symmetry log of sum of exp

Percentage Accurate: 53.1% → 98.1%

Time: 11.2s

Alternatives: 11

Speedup: 1.4×

• Could not converge on a ground truth

  1. a = 9.70784856138393e+263
  2. b = 1.4642641730716982e-137

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: 53.1% 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: 98.1% 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 48.8%

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

   1. *-un-lft-identity 48.8%

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

   2. exp-prod 48.8%

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

   3. expm1-log1p-u 45.8%

      \[\leadsto \log \left(e^{a} + {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(b\right)\right)\right)}}\right) \]

   4. expm1-undefine 45.8%

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

   5. pow-sub 45.8%

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

4. Applied egg-rr 45.8%

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

   1. exp-1-e 45.8%

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

   2. log1p-undefine 45.8%

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

   3. rem-exp-log 48.8%

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

   4. unpow1 48.8%

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

   5. exp-1-e 48.8%

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

6. Simplified 48.8%

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

   1. log1p-expm1-u 48.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(e^{a} + \frac{{e}^{\left(1 + b\right)}}{e}\right)\right)\right)} \]

   2. expm1-undefine 48.8%

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

   3. add-exp-log 48.8%

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

   4. pow1 48.8%

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

   5. pow-div 48.8%

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

   6. add-exp-log 45.8%

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

   7. expm1-undefine 45.8%

      \[\leadsto \mathsf{log1p}\left(\left(e^{a} + {e}^{\color{blue}{\left(\mathsf{expm1}\left(\log \left(1 + b\right)\right)\right)}}\right) - 1\right) \]

   8. log1p-define 45.8%

      \[\leadsto \mathsf{log1p}\left(\left(e^{a} + {e}^{\left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(b\right)}\right)\right)}\right) - 1\right) \]

   9. expm1-log1p-u 48.8%

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

   10. pow-to-exp 48.8%

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

   11. log-E 48.8%

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

   12. *-un-lft-identity 48.8%

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

8. Applied egg-rr 48.8%

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

   1. associate--l+ 49.6%

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

   2. expm1-undefine 74.4%

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

10. Simplified 74.4%

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

Alternative 2: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-91}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-91) (/ 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-91) {
		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-91) 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-91) {
		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-91:
		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-91)
		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-91)
		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-91], 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) < 1.00000000000000002e-91

   1. Initial program 8.8%

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

   3. Taylor expanded in b around 0 97.4%

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

      1. log1p-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 97.4%

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

   if 1.00000000000000002e-91 < (exp.f64 a)

   1. Initial program 64.8%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-91}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left({e}^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-132) (/ b (+ (exp a) 1.0)) (log1p (pow E b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-132) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p(pow(((double) M_E), b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-132) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p(Math.pow(Math.E, b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-132:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(math.pow(math.e, b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-132)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p((exp(1) ^ 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], 5e-132], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Power[E, b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.9999999999999999e-132

   1. Initial program 8.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 98.7%

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

   if 4.9999999999999999e-132 < (exp.f64 a)

   1. Initial program 64.4%

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

      1. *-un-lft-identity 64.4%

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

      2. exp-prod 64.4%

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

      3. expm1-log1p-u 60.3%

         \[\leadsto \log \left(e^{a} + {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(b\right)\right)\right)}}\right) \]

      4. expm1-undefine 60.3%

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

      5. pow-sub 60.3%

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

   4. Applied egg-rr 60.3%

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

      1. exp-1-e 60.3%

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

      2. log1p-undefine 60.3%

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

      3. rem-exp-log 64.4%

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

      4. unpow1 64.4%

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

      5. exp-1-e 64.4%

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

   6. Simplified 64.4%

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

   7. Taylor expanded in a around 0 61.6%

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

      1. log1p-define 61.6%

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

      2. log-E 61.6%

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

      3. distribute-lft-in 61.6%

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

      4. metadata-eval 61.6%

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

      5. metadata-eval 61.6%

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

      6. cancel-sign-sub-inv 61.6%

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

      7. e-exp-1 61.6%

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

      8. div-exp 61.6%

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

      9. sub-neg 61.6%

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

      10. neg-mul-1 61.6%

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

      11. remove-double-neg 61.6%

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

      12. +-commutative 61.6%

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

   9. Simplified 61.6%

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

       1. *-un-lft-identity 61.6%

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

       2. exp-prod 61.6%

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

       3. associate--l+ 61.6%

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

       4. metadata-eval 61.6%

          \[\leadsto \mathsf{log1p}\left({\left(e^{1}\right)}^{\left(b + \color{blue}{0}\right)}\right) \]

       5. +-rgt-identity 61.6%

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

       6. e-exp-1 61.6%

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

   11. Applied egg-rr 61.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left({e}^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-132) (/ b (+ (exp a) 1.0)) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-132) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-132) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-132:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-132)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(exp(b));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-132], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.9999999999999999e-132

   1. Initial program 8.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 98.7%

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

   if 4.9999999999999999e-132 < (exp.f64 a)

   1. Initial program 64.4%

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

   3. Taylor expanded in a around 0 61.6%

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

      1. log1p-define 61.6%

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

   5. Simplified 61.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-132)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (* b (+ 0.5 (* b 0.125))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-132) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (b * (0.5 + (b * 0.125)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 5d-132) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (b * (0.5d0 + (b * 0.125d0)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-132) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (b * (0.5 + (b * 0.125)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-132:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (b * (0.5 + (b * 0.125)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-132)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(b * Float64(0.5 + Float64(b * 0.125))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-132)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (b * (0.5 + (b * 0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-132], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * N[(0.5 + N[(b * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.9999999999999999e-132

