symmetry log of sum of exp

Percentage Accurate: 53.4% → 98.7%

Time: 15.2s

Alternatives: 13

Speedup: 2.8×

• could not determine a ground truth

  1. a = -1.3858873735053372e+185
  2. b = -6.952781937174182e+155

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: 53.4% 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

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

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 (+ 1.0 (exp a))) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} 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 / (1.0d0 + exp(a))
    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 / (1.0 + Math.exp(a));
	} 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 / (1.0 + math.exp(a))
	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(1.0 + exp(a)));
	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 / (1.0 + exp(a));
	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[(1.0 + N[Exp[a], $MachinePrecision]), $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 10.8%

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

   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 0.0 < (exp.f64 a)

   1. Initial program 70.5%

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

4. Final simplification 78.0%

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

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := 1 + e^{a}\\ \frac{b}{t_0} + \mathsf{fma}\left(\frac{1}{t_0} + \frac{-1}{{t_0}^{2}}, b \cdot \left(b \cdot 0.5\right), \mathsf{log1p}\left(e^{a}\right)\right) \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp a))))
   (+
    (/ b t_0)
    (fma
     (+ (/ 1.0 t_0) (/ -1.0 (pow t_0 2.0)))
     (* b (* b 0.5))
     (log1p (exp a))))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = 1.0 + exp(a);
	return (b / t_0) + fma(((1.0 / t_0) + (-1.0 / pow(t_0, 2.0))), (b * (b * 0.5)), log1p(exp(a)));
}

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(1.0 + exp(a))
	return Float64(Float64(b / t_0) + fma(Float64(Float64(1.0 / t_0) + Float64(-1.0 / (t_0 ^ 2.0))), Float64(b * Float64(b * 0.5)), log1p(exp(a))))
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]}, N[(N[(b / t$95$0), $MachinePrecision] + N[(N[(N[(1.0 / t$95$0), $MachinePrecision] + N[(-1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 55.3%

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

2. Taylor expanded in b around 0 76.3%

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

   1. +-commutative 76.3%

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

   2. +-commutative 76.3%

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

   3. associate-+l+ 76.3%

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

   4. associate-*r* 76.3%

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

   5. *-commutative 76.3%

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

   6. fma-def 76.3%

      \[\leadsto \frac{b}{1 + e^{a}} + \color{blue}{\mathsf{fma}\left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}, 0.5 \cdot {b}^{2}, \log \left(1 + e^{a}\right)\right)} \]

4. Simplified 76.3%

   \[\leadsto \color{blue}{\frac{b}{1 + e^{a}} + \mathsf{fma}\left(\frac{1}{1 + e^{a}} + \frac{-1}{{\left(1 + e^{a}\right)}^{2}}, b \cdot \left(b \cdot 0.5\right), \mathsf{log1p}\left(e^{a}\right)\right)} \]

5. Final simplification 76.3%

   \[\leadsto \frac{b}{1 + e^{a}} + \mathsf{fma}\left(\frac{1}{1 + e^{a}} + \frac{-1}{{\left(1 + e^{a}\right)}^{2}}, b \cdot \left(b \cdot 0.5\right), \mathsf{log1p}\left(e^{a}\right)\right) \]

Alternative 3: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 10.8%

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

   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 0.0 < (exp.f64 a)

   1. Initial program 70.5%

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

   2. Taylor expanded in b around 0 66.9%

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

      1. *-un-lft-identity 66.9%

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

      2. log-prod 66.9%

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

      3. metadata-eval 66.9%

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

      4. log1p-def 66.9%

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

      5. +-commutative 66.9%

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

   4. Applied egg-rr 66.9%

      \[\leadsto \color{blue}{0 + \mathsf{log1p}\left(e^{a} + b\right)} \]
   5. Step-by-step derivation

      1. +-lft-identity 66.9%

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

   6. Simplified 66.9%

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

4. Final simplification 75.3%

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

Alternative 4: 97.9% 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 55.3%

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

   1. add-sqr-sqrt 54.2%

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

   2. log-prod 54.6%

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

3. Applied egg-rr 54.6%

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

   1. log-prod 54.2%

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

   2. rem-square-sqrt 55.3%

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

   3. log1p-expm1 55.3%

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

   4. expm1-def 55.3%

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

   5. rem-exp-log 55.3%

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

   6. associate--l+ 55.4%

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

   7. expm1-def 78.0%

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

5. Simplified 78.0%

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

6. Final simplification 78.0%

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

Alternative 5: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := 2 + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log t_0 + \frac{a}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ b (* 0.5 (* b b))))))
   (if (<= (exp a) 0.0) (/ b (+ 1.0 (exp a))) (+ (log t_0) (/ a t_0)))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = 2.0 + (b + (0.5 * (b * b)));
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log(t_0) + (a / t_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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (b + (0.5d0 * (b * b)))
    if (exp(a) <= 0.0d0) then
        tmp = b / (1.0d0 + exp(a))
    else
        tmp = log(t_0) + (a / t_0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = 2.0 + (b + (0.5 * (b * b)));
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log(t_0) + (a / t_0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = 2.0 + (b + (0.5 * (b * b)))
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log(t_0) + (a / t_0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(2.0 + Float64(b + Float64(0.5 * Float64(b * b))))
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = Float64(log(t_0) + Float64(a / t_0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = 2.0 + (b + (0.5 * (b * b)));
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = b / (1.0 + exp(a));
	else
		tmp = log(t_0) + (a / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(2.0 + N[(b + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[t$95$0], $MachinePrecision] + N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 10.8%

