symmetry log of sum of exp

Percentage Accurate: 55.3% → 98.5%

Time: 21.4s

Alternatives: 22

Speedup: 2.8×

• could not determine a ground truth

  1. a = -5.6002193144842325e+35
  2. b = -2.776207728205147e+138

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: 55.3% 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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in b around 0 75.1%

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

   1. log1p-define 75.2%

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

5. Simplified 75.2%

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

6. Final simplification 75.2%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-156) (/ b (+ (exp a) 1.0)) (log (+ 1.0 (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-156) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((1.0 + 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) <= 1d-156) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((1.0d0 + 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) <= 1e-156) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((1.0 + Math.exp(b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 1e-156:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((1.0 + math.exp(b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 1e-156)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(1.0 + 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) <= 1e-156)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((1.0 + 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], 1e-156], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1.00000000000000004e-156

   1. Initial program 8.6%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 1.00000000000000004e-156 < (exp.f64 a)

   1. Initial program 69.4%

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

   3. Taylor expanded in a around 0 67.3%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-156) (/ b (+ (exp a) 1.0)) (log1p (exp a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-156) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p(exp(a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 1e-156) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p(Math.exp(a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 1e-156:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(math.exp(a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 1e-156)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(exp(a));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 1e-156], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1.00000000000000004e-156

   1. Initial program 8.6%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 1.00000000000000004e-156 < (exp.f64 a)

   1. Initial program 69.4%

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

   3. Taylor expanded in b around 0 66.1%

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

      1. log1p-define 66.1%

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

   5. Simplified 66.1%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-156) (/ b (+ (exp a) 1.0)) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-156) {
		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) <= 1e-156) {
		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) <= 1e-156:
		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) <= 1e-156)
		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], 1e-156], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1.00000000000000004e-156

   1. Initial program 8.6%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 1.00000000000000004e-156 < (exp.f64 a)

   1. Initial program 69.4%

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

   3. Taylor expanded in a around 0 67.3%

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

      1. log1p-define 67.4%

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

   5. Simplified 67.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right) + a \cdot \left(1 + a \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-156)
   (/ b (+ (exp a) 1.0))
   (log
    (+
     (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))
     (* a (+ 1.0 (* a 0.5)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-156) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(((2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))) + (a * (1.0 + (a * 0.5)))));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 1d-156) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(((2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0)))))) + (a * (1.0d0 + (a * 0.5d0)))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 1e-156) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(((2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))) + (a * (1.0 + (a * 0.5)))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 1e-156:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(((2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))) + (a * (1.0 + (a * 0.5)))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 1e-156)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))) + Float64(a * Float64(1.0 + Float64(a * 0.5)))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 1e-156)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(((2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))) + (a * (1.0 + (a * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 1e-156], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1.00000000000000004e-156

   1. Initial program 8.6%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 1.00000000000000004e-156 < (exp.f64 a)

   1. Initial program 69.4%

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

   3. Taylor expanded in a around 0 68.9%

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

      1. associate-+r+ 68.9%

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

      2. *-commutative 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in b around 0 65.9%

      \[\leadsto \log \left(\color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(0.5 + 0.16666666666666666 \cdot b\right)\right)\right)} + a \cdot \left(1 + a \cdot 0.5\right)\right) \]
   7. Step-by-step derivation

      1. *-commutative 65.9%

         \[\leadsto \log \left(\left(2 + b \cdot \left(1 + b \cdot \left(0.5 + \color{blue}{b \cdot 0.16666666666666666}\right)\right)\right) + a \cdot \left(1 + a \cdot 0.5\right)\right) \]

   8. Simplified 65.9%

      \[\leadsto \log \left(\color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)} + a \cdot \left(1 + a \cdot 0.5\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right) + a \cdot \left(1 + a \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(a \cdot \left(1 + a \cdot 0.5\right) + \left(2 + b \cdot \left(1 + b \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-156)
   (/ b (+ (exp a) 1.0))
   (log (+ (* a (+ 1.0 (* a 0.5))) (+ 2.0 (* b (+ 1.0 (* b 0.5))))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-156) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(((a * (1.0 + (a * 0.5))) + (2.0 + (b * (1.0 + (b * 0.5))))));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 1d-156) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(((a * (1.0d0 + (a * 0.5d0))) + (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 1e-156) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(((a * (1.0 + (a * 0.5))) + (2.0 + (b * (1.0 + (b * 0.5))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 1e-156:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(((a * (1.0 + (a * 0.5))) + (2.0 + (b * (1.0 + (b * 0.5))))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 1e-156)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(Float64(a * Float64(1.0 + Float64(a * 0.5))) + Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5))))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 1e-156)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(((a * (1.0 + (a * 0.5))) + (2.0 + (b * (1.0 + (b * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 1e-156], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(a * N[(1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1.00000000000000004e-156

