symmetry log of sum of exp

Percentage Accurate: 53.5% → 99.0%

Time: 19.3s

Alternatives: 11

Speedup: 2.8×

• could not determine a ground truth

  1. a = 5.681568507062136e+106
  2. b = 6.5330868485048055e-307

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-110}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-110) (/ 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) <= 2e-110) {
		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) <= 2d-110) 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) <= 2e-110) {
		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) <= 2e-110:
		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) <= 2e-110)
		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) <= 2e-110)
		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], 2e-110], 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) < 2.0000000000000001e-110

   1. Initial program 6.8%

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

   2. Taylor expanded in b around 0 96.6%

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

      1. log1p-def 96.6%

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

   4. Simplified 96.6%

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

   5. Taylor expanded in b around inf 96.6%

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

   if 2.0000000000000001e-110 < (exp.f64 a)

   1. Initial program 65.4%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-110}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

Alternative 2: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-110}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;e^{a} \leq 1:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-110)
   (/ b (+ (exp a) 1.0))
   (if (<= (exp a) 1.0) (log1p (exp a)) (log1p (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-110) {
		tmp = b / (exp(a) + 1.0);
	} else if (exp(a) <= 1.0) {
		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 (Math.exp(a) <= 2e-110) {
		tmp = b / (Math.exp(a) + 1.0);
	} else if (Math.exp(a) <= 1.0) {
		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 math.exp(a) <= 2e-110:
		tmp = b / (math.exp(a) + 1.0)
	elif math.exp(a) <= 1.0:
		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 (exp(a) <= 2e-110)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	elseif (exp(a) <= 1.0)
		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[N[Exp[a], $MachinePrecision], 2e-110], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[a], $MachinePrecision], 1.0], 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 (exp.f64 a) < 2.0000000000000001e-110

   1. Initial program 6.8%

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

   2. Taylor expanded in b around 0 96.6%

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

      1. log1p-def 96.6%

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

   4. Simplified 96.6%

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

   5. Taylor expanded in b around inf 96.6%

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

   if 2.0000000000000001e-110 < (exp.f64 a) < 1

   1. Initial program 64.9%

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

   2. Taylor expanded in b around 0 62.5%

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

      1. log1p-def 63.0%

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

   4. Simplified 63.0%

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

   if 1 < (exp.f64 a)

   1. Initial program 81.0%

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

   2. Taylor expanded in a around 0 33.3%

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

      1. log1p-def 33.3%

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

   4. Simplified 33.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-110}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;e^{a} \leq 1:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternative 3: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (log1p (+ (exp a) (expm1 b))))

C

assert(a < b);
double code(double a, double b) {
	return log1p((exp(a) + expm1(b)));
}

Java

assert a < b;
public static double code(double a, double b) {
	return Math.log1p((Math.exp(a) + Math.expm1(b)));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log1p((math.exp(a) + math.expm1(b)))

Julia

a, b = sort([a, b])
function code(a, b)
	return log1p(Float64(exp(a) + expm1(b)))
end

Wolfram

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

Derivation

1. Initial program 52.8%

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

   1. add-sqr-sqrt 51.8%

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

   2. log-prod 52.2%

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

3. Applied egg-rr 52.2%

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

   1. log-prod 51.8%

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

   2. rem-square-sqrt 52.8%

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

   3. log1p-expm1 52.8%

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

   4. expm1-def 52.8%

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

   5. rem-exp-log 52.8%

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

   6. associate--l+ 53.3%

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

   7. expm1-def 72.5%

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

5. Simplified 72.5%

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

6. Final simplification 72.5%

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

Alternative 4: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-110}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + \left(a + \left(b + 0.5 \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-110)
   (/ b (+ (exp a) 1.0))
   (log (+ 2.0 (+ a (+ b (* 0.5 (* a a))))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-110) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((2.0 + (a + (b + (0.5 * (a * a))))));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 2d-110) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((2.0d0 + (a + (b + (0.5d0 * (a * a))))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 2e-110) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((2.0 + (a + (b + (0.5 * (a * a))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 2e-110:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((2.0 + (a + (b + (0.5 * (a * a))))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-110)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(2.0 + Float64(a + Float64(b + Float64(0.5 * Float64(a * a))))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 2e-110)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((2.0 + (a + (b + (0.5 * (a * a))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 2e-110], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(2.0 + N[(a + N[(b + N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2.0000000000000001e-110

