symmetry log of sum of exp

Percentage Accurate: 89.6% → 97.3%

Time: 16.8s

Alternatives: 12

Speedup: 1.5×

• could not determine a ground truth

  1. a = 4.477272053044382e+69
  2. b = -7.944667550574141e+307

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: 89.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(e^{a}\right)\\ t_1 := e^{a} + 1\\ t_2 := \frac{b}{t_1}\\ \mathbf{if}\;b \leq -4 \cdot 10^{-67}:\\ \;\;\;\;t_0 + t_2\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\log t_1 + \left(t_2 + 0.5 \cdot \left({b}^{2} \cdot \left(\frac{1}{t_1} + \frac{-1}{{t_1}^{2}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (log1p (exp a))) (t_1 (+ (exp a) 1.0)) (t_2 (/ b t_1)))
   (if (<= b -4e-67)
     (+ t_0 t_2)
     (if (<= b 3.5e-84)
       t_0
       (+
        (log t_1)
        (+
         t_2
         (* 0.5 (* (pow b 2.0) (+ (/ 1.0 t_1) (/ -1.0 (pow t_1 2.0)))))))))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = log1p(exp(a));
	double t_1 = exp(a) + 1.0;
	double t_2 = b / t_1;
	double tmp;
	if (b <= -4e-67) {
		tmp = t_0 + t_2;
	} else if (b <= 3.5e-84) {
		tmp = t_0;
	} else {
		tmp = log(t_1) + (t_2 + (0.5 * (pow(b, 2.0) * ((1.0 / t_1) + (-1.0 / pow(t_1, 2.0))))));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = Math.log1p(Math.exp(a));
	double t_1 = Math.exp(a) + 1.0;
	double t_2 = b / t_1;
	double tmp;
	if (b <= -4e-67) {
		tmp = t_0 + t_2;
	} else if (b <= 3.5e-84) {
		tmp = t_0;
	} else {
		tmp = Math.log(t_1) + (t_2 + (0.5 * (Math.pow(b, 2.0) * ((1.0 / t_1) + (-1.0 / Math.pow(t_1, 2.0))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = math.log1p(math.exp(a))
	t_1 = math.exp(a) + 1.0
	t_2 = b / t_1
	tmp = 0
	if b <= -4e-67:
		tmp = t_0 + t_2
	elif b <= 3.5e-84:
		tmp = t_0
	else:
		tmp = math.log(t_1) + (t_2 + (0.5 * (math.pow(b, 2.0) * ((1.0 / t_1) + (-1.0 / math.pow(t_1, 2.0))))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = log1p(exp(a))
	t_1 = Float64(exp(a) + 1.0)
	t_2 = Float64(b / t_1)
	tmp = 0.0
	if (b <= -4e-67)
		tmp = Float64(t_0 + t_2);
	elseif (b <= 3.5e-84)
		tmp = t_0;
	else
		tmp = Float64(log(t_1) + Float64(t_2 + Float64(0.5 * Float64((b ^ 2.0) * Float64(Float64(1.0 / t_1) + Float64(-1.0 / (t_1 ^ 2.0)))))));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(b / t$95$1), $MachinePrecision]}, If[LessEqual[b, -4e-67], N[(t$95$0 + t$95$2), $MachinePrecision], If[LessEqual[b, 3.5e-84], t$95$0, N[(N[Log[t$95$1], $MachinePrecision] + N[(t$95$2 + N[(0.5 * N[(N[Power[b, 2.0], $MachinePrecision] * N[(N[(1.0 / t$95$1), $MachinePrecision] + N[(-1.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.99999999999999977e-67