   1. Initial program 8.6%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 1.00000000000000004e-156 < (exp.f64 a)

   1. Initial program 69.4%

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

   3. Taylor expanded in a around 0 68.9%

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

      1. associate-+r+ 68.9%

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

      2. *-commutative 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in b around 0 66.7%

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

      1. +-commutative 66.7%

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

   8. Simplified 66.7%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(a \cdot \left(1 + a \cdot 0.5\right) + \left(2 + b \cdot \left(1 + b \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(b \cdot 0.5 + a \cdot \left(0.5 - b \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.1)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (+ (* b 0.5) (* a (- 0.5 (* b 0.25)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.1) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + ((b * 0.5) + (a * (0.5 - (b * 0.25))));
	}
	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) <= 0.1d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + ((b * 0.5d0) + (a * (0.5d0 - (b * 0.25d0))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.1) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + ((b * 0.5) + (a * (0.5 - (b * 0.25))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.1:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + ((b * 0.5) + (a * (0.5 - (b * 0.25))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.1)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(Float64(b * 0.5) + Float64(a * Float64(0.5 - Float64(b * 0.25)))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.1)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + ((b * 0.5) + (a * (0.5 - (b * 0.25))));
	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], 0.1], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(N[(b * 0.5), $MachinePrecision] + N[(a * N[(0.5 - N[(b * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.10000000000000001

   1. Initial program 8.6%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 0.10000000000000001 < (exp.f64 a)

   1. Initial program 69.4%

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

   3. Taylor expanded in b around 0 66.9%

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

      1. log1p-define 66.9%

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

   5. Simplified 66.9%

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

   6. Taylor expanded in a around 0 66.6%

      \[\leadsto \color{blue}{\log 2 + \left(0.5 \cdot b + a \cdot \left(0.5 - 0.25 \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(b \cdot 0.5 + a \cdot \left(0.5 - b \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + \left(b + a \cdot \left(1 + a \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-156)
   (/ b (+ (exp a) 1.0))
   (log (+ 2.0 (+ b (* a (+ 1.0 (* a 0.5))))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-156) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((2.0 + (b + (a * (1.0 + (a * 0.5))))));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 1d-156) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((2.0d0 + (b + (a * (1.0d0 + (a * 0.5d0))))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 1e-156) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((2.0 + (b + (a * (1.0 + (a * 0.5))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 1e-156:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((2.0 + (b + (a * (1.0 + (a * 0.5))))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 1e-156)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(2.0 + Float64(b + Float64(a * Float64(1.0 + Float64(a * 0.5))))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 1e-156)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((2.0 + (b + (a * (1.0 + (a * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 1e-156], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(2.0 + N[(b + N[(a * N[(1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1.00000000000000004e-156

   1. Initial program 8.6%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 1.00000000000000004e-156 < (exp.f64 a)

   1. Initial program 69.4%

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

   3. Taylor expanded in b around 0 65.9%

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

      1. associate-+r+ 65.9%

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

      2. +-commutative 65.9%

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

   5. Simplified 65.9%

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

   6. Taylor expanded in a around 0 65.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + \left(b + a \cdot \left(1 + a \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-156)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (* a (+ 0.5 (* a 0.125))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-156) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (a * (0.5 + (a * 0.125)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 1d-156) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (a * (0.5d0 + (a * 0.125d0)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 1e-156) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (a * (0.5 + (a * 0.125)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 1e-156:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (a * (0.5 + (a * 0.125)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 1e-156)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(a * Float64(0.5 + Float64(a * 0.125))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 1e-156)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (a * (0.5 + (a * 0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 1e-156], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * N[(0.5 + N[(a * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1.00000000000000004e-156

   1. Initial program 8.6%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 1.00000000000000004e-156 < (exp.f64 a)

   1. Initial program 69.4%

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

   3. Taylor expanded in b around 0 66.1%

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

      1. log1p-define 66.1%

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

   5. Simplified 66.1%

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

   6. Taylor expanded in a around 0 65.8%

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

      1. *-commutative 65.8%

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

   8. Simplified 65.8%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-156)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (* b (+ 0.5 (* b 0.125))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-156) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (b * (0.5 + (b * 0.125)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 1d-156) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (b * (0.5d0 + (b * 0.125d0)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 1e-156) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (b * (0.5 + (b * 0.125)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 1e-156:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (b * (0.5 + (b * 0.125)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 1e-156)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(b * Float64(0.5 + Float64(b * 0.125))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 1e-156)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (b * (0.5 + (b * 0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 1e-156], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * N[(0.5 + N[(b * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1.00000000000000004e-156