   1. Initial program 6.8%

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

   2. Taylor expanded in b around 0 96.6%

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

      1. log1p-def 96.6%

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

   4. Simplified 96.6%

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

   5. Taylor expanded in b around inf 96.6%

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

   if 2.0000000000000001e-110 < (exp.f64 a)

   1. Initial program 65.4%

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

   2. Taylor expanded in b around 0 61.9%

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

   3. Taylor expanded in a around 0 61.3%

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

      1. unpow2 61.3%

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

   5. Simplified 61.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-110}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + \left(a + \left(b + 0.5 \cdot \left(a \cdot a\right)\right)\right)\right)\\ \end{array} \]

Alternative 5: 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 52.8%

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

2. Taylor expanded in b around 0 50.0%

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

   1. log1p-def 69.6%

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

   2. +-commutative 69.6%

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

4. Applied egg-rr 69.6%

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

5. Final simplification 69.6%

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

Alternative 6: 97.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right) + \frac{a}{b + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.35) (/ 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 (a <= -1.35) {
		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 (a <= (-1.35d0)) 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 (a <= -1.35) {
		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 a <= -1.35:
		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 (a <= -1.35)
		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 (a <= -1.35)
		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[a, -1.35], 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 a < -1.3500000000000001

   1. Initial program 6.8%

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

   2. Taylor expanded in b around 0 94.9%

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

      1. log1p-def 96.7%

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

   4. Simplified 96.7%

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

   5. Taylor expanded in b around inf 94.9%

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

   if -1.3500000000000001 < a

   1. Initial program 65.7%

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

   2. Taylor expanded in b around 0 62.2%

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

   3. Taylor expanded in a around 0 60.9%

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

      1. +-commutative 60.9%

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

      2. +-commutative 60.9%

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

   5. Simplified 60.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right) + \frac{a}{b + 2}\\ \end{array} \]

Alternative 7: 97.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 6.8%

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

   2. Taylor expanded in b around 0 94.9%

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

      1. log1p-def 96.7%

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

   4. Simplified 96.7%

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

   5. Taylor expanded in b around inf 94.9%

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

   if -1 < a

   1. Initial program 65.7%

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

   2. Taylor expanded in b around 0 62.2%

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

   3. Taylor expanded in a around 0 60.8%

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

      1. +-commutative 60.8%

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

      2. +-commutative 60.8%

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

      3. associate-+l+ 60.8%

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

   5. Simplified 60.8%

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

4. Final simplification 68.3%

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

Alternative 8: 49.2% 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 52.8%

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

2. Taylor expanded in a around 0 49.6%

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

   1. log1p-def 49.6%

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

4. Simplified 49.6%

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

5. Taylor expanded in b around 0 47.8%

   \[\leadsto \color{blue}{\log 2 + 0.5 \cdot b} \]
6. Step-by-step derivation

   1. *-commutative 47.8%

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

7. Simplified 47.8%

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

8. Final simplification 47.8%

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

Alternative 9: 49.0% 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 52.8%

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

2. Taylor expanded in b around 0 50.0%

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

3. Taylor expanded in a around 0 47.1%

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

   1. +-commutative 47.1%

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

5. Simplified 47.1%

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

6. Final simplification 47.1%

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

Alternative 10: 49.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

2. Taylor expanded in b around 0 50.0%

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

3. Taylor expanded in a around 0 47.1%

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

   1. +-commutative 47.1%

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

5. Simplified 47.1%

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

   1. log1p-expm1-u 47.1%

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

   2. expm1-udef 47.1%

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

   3. add-exp-log 47.1%

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

7. Applied egg-rr 47.1%

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

   1. associate--l+ 47.1%

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

   2. metadata-eval 47.1%

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

9. Simplified 47.1%

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

10. Final simplification 47.1%

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

Alternative 11: 48.5% 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 52.8%

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

2. Taylor expanded in a around 0 49.6%

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

   1. log1p-def 49.6%

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

4. Simplified 49.6%

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

5. Taylor expanded in b around 0 47.8%

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

6. Final simplification 47.8%

   \[\leadsto \log 2 \]

Reproduce

herbie shell --seed 2023287 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))