symmetry log of sum of exp

Percentage Accurate: 54.0% → 98.7%

Time: 14.7s

Alternatives: 14

Speedup: 2.9×

• could not determine a ground truth

  1. a = 1.7473783217681749e-118
  2. b = 3.5907887979913135e+296

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: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

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

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 12.4%

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

   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 63.2%

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

4. Final simplification 70.1%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ 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 (exp(a) <= 0.0) {
		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 (exp(a) <= 0.0d0) 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 (Math.exp(a) <= 0.0) {
		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 math.exp(a) <= 0.0:
		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 (exp(a) <= 0.0)
		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 (exp(a) <= 0.0)
		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[N[Exp[a], $MachinePrecision], 0.0], 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 (exp.f64 a) < 0.0

   1. Initial program 12.4%

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

   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 63.2%

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

   2. Taylor expanded in b around 0 60.2%

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

      1. unpow2 60.2%

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

   4. Simplified 60.2%

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

4. Final simplification 67.7%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.2:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.2) (/ b (+ (exp a) 1.0)) (log1p (+ a (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.2) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p((a + exp(b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.2) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p((a + Math.exp(b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.2:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p((a + math.exp(b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.2)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(Float64(a + exp(b)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.2], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(a + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.20000000000000001

   1. Initial program 12.4%

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

   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 0.20000000000000001 < (exp.f64 a)

   1. Initial program 63.2%

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

      1. add-sqr-sqrt 61.8%

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

      2. log-prod 62.4%

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

   3. Applied egg-rr 62.4%

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

      1. log-prod 61.8%

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

      2. rem-square-sqrt 63.2%

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

      3. log1p-expm1 63.2%

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

      4. expm1-def 63.2%

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

      5. rem-exp-log 63.2%

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

      6. associate--l+ 63.3%

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

      7. expm1-def 63.3%

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

   5. Simplified 63.3%

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

   6. Taylor expanded in a around 0 98.2%

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

      1. +-commutative 98.2%

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

   8. Simplified 98.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.2:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + e^{b}\right)\\ \end{array} \]

Alternative 4: 97.5% accurate, 1.0× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ b (+ (exp a) 1.0)) (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 {
		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 {
		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)
	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));
	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], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 12.4%

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

   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 63.2%

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

   2. Taylor expanded in a around 0 61.3%

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

      1. log1p-def 61.3%

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

   4. Simplified 61.3%

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

4. Final simplification 68.6%

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

Alternative 5: 97.8% 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 53.7%

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

   1. add-sqr-sqrt 52.5%

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

   2. log-prod 53.0%

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

3. Applied egg-rr 53.0%

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

   1. log-prod 52.5%

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

   2. rem-square-sqrt 53.7%

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

   3. log1p-expm1 53.7%

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

   4. expm1-def 53.7%

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

   5. rem-exp-log 53.7%

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

   6. associate--l+ 53.7%

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

   7. expm1-def 69.4%

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

5. Simplified 69.4%

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

6. Final simplification 69.4%

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

Alternative 6: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.2:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + \left(a + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.2)
   (/ b (+ (exp a) 1.0))
   (log (+ 2.0 (+ a (+ b (* 0.5 (* b b))))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.2) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((2.0 + (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 (exp(a) <= 0.2d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((2.0d0 + (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 (Math.exp(a) <= 0.2) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((2.0 + (a + (b + (0.5 * (b * b))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.2:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((2.0 + (a + (b + (0.5 * (b * b))))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.2)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(2.0 + Float64(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 (exp(a) <= 0.2)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((2.0 + (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[N[Exp[a], $MachinePrecision], 0.2], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(2.0 + N[(a + N[(b + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.20000000000000001

   1. Initial program 12.4%

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

   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 0.20000000000000001 < (exp.f64 a)