   1. Initial program 8.6%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 1.00000000000000004e-156 < (exp.f64 a)

   1. Initial program 69.4%

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

   3. Taylor expanded in a around 0 67.3%

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

   4. Taylor expanded in b around 0 65.6%

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

      1. *-commutative 65.6%

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

   6. Simplified 65.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{b + 2} + \log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.1)
   (/ b (+ (exp a) 1.0))
   (+ (/ a (+ b 2.0)) (log (+ b 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.1) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = (a / (b + 2.0)) + log((b + 2.0));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.1d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = (a / (b + 2.0d0)) + log((b + 2.0d0))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.1) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = (a / (b + 2.0)) + Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.1:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = (a / (b + 2.0)) + math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.1)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(Float64(a / Float64(b + 2.0)) + log(Float64(b + 2.0)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.1)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = (a / (b + 2.0)) + log((b + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.1], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a / N[(b + 2.0), $MachinePrecision]), $MachinePrecision] + N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.10000000000000001

   1. Initial program 8.6%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 0.10000000000000001 < (exp.f64 a)

   1. Initial program 69.4%

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

   3. Taylor expanded in b around 0 65.9%

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

      1. associate-+r+ 65.9%

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

      2. +-commutative 65.9%

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

   5. Simplified 65.9%

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

   6. Taylor expanded in a around 0 65.6%

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

   7. Taylor expanded in a around 0 65.6%

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

      1. +-commutative 65.6%

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

      2. +-commutative 65.6%

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

   9. Simplified 65.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{b + 2} + \log \left(b + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right) + \frac{a}{b + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.1)
   (/ b (+ (exp a) 1.0))
   (+ (log1p (+ b 1.0)) (/ a (+ b 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.1) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p((b + 1.0)) + (a / (b + 2.0));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.1) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p((b + 1.0)) + (a / (b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.1:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p((b + 1.0)) + (a / (b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.1)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log1p(Float64(b + 1.0)) + Float64(a / Float64(b + 2.0)));
	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], 0.1], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(b + 1.0), $MachinePrecision]], $MachinePrecision] + N[(a / N[(b + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.10000000000000001

   1. Initial program 8.6%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 0.10000000000000001 < (exp.f64 a)

   1. Initial program 69.4%

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

   3. Taylor expanded in b around 0 65.9%

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

      1. associate-+r+ 65.9%

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

      2. +-commutative 65.9%

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

   5. Simplified 65.9%

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

   6. Taylor expanded in a around 0 65.6%

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

   7. Taylor expanded in a around 0 65.6%

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

      1. +-commutative 65.6%

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

      2. +-commutative 65.6%

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

   9. Simplified 65.6%

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

       1. log1p-expm1-u 65.6%

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

       2. expm1-undefine 65.6%

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

       3. add-exp-log 65.6%

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

   11. Applied egg-rr 65.6%

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

       1. associate--l+ 65.6%

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

       2. metadata-eval 65.6%

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

   13. Simplified 65.6%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(b + 1\right)} + \frac{a}{b + 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.1:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right) + \frac{a}{b + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.5 + \frac{\log 2}{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-156) (/ b (+ (exp a) 1.0)) (* b (+ 0.5 (/ (log 2.0) b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-156) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = b * (0.5 + (log(2.0) / 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) <= 1d-156) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = b * (0.5d0 + (log(2.0d0) / 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) <= 1e-156) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = b * (0.5 + (Math.log(2.0) / b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 1e-156:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = b * (0.5 + (math.log(2.0) / b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 1e-156)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(b * Float64(0.5 + Float64(log(2.0) / b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 1e-156)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = b * (0.5 + (log(2.0) / 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], 1e-156], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(b * N[(0.5 + N[(N[Log[2.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1.00000000000000004e-156

   1. Initial program 8.6%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 1.00000000000000004e-156 < (exp.f64 a)

   1. Initial program 69.4%

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

   3. Taylor expanded in b around 0 66.9%

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

      1. log1p-define 66.9%

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

   5. Simplified 66.9%

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

   6. Taylor expanded in a around 0 65.6%

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

   7. Taylor expanded in b around inf 65.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-156}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.5 + \frac{\log 2}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 96.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in b around 0 51.1%