   1. Initial program 85.2%

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

   3. Taylor expanded in b around 0 29.9%

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

      1. log1p-def 30.1%

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

   5. Simplified 30.1%

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

   if -3.99999999999999977e-67 < b < 3.5000000000000001e-84

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

   5. Simplified 100.0%

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

   if 3.5000000000000001e-84 < b

   1. Initial program 70.9%

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

   3. Taylor expanded in b around 0 93.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right) + \frac{b}{e^{a} + 1}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-84}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + 1\right) + \left(\frac{b}{e^{a} + 1} + 0.5 \cdot \left({b}^{2} \cdot \left(\frac{1}{e^{a} + 1} + \frac{-1}{{\left(e^{a} + 1\right)}^{2}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(e^{a}\right)\\ \mathbf{if}\;b \leq -4 \cdot 10^{-67} \lor \neg \left(b \leq 3.5 \cdot 10^{-84}\right):\\ \;\;\;\;t_0 + \frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (log1p (exp a))))
   (if (or (<= b -4e-67) (not (<= b 3.5e-84)))
     (+ t_0 (/ b (+ (exp a) 1.0)))
     t_0)))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = log1p(exp(a));
	double tmp;
	if ((b <= -4e-67) || !(b <= 3.5e-84)) {
		tmp = t_0 + (b / (exp(a) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = Math.log1p(Math.exp(a));
	double tmp;
	if ((b <= -4e-67) || !(b <= 3.5e-84)) {
		tmp = t_0 + (b / (Math.exp(a) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = math.log1p(math.exp(a))
	tmp = 0
	if (b <= -4e-67) or not (b <= 3.5e-84):
		tmp = t_0 + (b / (math.exp(a) + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = log1p(exp(a))
	tmp = 0.0
	if ((b <= -4e-67) || !(b <= 3.5e-84))
		tmp = Float64(t_0 + Float64(b / Float64(exp(a) + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[b, -4e-67], N[Not[LessEqual[b, 3.5e-84]], $MachinePrecision]], N[(t$95$0 + N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if b < -3.99999999999999977e-67 or 3.5000000000000001e-84 < b

   1. Initial program 81.3%

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

   3. Taylor expanded in b around 0 46.6%

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

      1. log1p-def 46.8%

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

   5. Simplified 46.8%

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

   if -3.99999999999999977e-67 < b < 3.5000000000000001e-84

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-67} \lor \neg \left(b \leq 3.5 \cdot 10^{-84}\right):\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right) + \frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b -2.3e-66) (/ b (+ (exp a) 1.0)) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= -2.3e-66) {
		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 (b <= (-2.3d-66)) 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 (b <= -2.3e-66) {
		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 b <= -2.3e-66:
		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 (b <= -2.3e-66)
		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 (b <= -2.3e-66)
		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[b, -2.3e-66], 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 b < -2.29999999999999992e-66

   1. Initial program 85.2%

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

   3. Taylor expanded in b around 0 29.9%

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

      1. log1p-def 30.1%

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

   5. Simplified 30.1%

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

   6. Taylor expanded in b around inf 10.4%

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

   if -2.29999999999999992e-66 < b

   1. Initial program 95.0%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-66}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{2 + \left(a + 0.5 \cdot {a}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + 0.5 \cdot \left(b + a\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-27)
   (/ b (+ 2.0 (+ a (* 0.5 (pow a 2.0)))))
   (+ (log 2.0) (* 0.5 (+ b a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-27) {
		tmp = b / (2.0 + (a + (0.5 * pow(a, 2.0))));
	} else {
		tmp = log(2.0) + (0.5 * (b + a));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 2d-27) then
        tmp = b / (2.0d0 + (a + (0.5d0 * (a ** 2.0d0))))
    else
        tmp = log(2.0d0) + (0.5d0 * (b + a))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 2e-27) {
		tmp = b / (2.0 + (a + (0.5 * Math.pow(a, 2.0))));
	} else {
		tmp = Math.log(2.0) + (0.5 * (b + a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 2e-27:
		tmp = b / (2.0 + (a + (0.5 * math.pow(a, 2.0))))
	else:
		tmp = math.log(2.0) + (0.5 * (b + a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-27)
		tmp = Float64(b / Float64(2.0 + Float64(a + Float64(0.5 * (a ^ 2.0)))));
	else
		tmp = Float64(log(2.0) + Float64(0.5 * Float64(b + a)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 2e-27)
		tmp = b / (2.0 + (a + (0.5 * (a ^ 2.0))));
	else
		tmp = log(2.0) + (0.5 * (b + a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 2e-27], N[(b / N[(2.0 + N[(a + N[(0.5 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(0.5 * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2.0000000000000001e-27

   1. Initial program 77.9%

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

   3. Taylor expanded in b around 0 29.1%

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

      1. log1p-def 29.4%

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

   5. Simplified 29.4%

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

   6. Taylor expanded in b around inf 29.1%

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

   7. Taylor expanded in a around 0 65.2%

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

   if 2.0000000000000001e-27 < (exp.f64 a)

   1. Initial program 95.7%

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

   3. Taylor expanded in b around 0 70.2%

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

      1. log1p-def 70.2%

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

   5. Simplified 70.2%

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

   6. Taylor expanded in a around 0 70.2%

      \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{2}} \]

   7. Taylor expanded in a around 0 69.8%

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

      1. distribute-lft-out 69.8%

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

   9. Simplified 69.8%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{2 + \left(a + 0.5 \cdot {a}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + 0.5 \cdot \left(b + a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.6% accurate, 1.4× speedup