   1. Initial program 63.2%

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

   2. Taylor expanded in a around 0 62.6%

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

      1. associate-+r+ 62.6%

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

      2. +-commutative 62.6%

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

   4. Simplified 62.6%

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

   5. Taylor expanded in b around 0 59.6%

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

      1. unpow2 59.6%

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

   7. Simplified 59.6%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.2:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + \left(a + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 97.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(((b + (0.5 * (b * b))) + 2.0))
	return tmp

Julia

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 12.4%

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

   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 63.2%

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

   2. Taylor expanded in a around 0 61.3%

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

   3. Taylor expanded in b around 0 58.9%

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

      1. unpow2 58.9%

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

   5. Simplified 58.9%

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

4. Final simplification 66.6%

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

Alternative 8: 56.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -135:\\ \;\;\;\;b \cdot \left(0.5 + b \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -135.0) (* b (+ 0.5 (* b 0.125))) (+ (* b 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -135.0) {
		tmp = b * (0.5 + (b * 0.125));
	} else {
		tmp = (b * 0.5) + log(2.0);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-135.0d0)) then
        tmp = b * (0.5d0 + (b * 0.125d0))
    else
        tmp = (b * 0.5d0) + log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -135.0) {
		tmp = b * (0.5 + (b * 0.125));
	} else {
		tmp = (b * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -135.0:
		tmp = b * (0.5 + (b * 0.125))
	else:
		tmp = (b * 0.5) + math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -135.0)
		tmp = Float64(b * Float64(0.5 + Float64(b * 0.125)));
	else
		tmp = Float64(Float64(b * 0.5) + log(2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -135.0)
		tmp = b * (0.5 + (b * 0.125));
	else
		tmp = (b * 0.5) + log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -135.0], N[(b * N[(0.5 + N[(b * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -135

   1. Initial program 12.4%

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

   2. Taylor expanded in a around 0 6.3%

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

   3. Taylor expanded in b around 0 4.9%

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

   4. Taylor expanded in b around inf 18.3%

      \[\leadsto \color{blue}{0.5 \cdot b + 0.125 \cdot {b}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 18.3%

         \[\leadsto \color{blue}{b \cdot 0.5} + 0.125 \cdot {b}^{2} \]

      2. *-commutative 18.3%

         \[\leadsto b \cdot 0.5 + \color{blue}{{b}^{2} \cdot 0.125} \]

      3. unpow2 18.3%

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

      4. associate-*l* 18.3%

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

      5. *-commutative 18.3%

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

      6. distribute-lft-out 18.3%

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

      7. *-commutative 18.3%

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

   6. Simplified 18.3%

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

   if -135 < a

   1. Initial program 63.2%

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

   2. Taylor expanded in a around 0 61.3%

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

   3. Taylor expanded in b around 0 58.7%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -135:\\ \;\;\;\;b \cdot \left(0.5 + b \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 9: 96.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -43:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -43.0) (/ b (+ (exp a) 1.0)) (+ (* b 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -43.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = (b * 0.5) + log(2.0);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-43.0d0)) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = (b * 0.5d0) + log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -43.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = (b * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -43.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = (b * 0.5) + math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -43.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(Float64(b * 0.5) + log(2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -43.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = (b * 0.5) + log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -43.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(b * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -43

   1. Initial program 12.4%

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

   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 -43 < a

   1. Initial program 63.2%

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

   2. Taylor expanded in a around 0 61.3%

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

   3. Taylor expanded in b around 0 58.7%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -43:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 10: 56.6% accurate, 2.9× speedup

Math

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

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 (* b 0.125))) (log (+ a 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = b * (0.5 + (b * 0.125));
	} 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 + (b * 0.125d0))
    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 + (b * 0.125));
	} 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 + (b * 0.125))
	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 * Float64(0.5 + Float64(b * 0.125)));
	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 + (b * 0.125));
	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 * N[(0.5 + N[(b * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(a + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 12.4%