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

   1. associate-+r+ 51.1%

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

   2. +-commutative 51.1%

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

5. Simplified 51.1%

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

6. Taylor expanded in a around inf 51.1%

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

   1. log1p-define 73.7%

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

8. Simplified 73.7%

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

9. Final simplification 73.7%

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

Alternative 15: 59.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(a + \left(b + 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b -4e-213) (log1p b) (log (+ a (+ b 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= -4e-213) {
		tmp = log1p(b);
	} else {
		tmp = log((a + (b + 2.0)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= -4e-213) {
		tmp = Math.log1p(b);
	} else {
		tmp = Math.log((a + (b + 2.0)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= -4e-213:
		tmp = math.log1p(b)
	else:
		tmp = math.log((a + (b + 2.0)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= -4e-213)
		tmp = log1p(b);
	else
		tmp = log(Float64(a + Float64(b + 2.0)));
	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, -4e-213], N[Log[1 + b], $MachinePrecision], N[Log[N[(a + N[(b + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.9999999999999998e-213

   1. Initial program 36.7%

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

   3. Taylor expanded in b around 0 32.1%

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

      1. associate-+r+ 32.1%

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

      2. +-commutative 32.1%

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

   5. Simplified 32.1%

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

   6. Taylor expanded in a around inf 32.1%

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

      1. log1p-define 51.2%

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

   8. Simplified 51.2%

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

   9. Taylor expanded in b around inf 21.0%

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

   if -3.9999999999999998e-213 < b

   1. Initial program 69.7%

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

   3. Taylor expanded in b around 0 67.9%

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

      1. associate-+r+ 67.9%

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

      2. +-commutative 67.9%

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

   5. Simplified 67.9%

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

   6. Taylor expanded in a around 0 65.3%

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

      1. associate-+r+ 65.3%

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

      2. +-commutative 65.3%

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

      3. associate-+l+ 65.3%

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

   8. Simplified 65.3%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(a + \left(b + 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + \left(a + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b -3.4e-213) (log1p b) (log1p (+ b (+ a 1.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= -3.4e-213) {
		tmp = log1p(b);
	} else {
		tmp = log1p((b + (a + 1.0)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= -3.4e-213) {
		tmp = Math.log1p(b);
	} else {
		tmp = Math.log1p((b + (a + 1.0)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= -3.4e-213:
		tmp = math.log1p(b)
	else:
		tmp = math.log1p((b + (a + 1.0)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= -3.4e-213)
		tmp = log1p(b);
	else
		tmp = log1p(Float64(b + Float64(a + 1.0)));
	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, -3.4e-213], N[Log[1 + b], $MachinePrecision], N[Log[1 + N[(b + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.4000000000000002e-213

   1. Initial program 36.7%

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

   3. Taylor expanded in b around 0 32.1%

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

      1. associate-+r+ 32.1%

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

      2. +-commutative 32.1%

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

   5. Simplified 32.1%

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

   6. Taylor expanded in a around inf 32.1%

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

      1. log1p-define 51.2%

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

   8. Simplified 51.2%

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

   9. Taylor expanded in b around inf 21.0%

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

   if -3.4000000000000002e-213 < b

   1. Initial program 69.7%

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

   3. Taylor expanded in b around 0 67.9%

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

      1. associate-+r+ 67.9%

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

      2. +-commutative 67.9%

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

   5. Simplified 67.9%

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

   6. Taylor expanded in a around inf 67.9%

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

      1. log1p-define 93.6%

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

   8. Simplified 93.6%

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

   9. Taylor expanded in a around 0 65.3%

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

       1. associate-+r+ 65.3%

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

       2. +-commutative 65.3%

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

   11. Simplified 65.3%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{b + \left(1 + a\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + \left(a + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b -3.1e-213) (log1p b) (+ (log 2.0) (* a 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= -3.1e-213) {
		tmp = log1p(b);
	} else {
		tmp = log(2.0) + (a * 0.5);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= -3.1e-213) {
		tmp = Math.log1p(b);
	} else {
		tmp = Math.log(2.0) + (a * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= -3.1e-213:
		tmp = math.log1p(b)
	else:
		tmp = math.log(2.0) + (a * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= -3.1e-213)
		tmp = log1p(b);
	else
		tmp = Float64(log(2.0) + Float64(a * 0.5));
	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, -3.1e-213], N[Log[1 + b], $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.0999999999999998e-213