Math

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

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 (+ a 2.0)) (+ (log 2.0) (* 0.5 (+ b a)))))

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (a + 2.0d0)
    else
        tmp = log(2.0d0) + (0.5d0 * (b + a))
    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 / (a + 2.0);
	} else {
		tmp = Math.log(2.0) + (0.5 * (b + a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (a + 2.0)
	else:
		tmp = math.log(2.0) + (0.5 * (b + 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(a + 2.0));
	else
		tmp = Float64(log(2.0) + Float64(0.5 * Float64(b + a)));
	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 / (a + 2.0);
	else
		tmp = log(2.0) + (0.5 * (b + a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(a + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(0.5 * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 78.9%

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

   3. Taylor expanded in b around 0 28.2%

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

      1. log1p-def 28.2%

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

   5. Simplified 28.2%

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

   6. Taylor expanded in b around inf 28.2%

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

   7. Taylor expanded in a around 0 48.0%

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

      1. +-commutative 48.0%

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

   9. Simplified 48.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 95.4%

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

   3. Taylor expanded in b around 0 70.3%

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

      1. log1p-def 70.4%

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

   5. Simplified 70.4%

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

   6. Taylor expanded in a around 0 70.0%

      \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{2}} \]

   7. Taylor expanded in a around 0 69.5%

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

      1. distribute-lft-out 69.5%

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

   9. Simplified 69.5%

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

4. Final simplification 64.8%

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

Alternative 6: 71.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + \left(a + 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-27) (/ b (+ a 2.0)) (log (+ b (+ a 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-27) {
		tmp = b / (a + 2.0);
	} else {
		tmp = log((b + (a + 2.0)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 2d-27) then
        tmp = b / (a + 2.0d0)
    else
        tmp = log((b + (a + 2.0d0)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 2e-27) {
		tmp = b / (a + 2.0);
	} else {
		tmp = Math.log((b + (a + 2.0)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 2e-27:
		tmp = b / (a + 2.0)
	else:
		tmp = math.log((b + (a + 2.0)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-27)
		tmp = Float64(b / Float64(a + 2.0));
	else
		tmp = log(Float64(b + Float64(a + 2.0)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 2e-27)
		tmp = b / (a + 2.0);
	else
		tmp = log((b + (a + 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 2e-27], N[(b / N[(a + 2.0), $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) < 2.0000000000000001e-27

   1. Initial program 77.9%

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

   3. Taylor expanded in b around 0 29.1%

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

      1. log1p-def 29.4%

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

   5. Simplified 29.4%

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

   6. Taylor expanded in b around inf 29.1%

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

   7. Taylor expanded in a around 0 47.1%

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

      1. +-commutative 47.1%

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

   9. Simplified 47.1%

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

   if 2.0000000000000001e-27 < (exp.f64 a)

   1. Initial program 95.7%

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

   3. Taylor expanded in a around 0 94.4%

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

      1. +-commutative 94.4%

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

   5. Simplified 94.4%

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

   6. Taylor expanded in b around 0 69.0%

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

      1. +-commutative 69.0%

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

      2. +-commutative 69.0%

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

      3. associate-+l+ 69.0%

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

   8. Simplified 69.0%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + \left(a + 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-66}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \left(b + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b -2.7e-66) (/ b (+ (exp a) 1.0)) (log (+ (exp a) (+ b 1.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= -2.7e-66) {
		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 (b <= (-2.7d-66)) 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 (b <= -2.7e-66) {
		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 b <= -2.7e-66:
		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 (b <= -2.7e-66)
		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 (b <= -2.7e-66)
		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[b, -2.7e-66], 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 b < -2.69999999999999996e-66

   1. Initial program 85.2%

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

   3. Taylor expanded in b around 0 29.9%

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

      1. log1p-def 30.1%

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

   5. Simplified 30.1%

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

   6. Taylor expanded in b around inf 10.4%

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

   if -2.69999999999999996e-66 < b

   1. Initial program 95.0%

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

   3. Taylor expanded in b around 0 94.9%

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

      1. associate-+r+ 94.9%

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

      2. +-commutative 94.9%

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

   5. Simplified 94.9%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-66}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \left(b + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-66}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b -2.7e-66) (/ b (+ (exp a) 1.0)) (log1p (exp a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= -2.7e-66) {
		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 (b <= -2.7e-66) {
		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 b <= -2.7e-66:
		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 (b <= -2.7e-66)
		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[b, -2.7e-66], 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 b < -2.69999999999999996e-66

