symmetry log of sum of exp

Percentage Accurate: 53.8% → 98.7%

Time: 15.5s

Alternatives: 13

Speedup: 2.8×

• could not determine a ground truth

  1. a = 1.1481652993734254e+281
  2. b = -1.78659524890858e-104

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: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 13.1%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 69.4%

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

4. Final simplification 76.8%

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

Alternative 2: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right) + \frac{1}{\frac{2}{b}}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (+ (log1p (exp a)) (/ 1.0 (/ 2.0 b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p(exp(a)) + (1.0 / (2.0 / b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p(Math.exp(a)) + (1.0 / (2.0 / b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(math.exp(a)) + (1.0 / (2.0 / b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log1p(exp(a)) + Float64(1.0 / Float64(2.0 / b)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(2.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 13.1%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 69.4%

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

   2. Taylor expanded in b around 0 66.0%

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

      1. log1p-def 66.0%

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

   4. Simplified 66.0%

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

      1. clear-num 66.0%

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

      2. inv-pow 66.0%

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

   6. Applied egg-rr 66.0%

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

   7. Taylor expanded in a around 0 66.0%

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

      1. unpow-1 66.0%

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

   9. Applied egg-rr 66.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right) + \frac{1}{\frac{2}{b}}\\ \end{array} \]

Alternative 3: 98.2% 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 55.8%

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

2. Taylor expanded in b around 0 74.2%

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

   1. log1p-def 74.3%

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

4. Simplified 74.3%

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

5. Final simplification 74.3%

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

Alternative 4: 97.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((b + 2.0d0)) + (a / (b + 2.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(Float64(b + 2.0)) + Float64(a / 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) <= 0.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((b + 2.0)) + (a / (b + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision] + N[(a / N[(b + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 13.1%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 69.4%

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

   2. Taylor expanded in b around 0 65.2%

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

   3. Taylor expanded in a around 0 64.3%

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

4. Final simplification 73.0%

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

Alternative 5: 97.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 13.1%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 69.4%

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

   2. Taylor expanded in b around 0 65.2%

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

   3. Taylor expanded in a around 0 64.3%

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

   4. Taylor expanded in b around 0 64.3%

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

4. Final simplification 73.0%

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

Alternative 6: 97.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + b\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1e+25) (/ b (+ (exp a) 1.0)) (log1p (+ (exp a) b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1e+25) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p((exp(a) + b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1e+25) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p((Math.exp(a) + b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1e+25:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p((math.exp(a) + b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1e+25)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(Float64(exp(a) + 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, -1e+25], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(N[Exp[a], $MachinePrecision] + b), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.00000000000000009e25

   1. Initial program 13.8%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 100.0%

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

   if -1.00000000000000009e25 < a

   1. Initial program 67.5%

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

   2. Taylor expanded in b around 0 63.4%

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

      1. log1p-def 66.2%

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

      2. +-commutative 66.2%

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

   4. Applied egg-rr 66.2%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + b\right)\\ \end{array} \]

Alternative 7: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -108:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -108.0) (/ b (+ (exp a) 1.0)) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -108.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -108.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -108.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -108.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(exp(b));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -108.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -108

   1. Initial program 13.1%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 100.0%

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

   if -108 < a

   1. Initial program 69.4%

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

   2. Taylor expanded in a around 0 66.4%

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

      1. log1p-def 66.4%

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

   4. Simplified 66.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -108:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternative 8: 97.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 13.1%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 100.0%

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

   if -1 < a

   1. Initial program 69.4%

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

   2. Taylor expanded in b around 0 65.2%

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

   3. Taylor expanded in a around 0 64.2%

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

      1. log1p-expm1-u 64.2%

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

      2. expm1-udef 64.2%

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

      3. add-exp-log 64.2%

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

      4. +-commutative 64.2%

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

      5. +-commutative 64.2%

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

      6. associate-+l+ 64.2%

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

   5. Applied egg-rr 64.2%

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

      1. associate-+r+ 64.2%

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

      2. +-commutative 64.2%

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

      3. associate--l+ 64.3%

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

      4. +-commutative 64.3%

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

      5. metadata-eval 64.3%

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

   7. Simplified 64.3%

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

4. Final simplification 72.9%

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

Alternative 9: 49.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

2. Taylor expanded in b around 0 74.2%

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

   1. log1p-def 74.3%

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

4. Simplified 74.3%

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

5. Taylor expanded in a around 0 49.9%

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

6. Final simplification 49.9%

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

Alternative 10: 48.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

2. Taylor expanded in b around 0 51.4%

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

3. Taylor expanded in a around 0 49.1%

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

4. Final simplification 49.1%

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

Alternative 11: 48.3% 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 55.8%

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

2. Taylor expanded in a around 0 51.5%

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

   1. log1p-def 51.5%

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

4. Simplified 51.5%

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

5. Taylor expanded in b around 0 49.7%

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

6. Final simplification 49.7%

   \[\leadsto \log 2 \]

Alternative 12: 2.6% accurate, 101.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

2. Taylor expanded in b around 0 51.4%

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

3. Taylor expanded in a around 0 49.4%

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

4. Taylor expanded in a around inf 4.0%

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

   1. +-commutative 4.0%

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

6. Simplified 4.0%

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

7. Taylor expanded in b around 0 7.4%

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

   1. *-commutative 7.4%

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

9. Simplified 7.4%

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

10. Final simplification 7.4%

    \[\leadsto a \cdot 0.5 \]

Alternative 13: 2.6% accurate, 101.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

2. Taylor expanded in b around 0 51.4%

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

3. Taylor expanded in a around 0 49.4%

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

4. Taylor expanded in a around inf 4.0%

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

   1. +-commutative 4.0%

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

6. Simplified 4.0%

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

7. Taylor expanded in b around inf 3.9%

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

8. Final simplification 3.9%

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

Reproduce

herbie shell --seed 2023331 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))