   1. Initial program 8.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 98.7%

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

   if 4.9999999999999999e-132 < (exp.f64 a)

   1. Initial program 64.4%

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

   3. Taylor expanded in a around 0 61.6%

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

   4. Taylor expanded in b around 0 58.9%

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

      1. *-commutative 58.9%

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

   6. Simplified 58.9%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-132) (/ b (+ (exp a) 1.0)) (+ (* b 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-132) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = (b * 0.5) + 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 (exp(a) <= 5d-132) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = (b * 0.5d0) + log(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) <= 5e-132) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = (b * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-132:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = (b * 0.5) + math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-132)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(Float64(b * 0.5) + log(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) <= 5e-132)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = (b * 0.5) + 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[N[Exp[a], $MachinePrecision], 5e-132], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(b * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.9999999999999999e-132

   1. Initial program 8.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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-define 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 98.7%

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

   if 4.9999999999999999e-132 < (exp.f64 a)

   1. Initial program 64.4%

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

   3. Taylor expanded in b around 0 59.3%

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

      1. log1p-define 59.8%

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

   5. Simplified 59.8%

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

   6. Taylor expanded in a around 0 58.2%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-132}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{log1p}\left(e^{a} + b \cdot \left(1 + b \cdot 0.5\right)\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (log1p (+ (exp a) (* b (+ 1.0 (* b 0.5))))))

C

assert(a < b);
double code(double a, double b) {
	return log1p((exp(a) + (b * (1.0 + (b * 0.5)))));
}

Java

assert a < b;
public static double code(double a, double b) {
	return Math.log1p((Math.exp(a) + (b * (1.0 + (b * 0.5)))));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log1p((math.exp(a) + (b * (1.0 + (b * 0.5)))))

Julia

a, b = sort([a, b])
function code(a, b)
	return log1p(Float64(exp(a) + Float64(b * Float64(1.0 + Float64(b * 0.5)))))
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[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.8%

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

   1. *-un-lft-identity 48.8%

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

   2. exp-prod 48.8%

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

   3. expm1-log1p-u 45.8%

      \[\leadsto \log \left(e^{a} + {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(b\right)\right)\right)}}\right) \]

   4. expm1-undefine 45.8%

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

   5. pow-sub 45.8%

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

4. Applied egg-rr 45.8%

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

   1. exp-1-e 45.8%

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

   2. log1p-undefine 45.8%

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

   3. rem-exp-log 48.8%

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

   4. unpow1 48.8%

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

   5. exp-1-e 48.8%

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

6. Simplified 48.8%

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

   1. log1p-expm1-u 48.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(e^{a} + \frac{{e}^{\left(1 + b\right)}}{e}\right)\right)\right)} \]

   2. expm1-undefine 48.8%

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

   3. add-exp-log 48.8%

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

   4. pow1 48.8%

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

   5. pow-div 48.8%

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

   6. add-exp-log 45.8%

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

   7. expm1-undefine 45.8%

      \[\leadsto \mathsf{log1p}\left(\left(e^{a} + {e}^{\color{blue}{\left(\mathsf{expm1}\left(\log \left(1 + b\right)\right)\right)}}\right) - 1\right) \]

   8. log1p-define 45.8%

      \[\leadsto \mathsf{log1p}\left(\left(e^{a} + {e}^{\left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(b\right)}\right)\right)}\right) - 1\right) \]

   9. expm1-log1p-u 48.8%

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

   10. pow-to-exp 48.8%

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

   11. log-E 48.8%

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

   12. *-un-lft-identity 48.8%

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

8. Applied egg-rr 48.8%

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

   1. associate--l+ 49.6%

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

   2. expm1-undefine 74.4%

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

10. Simplified 74.4%

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

11. Taylor expanded in b around 0 71.1%

    \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{a} + b \cdot \left(1 + 0.5 \cdot b\right)}\right) \]
12. Step-by-step derivation

    1. *-commutative 71.1%

       \[\leadsto \mathsf{log1p}\left(e^{a} + b \cdot \left(1 + \color{blue}{b \cdot 0.5}\right)\right) \]

13. Simplified 71.1%

    \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{a} + b \cdot \left(1 + b \cdot 0.5\right)}\right) \]
14. Add Preprocessing

Alternative 8: 48.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.8%

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

3. Taylor expanded in b around 0 70.4%

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

   1. log1p-define 71.1%

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

5. Simplified 71.1%

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

6. Taylor expanded in a around 0 43.0%

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

7. Final simplification 43.0%

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

Alternative 9: 48.5% 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 48.8%

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

3. Taylor expanded in a around 0 45.4%

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

4. Taylor expanded in b around 0 42.2%

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

5. Final simplification 42.2%

   \[\leadsto \log \left(b + 2\right) \]
6. Add Preprocessing

Alternative 10: 47.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.8%

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

3. Taylor expanded in b around 0 43.9%

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

   1. log1p-define 44.7%

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

5. Simplified 44.7%

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

6. Taylor expanded in a around 0 42.7%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{1}\right) \]
7. Add Preprocessing

Alternative 11: 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 48.8%

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

3. Taylor expanded in b around 0 43.9%

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

   1. log1p-define 44.7%

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

5. Simplified 44.7%

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

6. Taylor expanded in a around 0 42.8%

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

7. Taylor expanded in a around inf 7.5%

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

   1. *-commutative 7.5%

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

9. Simplified 7.5%

   \[\leadsto \color{blue}{a \cdot 0.5} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))