   1. Initial program 85.2%

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

   3. Taylor expanded in b around 0 29.9%

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

      1. log1p-def 30.1%

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

   5. Simplified 30.1%

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

   6. Taylor expanded in b around inf 10.4%

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

   if -2.69999999999999996e-66 < b

   1. Initial program 95.0%

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

   3. Taylor expanded in b around 0 94.3%

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

      1. log1p-def 94.3%

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

   5. Simplified 94.3%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-66}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1700000000:\\ \;\;\;\;\frac{b}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1700000000.0) (/ b (+ a 2.0)) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1700000000.0) {
		tmp = b / (a + 2.0);
	} else {
		tmp = log((b + 2.0));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1700000000.0d0)) then
        tmp = b / (a + 2.0d0)
    else
        tmp = log((b + 2.0d0))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1700000000.0) {
		tmp = b / (a + 2.0);
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1700000000.0:
		tmp = b / (a + 2.0)
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1700000000.0)
		tmp = Float64(b / Float64(a + 2.0));
	else
		tmp = log(Float64(b + 2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1700000000.0)
		tmp = b / (a + 2.0);
	else
		tmp = log((b + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1700000000.0], N[(b / N[(a + 2.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.7e9

   1. Initial program 78.5%

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

   3. Taylor expanded in b around 0 28.6%

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

      1. log1p-def 28.6%

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

   5. Simplified 28.6%

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

   6. Taylor expanded in b around inf 28.6%

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

   7. Taylor expanded in a around 0 48.8%

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

      1. +-commutative 48.8%

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

   9. Simplified 48.8%

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

   if -1.7e9 < a

   1. Initial program 95.4%

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

   3. Taylor expanded in a around 0 93.4%

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

      1. +-commutative 93.4%

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

   5. Simplified 93.4%

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

   6. Taylor expanded in b around 0 68.3%

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

      1. +-commutative 68.3%

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

      2. +-commutative 68.3%

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

      3. associate-+l+ 68.3%

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

   8. Simplified 68.3%

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

   9. Taylor expanded in a around 0 68.0%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1700000000:\\ \;\;\;\;\frac{b}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -4200000000:\\ \;\;\;\;\frac{b}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -4200000000.0) (/ b (+ a 2.0)) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -4200000000.0) {
		tmp = b / (a + 2.0);
	} 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 <= (-4200000000.0d0)) then
        tmp = b / (a + 2.0d0)
    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 <= -4200000000.0) {
		tmp = b / (a + 2.0);
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -4200000000.0:
		tmp = b / (a + 2.0)
	else:
		tmp = math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -4200000000.0)
		tmp = Float64(b / Float64(a + 2.0));
	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 <= -4200000000.0)
		tmp = b / (a + 2.0);
	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, -4200000000.0], N[(b / N[(a + 2.0), $MachinePrecision]), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.2e9

   1. Initial program 78.5%

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

   3. Taylor expanded in b around 0 28.6%

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

      1. log1p-def 28.6%

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

   5. Simplified 28.6%

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

   6. Taylor expanded in b around inf 28.6%

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

   7. Taylor expanded in a around 0 48.8%

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

      1. +-commutative 48.8%

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

   9. Simplified 48.8%

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

   if -4.2e9 < a

   1. Initial program 95.4%

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

   3. Taylor expanded in b around 0 69.5%

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

      1. log1p-def 69.5%

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

   5. Simplified 69.5%

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

   6. Taylor expanded in a around 0 68.2%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4200000000:\\ \;\;\;\;\frac{b}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.3% accurate, 60.6× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return b / (a + 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 / (a + 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

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

3. Taylor expanded in b around 0 61.2%

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

   1. log1p-def 61.3%

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

5. Simplified 61.3%

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

6. Taylor expanded in b around inf 9.2%

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

7. Taylor expanded in a around 0 13.1%

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

   1. +-commutative 13.1%

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

9. Simplified 13.1%

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

10. Final simplification 13.1%

    \[\leadsto \frac{b}{a + 2} \]
11. Add Preprocessing

Alternative 12: 7.1% accurate, 101.0× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return 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 = b / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.8%

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

3. Taylor expanded in b around 0 61.2%

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

   1. log1p-def 61.3%

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

5. Simplified 61.3%

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

6. Taylor expanded in b around inf 9.2%

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

7. Taylor expanded in a around 0 4.8%

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

8. Final simplification 4.8%

   \[\leadsto \frac{b}{2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024023 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))