symmetry log of sum of exp

Percentage Accurate: 53.7% → 98.7%

Time: 18.1s

Alternatives: 10

Speedup: 2.9×

• could not determine a ground truth

  1. a = -1.2024655073235916e+253
  2. b = 5.284330945093561e+37

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -280:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;a \leq -5 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -280.0)
   (/ b (+ (exp a) 1.0))
   (if (<= a -5e-162) (log1p (exp a)) (log1p (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -280.0) {
		tmp = b / (exp(a) + 1.0);
	} else if (a <= -5e-162) {
		tmp = log1p(exp(a));
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -280.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else if (a <= -5e-162) {
		tmp = Math.log1p(Math.exp(a));
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -280.0:
		tmp = b / (math.exp(a) + 1.0)
	elif a <= -5e-162:
		tmp = math.log1p(math.exp(a))
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -280.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	elseif (a <= -5e-162)
		tmp = log1p(exp(a));
	else
		tmp = log1p(exp(b));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -280.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -5e-162], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -280

   1. Initial program 9.4%

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

   3. Taylor expanded in b around 0 98.7%

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

      1. log1p-def 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in b around inf 98.7%

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

   if -280 < a < -5.00000000000000014e-162

   1. Initial program 76.4%

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

   3. Taylor expanded in b around 0 73.3%

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

      1. log1p-def 73.3%

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

   5. Simplified 73.3%

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

   if -5.00000000000000014e-162 < a

   1. Initial program 71.6%

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

   3. Taylor expanded in a around 0 69.0%

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

      1. log1p-def 69.7%

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

   5. Simplified 69.7%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -280:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;a \leq -5 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-22}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-22) (/ b (+ (exp a) 1.0)) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-22) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((exp(a) + exp(b)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 1d-22) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((exp(a) + exp(b)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 1e-22) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((Math.exp(a) + Math.exp(b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 1e-22:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((math.exp(a) + math.exp(b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 1e-22)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(exp(a) + exp(b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 1e-22)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((exp(a) + exp(b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 1e-22], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1e-22

   1. Initial program 9.4%

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

   3. Taylor expanded in b around 0 98.7%

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

      1. log1p-def 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in b around inf 98.7%

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

   if 1e-22 < (exp.f64 a)

   1. Initial program 72.8%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-22}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% 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 78.7%

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

   1. log1p-def 78.7%

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

5. Simplified 78.7%

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

6. Final simplification 78.7%

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

Alternative 4: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -420:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -420.0) (/ b (+ (exp a) 1.0)) (log1p (exp a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -420.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p(exp(a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -420.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p(Math.exp(a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -420.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(math.exp(a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -420.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(exp(a));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -420.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -420

   1. Initial program 9.4%

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

   3. Taylor expanded in b around 0 98.7%

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

      1. log1p-def 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in b around inf 98.7%

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

   if -420 < a

   1. Initial program 72.8%

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

   3. Taylor expanded in b around 0 70.0%

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

      1. log1p-def 70.1%

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

   5. Simplified 70.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -420:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.36:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.36) (/ b 2.0) (+ (log 2.0) (* a 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.36) {
		tmp = b / 2.0;
	} else {
		tmp = log(2.0) + (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 (a <= (-1.36d0)) then
        tmp = b / 2.0d0
    else
        tmp = log(2.0d0) + (a * 0.5d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.36) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log(2.0) + (a * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.36:
		tmp = b / 2.0
	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 (a <= -1.36)
		tmp = Float64(b / 2.0);
	else
		tmp = Float64(log(2.0) + 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 (a <= -1.36)
		tmp = b / 2.0;
	else
		tmp = log(2.0) + (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[a, -1.36], N[(b / 2.0), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3600000000000001

   1. Initial program 9.4%

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

   3. Taylor expanded in b around 0 98.7%

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

      1. log1p-def 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in b around inf 98.7%

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

   7. Taylor expanded in a around 0 18.6%

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

   if -1.3600000000000001 < a

   1. Initial program 72.8%

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

   3. Taylor expanded in b around 0 70.0%

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

      1. log1p-def 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in a around 0 68.3%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.36:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.36:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.36) (/ b (+ (exp a) 1.0)) (+ (log 2.0) (* a 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.36) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (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 (a <= (-1.36d0)) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (a * 0.5d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.36) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (a * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.36:
		tmp = b / (math.exp(a) + 1.0)
	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 (a <= -1.36)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + 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 (a <= -1.36)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (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[a, -1.36], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3600000000000001

   1. Initial program 9.4%

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

   3. Taylor expanded in b around 0 98.7%

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

      1. log1p-def 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in b around inf 98.7%

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

   if -1.3600000000000001 < a

   1. Initial program 72.8%

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

   3. Taylor expanded in b around 0 70.0%

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

      1. log1p-def 70.1%

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

   5. Simplified 70.1%

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

   6. Taylor expanded in a around 0 68.3%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.36:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 9.4%

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

   3. Taylor expanded in b around 0 98.7%

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

      1. log1p-def 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in b around inf 98.7%

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

   7. Taylor expanded in a around 0 18.6%

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

   if -1 < a

   1. Initial program 72.8%

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

   3. Taylor expanded in b around 0 70.0%

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

   4. Taylor expanded in a around 0 68.2%

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

4. Final simplification 53.7%

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

Alternative 8: 56.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 9.4%

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

   3. Taylor expanded in b around 0 98.7%

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

      1. log1p-def 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in b around inf 98.7%

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

   7. Taylor expanded in a around 0 18.6%

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

   if -1 < a

   1. Initial program 72.8%

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

   3. Taylor expanded in b around 0 70.0%

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

   4. Taylor expanded in a around 0 68.2%

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

      1. log1p-expm1-u 68.2%

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

      2. expm1-udef 68.2%

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

      3. add-exp-log 68.2%

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

      4. +-commutative 68.2%

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

   6. Applied egg-rr 68.2%

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

      1. associate--l+ 68.2%

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

      2. metadata-eval 68.2%

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

   8. Simplified 68.2%

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

4. Final simplification 53.7%

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

Alternative 9: 56.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -118:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

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

C

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

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -118.0:
		tmp = b / 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 <= -118.0)
		tmp = Float64(b / 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 <= -118.0)
		tmp = b / 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, -118.0], N[(b / 2.0), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -118

   1. Initial program 9.4%

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

   3. Taylor expanded in b around 0 98.7%

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

      1. log1p-def 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in b around inf 98.7%

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

   7. Taylor expanded in a around 0 18.6%

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

   if -118 < a

   1. Initial program 72.8%

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

   3. Taylor expanded in b around 0 70.0%

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

   4. Taylor expanded in a around 0 67.1%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -118:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 12.0% 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 78.7%

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

   1. log1p-def 78.7%

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

5. Simplified 78.7%

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

6. Taylor expanded in b around inf 31.2%

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

7. Taylor expanded in a around 0 7.7%

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

8. Final simplification 7.7%

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

Reproduce

herbie shell --seed 2024031 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))