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

   2. Taylor expanded in a around 0 6.3%

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

   3. Taylor expanded in b around 0 4.9%

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

   4. Taylor expanded in b around inf 18.3%

      \[\leadsto \color{blue}{0.5 \cdot b + 0.125 \cdot {b}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 18.3%

         \[\leadsto \color{blue}{b \cdot 0.5} + 0.125 \cdot {b}^{2} \]

      2. *-commutative 18.3%

         \[\leadsto b \cdot 0.5 + \color{blue}{{b}^{2} \cdot 0.125} \]

      3. unpow2 18.3%

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

      4. associate-*l* 18.3%

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

      5. *-commutative 18.3%

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

      6. distribute-lft-out 18.3%

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

      7. *-commutative 18.3%

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

   6. Simplified 18.3%

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

   if -1 < a

   1. Initial program 63.2%

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

   2. Taylor expanded in a around 0 62.6%

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

      1. associate-+r+ 62.6%

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

      2. +-commutative 62.6%

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

   4. Simplified 62.6%

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

   5. Taylor expanded in b around 0 59.0%

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

      1. +-commutative 59.0%

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

   7. Simplified 59.0%

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

4. Final simplification 51.4%

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

Alternative 11: 56.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -110:\\ \;\;\;\;b \cdot \left(0.5 + b \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -110.0) (* b (+ 0.5 (* b 0.125))) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -110.0) {
		tmp = b * (0.5 + (b * 0.125));
	} 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 <= (-110.0d0)) then
        tmp = b * (0.5d0 + (b * 0.125d0))
    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 <= -110.0) {
		tmp = b * (0.5 + (b * 0.125));
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -110.0:
		tmp = b * (0.5 + (b * 0.125))
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -110.0)
		tmp = Float64(b * Float64(0.5 + Float64(b * 0.125)));
	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 <= -110.0)
		tmp = b * (0.5 + (b * 0.125));
	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, -110.0], N[(b * N[(0.5 + N[(b * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -110

   1. Initial program 12.4%

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

   2. Taylor expanded in a around 0 6.3%

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

   3. Taylor expanded in b around 0 4.9%

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

   4. Taylor expanded in b around inf 18.3%

      \[\leadsto \color{blue}{0.5 \cdot b + 0.125 \cdot {b}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 18.3%

         \[\leadsto \color{blue}{b \cdot 0.5} + 0.125 \cdot {b}^{2} \]

      2. *-commutative 18.3%

         \[\leadsto b \cdot 0.5 + \color{blue}{{b}^{2} \cdot 0.125} \]

      3. unpow2 18.3%

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

      4. associate-*l* 18.3%

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

      5. *-commutative 18.3%

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

      6. distribute-lft-out 18.3%

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

      7. *-commutative 18.3%

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

   6. Simplified 18.3%

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

   if -110 < a

   1. Initial program 63.2%

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

   2. Taylor expanded in a around 0 61.3%

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

   3. Taylor expanded in b around 0 57.6%

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

      1. +-commutative 57.6%

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

   5. Simplified 57.6%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -110:\\ \;\;\;\;b \cdot \left(0.5 + b \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]

Alternative 12: 56.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -118:\\ \;\;\;\;b \cdot \left(0.5 + b \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -118.0) (* b (+ 0.5 (* b 0.125))) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -118.0) {
		tmp = b * (0.5 + (b * 0.125));
	} 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 <= (-118.0d0)) then
        tmp = b * (0.5d0 + (b * 0.125d0))
    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 <= -118.0) {
		tmp = b * (0.5 + (b * 0.125));
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -118.0:
		tmp = b * (0.5 + (b * 0.125))
	else:
		tmp = math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -118.0)
		tmp = Float64(b * Float64(0.5 + Float64(b * 0.125)));
	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 <= -118.0)
		tmp = b * (0.5 + (b * 0.125));
	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, -118.0], N[(b * N[(0.5 + N[(b * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -118