   1. Initial program 36.7%

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

   3. Taylor expanded in b around 0 32.1%

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

      1. associate-+r+ 32.1%

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

      2. +-commutative 32.1%

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

   5. Simplified 32.1%

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

   6. Taylor expanded in a around inf 32.1%

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

      1. log1p-define 51.2%

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

   8. Simplified 51.2%

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

   9. Taylor expanded in b around inf 21.0%

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

   if -3.0999999999999998e-213 < b

   1. Initial program 69.7%

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

   3. Taylor expanded in b around 0 66.5%

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

      1. log1p-define 66.6%

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

   5. Simplified 66.6%

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

   6. Taylor expanded in a around 0 65.1%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 61.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b -3.6e-213) (log1p b) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= -3.6e-213) {
		tmp = log1p(b);
	} else {
		tmp = log((b + 2.0));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= -3.6e-213) {
		tmp = Math.log1p(b);
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= -3.6e-213:
		tmp = math.log1p(b)
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= -3.6e-213)
		tmp = log1p(b);
	else
		tmp = log(Float64(b + 2.0));
	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, -3.6e-213], N[Log[1 + b], $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.6000000000000001e-213

   1. Initial program 36.7%

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

   3. Taylor expanded in b around 0 32.1%

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

      1. associate-+r+ 32.1%

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

      2. +-commutative 32.1%

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

   5. Simplified 32.1%

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

   6. Taylor expanded in a around inf 32.1%

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

      1. log1p-define 51.2%

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

   8. Simplified 51.2%

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

   9. Taylor expanded in b around inf 21.0%

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

   if -3.6000000000000001e-213 < b

   1. Initial program 69.7%

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

   3. Taylor expanded in a around 0 66.8%

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

   4. Taylor expanded in b around 0 65.5%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 61.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b -3.4e-213) (log1p b) (log1p (+ b 1.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= -3.4e-213) {
		tmp = log1p(b);
	} else {
		tmp = log1p((b + 1.0));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= -3.4e-213) {
		tmp = Math.log1p(b);
	} else {
		tmp = Math.log1p((b + 1.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if b <= -3.4e-213:
		tmp = math.log1p(b)
	else:
		tmp = math.log1p((b + 1.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (b <= -3.4e-213)
		tmp = log1p(b);
	else
		tmp = log1p(Float64(b + 1.0));
	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, -3.4e-213], N[Log[1 + b], $MachinePrecision], N[Log[1 + N[(b + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.4000000000000002e-213

   1. Initial program 36.7%

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

   3. Taylor expanded in b around 0 32.1%

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

      1. associate-+r+ 32.1%

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

      2. +-commutative 32.1%

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

   5. Simplified 32.1%

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

   6. Taylor expanded in a around inf 32.1%

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

      1. log1p-define 51.2%

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

   8. Simplified 51.2%

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

   9. Taylor expanded in b around inf 21.0%

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

   if -3.4000000000000002e-213 < b

   1. Initial program 69.7%

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

   3. Taylor expanded in b around 0 67.9%

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

      1. associate-+r+ 67.9%

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

      2. +-commutative 67.9%

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

   5. Simplified 67.9%

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

   6. Taylor expanded in a around inf 67.9%

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

      1. log1p-define 93.6%

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

   8. Simplified 93.6%

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

   9. Taylor expanded in a around 0 65.5%

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

       1. +-commutative 65.5%

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

   11. Simplified 65.5%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{b + 1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{-213}:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in b around 0 75.1%

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

   1. log1p-define 75.2%

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

5. Simplified 75.2%

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

6. Taylor expanded in a around 0 50.2%

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

7. Final simplification 50.2%

   \[\leadsto b \cdot 0.5 + \log 2 \]
8. Add Preprocessing

Alternative 21: 50.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in b around 0 50.8%

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

   1. log1p-define 50.9%

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

5. Simplified 50.9%

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

6. Taylor expanded in a around 0 49.6%

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

7. Final simplification 49.6%

   \[\leadsto \log 2 \]
8. Add Preprocessing

Alternative 22: 11.8% 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 54.2%

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

3. Taylor expanded in b around 0 75.1%

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

   1. log1p-define 75.2%

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

5. Simplified 75.2%

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

6. Taylor expanded in b around inf 27.9%

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

7. Taylor expanded in a around 0 7.6%

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

8. Final simplification 7.6%

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

Reproduce

herbie shell --seed 2024066 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))