symmetry log of sum of exp

Percentage Accurate: 54.4% → 98.4%

Time: 18.0s

Alternatives: 11

Speedup: 2.9×

• could not determine a ground truth

  1. a = -4.51736353577865e+250
  2. b = 9.465938555771164e+26

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: 54.4% 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: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := 1 + e^{a}\\ t_1 := \frac{1}{t_0}\\ t_2 := \frac{-1}{{t_0}^{2}}\\ \log t_0 + \left(0.16666666666666666 \cdot \left({b}^{3} \cdot \left(\left(2 \cdot \frac{1}{{t_0}^{3}} + t_1\right) + 3 \cdot t_2\right)\right) + \left(0.5 \cdot \left({b}^{2} \cdot \left(t_1 + t_2\right)\right) + \frac{b}{t_0}\right)\right) \end{array} \end{array} \]

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))) (t_1 (/ 1.0 t_0)) (t_2 (/ -1.0 (pow t_0 2.0))))
   (+
    (log t_0)
    (+
     (*
      0.16666666666666666
      (* (pow b 3.0) (+ (+ (* 2.0 (/ 1.0 (pow t_0 3.0))) t_1) (* 3.0 t_2))))
     (+ (* 0.5 (* (pow b 2.0) (+ t_1 t_2))) (/ b t_0))))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = 1.0 + exp(a);
	double t_1 = 1.0 / t_0;
	double t_2 = -1.0 / pow(t_0, 2.0);
	return log(t_0) + ((0.16666666666666666 * (pow(b, 3.0) * (((2.0 * (1.0 / pow(t_0, 3.0))) + t_1) + (3.0 * t_2)))) + ((0.5 * (pow(b, 2.0) * (t_1 + t_2))) + (b / t_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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = 1.0d0 + exp(a)
    t_1 = 1.0d0 / t_0
    t_2 = (-1.0d0) / (t_0 ** 2.0d0)
    code = log(t_0) + ((0.16666666666666666d0 * ((b ** 3.0d0) * (((2.0d0 * (1.0d0 / (t_0 ** 3.0d0))) + t_1) + (3.0d0 * t_2)))) + ((0.5d0 * ((b ** 2.0d0) * (t_1 + t_2))) + (b / t_0)))
end function

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = 1.0 + Math.exp(a);
	double t_1 = 1.0 / t_0;
	double t_2 = -1.0 / Math.pow(t_0, 2.0);
	return Math.log(t_0) + ((0.16666666666666666 * (Math.pow(b, 3.0) * (((2.0 * (1.0 / Math.pow(t_0, 3.0))) + t_1) + (3.0 * t_2)))) + ((0.5 * (Math.pow(b, 2.0) * (t_1 + t_2))) + (b / t_0)));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = 1.0 + math.exp(a)
	t_1 = 1.0 / t_0
	t_2 = -1.0 / math.pow(t_0, 2.0)
	return math.log(t_0) + ((0.16666666666666666 * (math.pow(b, 3.0) * (((2.0 * (1.0 / math.pow(t_0, 3.0))) + t_1) + (3.0 * t_2)))) + ((0.5 * (math.pow(b, 2.0) * (t_1 + t_2))) + (b / t_0)))

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(1.0 + exp(a))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(-1.0 / (t_0 ^ 2.0))
	return Float64(log(t_0) + Float64(Float64(0.16666666666666666 * Float64((b ^ 3.0) * Float64(Float64(Float64(2.0 * Float64(1.0 / (t_0 ^ 3.0))) + t_1) + Float64(3.0 * t_2)))) + Float64(Float64(0.5 * Float64((b ^ 2.0) * Float64(t_1 + t_2))) + Float64(b / t_0))))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	t_0 = 1.0 + exp(a);
	t_1 = 1.0 / t_0;
	t_2 = -1.0 / (t_0 ^ 2.0);
	tmp = log(t_0) + ((0.16666666666666666 * ((b ^ 3.0) * (((2.0 * (1.0 / (t_0 ^ 3.0))) + t_1) + (3.0 * t_2)))) + ((0.5 * ((b ^ 2.0) * (t_1 + t_2))) + (b / t_0)));
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]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[Log[t$95$0], $MachinePrecision] + N[(N[(0.16666666666666666 * N[(N[Power[b, 3.0], $MachinePrecision] * N[(N[(N[(2.0 * N[(1.0 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + N[(3.0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[(N[Power[b, 2.0], $MachinePrecision] * N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 60.2%

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

2. Taylor expanded in b around 0 75.0%

   \[\leadsto \color{blue}{\log \left(1 + e^{a}\right) + \left(0.16666666666666666 \cdot \left({b}^{3} \cdot \left(\left(2 \cdot \frac{1}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{1 + e^{a}}\right) - 3 \cdot \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \left(0.5 \cdot \left({b}^{2} \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{b}{1 + e^{a}}\right)\right)} \]

3. Final simplification 75.0%

   \[\leadsto \log \left(1 + e^{a}\right) + \left(0.16666666666666666 \cdot \left({b}^{3} \cdot \left(\left(2 \cdot \frac{1}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{1 + e^{a}}\right) + 3 \cdot \frac{-1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \left(0.5 \cdot \left({b}^{2} \cdot \left(\frac{1}{1 + e^{a}} + \frac{-1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{b}{1 + e^{a}}\right)\right) \]

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

2. Taylor expanded in b around 0 75.0%

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

   1. log1p-def 75.0%

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

4. Simplified 75.0%

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

5. Final simplification 75.0%

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

Alternative 3: 58.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (* b 0.5) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b * 0.5;
	} 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) <= 0.0) {
		tmp = b * 0.5;
	} 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) <= 0.0:
		tmp = b * 0.5
	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) <= 0.0)
		tmp = Float64(b * 0.5);
	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], 0.0], N[(b * 0.5), $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 11.6%

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

      1. add-cbrt-cube 11.6%

         \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)}\right)} \]

      2. pow1/3 11.6%

         \[\leadsto \log \color{blue}{\left({\left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)}^{0.3333333333333333}\right)} \]

      3. log-pow 11.6%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)} \]

      4. pow3 11.6%

         \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(e^{a} + e^{b}\right)}^{3}\right)} \]

      5. log-pow 11.6%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

   3. Applied egg-rr 11.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

   4. