181 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))