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

   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 0.0 < (exp.f64 a)

   1. Initial program 70.5%

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

   2. Taylor expanded in b around 0 68.1%

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

      1. +-commutative 68.1%

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

      2. associate-+l+ 68.1%

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

      3. associate-+l+ 68.1%

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

      4. *-commutative 68.1%

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

      5. unpow2 68.1%

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

      6. associate-*l* 68.1%

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

      7. fma-def 68.1%

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

   4. Simplified 68.1%

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

   5. Taylor expanded in a around 0 67.4%

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

      1. unpow2 67.4%

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

      2. unpow2 67.4%

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

   7. Simplified 67.4%

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

4. Final simplification 75.7%

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

Alternative 6: 97.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} 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) <= 0.0d0) then
        tmp = b / (1.0d0 + exp(a))
    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) <= 0.0) {
		tmp = b / (1.0 + Math.exp(a));
	} 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) <= 0.0:
		tmp = b / (1.0 + math.exp(a))
	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) <= 0.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	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) <= 0.0)
		tmp = b / (1.0 + exp(a));
	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], 0.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $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) < 0.0

   1. Initial program 10.8%

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

   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 0.0 < (exp.f64 a)

   1. Initial program 70.5%

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

   2. Taylor expanded in b around 0 66.9%

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

   3. Taylor expanded in a around 0 66.6%

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

      1. +-commutative 66.6%

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

      2. *-commutative 66.6%

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

      3. unpow2 66.6%

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

   5. Simplified 66.6%

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

4. Final simplification 75.1%

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

Alternative 7: 96.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Derivation

1. Initial program 55.3%

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

2. Taylor expanded in b around 0 53.0%

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

   1. +-commutative 53.0%

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

   2. associate-+l+ 52.9%

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

   3. associate-+l+ 52.9%

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

   4. *-commutative 52.9%

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

   5. unpow2 52.9%

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

   6. associate-*l* 52.9%

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

   7. fma-def 52.9%

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

4. Simplified 52.9%

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

5. Taylor expanded in a around inf 53.0%

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

   1. log1p-def 75.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(b + \left(e^{a} + 0.5 \cdot {b}^{2}\right)\right)} \]

   2. associate-+r+ 75.6%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(b + e^{a}\right) + 0.5 \cdot {b}^{2}}\right) \]

   3. +-commutative 75.6%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(e^{a} + b\right)} + 0.5 \cdot {b}^{2}\right) \]

   4. unpow2 75.6%

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

7. Simplified 75.6%

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

8. Final simplification 75.6%

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

Alternative 8: 97.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right) + \frac{a}{b + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.4) (/ b (+ 1.0 (exp a))) (+ (log (+ b 2.0)) (/ a (+ b 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.4) {
		tmp = b / (1.0 + exp(a));
	} 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 <= (-1.4d0)) then
        tmp = b / (1.0d0 + exp(a))
    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 <= -1.4) {
		tmp = b / (1.0 + Math.exp(a));
	} 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 <= -1.4:
		tmp = b / (1.0 + math.exp(a))
	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 <= -1.4)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	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 <= -1.4)
		tmp = b / (1.0 + exp(a));
	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, -1.4], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $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 < -1.3999999999999999

   1. Initial program 10.8%

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

   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 -1.3999999999999999 < a

   1. Initial program 70.5%

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

   2. Taylor expanded in b around 0 66.9%

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

   3. Taylor expanded in a around 0 66.2%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right) + \frac{a}{b + 2}\\ \end{array} \]

Alternative 9: 97.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Python

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 10.8%

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

   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 < a

   1. Initial program 70.5%

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

   2. Taylor expanded in b around 0 66.9%

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

   3. Taylor expanded in a around 0 66.1%

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

      1. associate-+r+ 66.1%

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

      2. +-commutative 66.1%

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

      3. associate-+l+ 66.1%

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

   5. Simplified 66.1%

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

4. Final simplification 74.7%

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

Alternative 10: 48.8% 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 55.3%

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

2. Taylor expanded in b around 0 76.0%

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

   1. log1p-def 76.0%

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

4. Simplified 76.0%

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

5. Taylor expanded in a around 0 50.2%

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

6. Final simplification 50.2%

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

Alternative 11: 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 55.3%

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

2. Taylor expanded in b around 0 51.9%

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

3. Taylor expanded in a around 0 49.5%

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

4. Final simplification 49.5%

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

Alternative 12: 48.5% 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 55.3%

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

2. Taylor expanded in b around 0 51.9%

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

3. Taylor expanded in a around 0 49.5%

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

   1. log1p-expm1-u 49.5%

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

   2. expm1-udef 49.5%

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

   3. add-exp-log 49.5%

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

   4. +-commutative 49.5%

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

5. Applied egg-rr 49.5%

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

   1. associate--l+ 49.5%

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

   2. metadata-eval 49.5%

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

7. Simplified 49.5%

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

8. Final simplification 49.5%

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

Alternative 13: 48.0% 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 55.3%

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

2. Taylor expanded in b around 0 51.9%

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

3. Taylor expanded in a around 0 49.5%

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

4. Taylor expanded in b around 0 49.7%

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

5. Final simplification 49.7%

   \[\leadsto \log 2 \]

Reproduce

herbie shell --seed 2023268 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))