   1. Initial program 12.4%

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

   2. Taylor expanded in a around 0 6.3%

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

   3. Taylor expanded in b around 0 4.9%

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

   4. Taylor expanded in b around inf 18.3%

      \[\leadsto \color{blue}{0.5 \cdot b + 0.125 \cdot {b}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 18.3%

         \[\leadsto \color{blue}{b \cdot 0.5} + 0.125 \cdot {b}^{2} \]

      2. *-commutative 18.3%

         \[\leadsto b \cdot 0.5 + \color{blue}{{b}^{2} \cdot 0.125} \]

      3. unpow2 18.3%

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

      4. associate-*l* 18.3%

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

      5. *-commutative 18.3%

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

      6. distribute-lft-out 18.3%

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

      7. *-commutative 18.3%

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

   6. Simplified 18.3%

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

   if -118 < a

   1. Initial program 63.2%

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

   2. Taylor expanded in a around 0 61.3%

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

   3. Taylor expanded in b around 0 58.3%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -118:\\ \;\;\;\;b \cdot \left(0.5 + b \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 13: 11.7% accurate, 43.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ b \cdot \left(0.5 + b \cdot 0.125\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* b (+ 0.5 (* b 0.125))))

C

assert(a < b);
double code(double a, double b) {
	return b * (0.5 + (b * 0.125));
}

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 + (b * 0.125d0))
end function

Java

assert a < b;
public static double code(double a, double b) {
	return b * (0.5 + (b * 0.125));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return b * (0.5 + (b * 0.125))

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(b * Float64(0.5 + Float64(b * 0.125)))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = b * (0.5 + (b * 0.125));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(b * N[(0.5 + N[(b * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

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

2. Taylor expanded in a around 0 51.0%

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

3. Taylor expanded in b around 0 48.7%

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

4. Taylor expanded in b around inf 6.2%

   \[\leadsto \color{blue}{0.5 \cdot b + 0.125 \cdot {b}^{2}} \]
5. Step-by-step derivation

   1. *-commutative 6.2%

      \[\leadsto \color{blue}{b \cdot 0.5} + 0.125 \cdot {b}^{2} \]

   2. *-commutative 6.2%

      \[\leadsto b \cdot 0.5 + \color{blue}{{b}^{2} \cdot 0.125} \]

   3. unpow2 6.2%

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

   4. associate-*l* 6.2%

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

   5. *-commutative 6.2%

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

   6. distribute-lft-out 6.2%

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

   7. *-commutative 6.2%

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

6. Simplified 6.2%

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

7. Final simplification 6.2%

   \[\leadsto b \cdot \left(0.5 + b \cdot 0.125\right) \]

Alternative 14: 5.3% accurate, 60.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ b \cdot \left(b \cdot 0.125\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* b (* b 0.125)))

C

assert(a < b);
double code(double a, double b) {
	return b * (b * 0.125);
}

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 * (b * 0.125d0)
end function

Java

assert a < b;
public static double code(double a, double b) {
	return b * (b * 0.125);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return b * (b * 0.125)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(b * Float64(b * 0.125))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = b * (b * 0.125);
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(b * N[(b * 0.125), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.7%

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

2. Taylor expanded in a around 0 51.0%

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

3. Taylor expanded in b around 0 48.7%

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

4. Taylor expanded in b around inf 4.2%

   \[\leadsto \color{blue}{0.125 \cdot {b}^{2}} \]
5. Step-by-step derivation

   1. *-commutative 4.2%

      \[\leadsto \color{blue}{{b}^{2} \cdot 0.125} \]

   2. unpow2 4.2%

      \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot 0.125 \]

   3. associate-*l* 4.2%

      \[\leadsto \color{blue}{b \cdot \left(b \cdot 0.125\right)} \]

6. Simplified 4.2%

   \[\leadsto \color{blue}{b \cdot \left(b \cdot 0.125\right)} \]

7. Final simplification 4.2%

   \[\leadsto b \cdot \left(b \cdot 0.125\right) \]

Reproduce

herbie shell --seed 2023189 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))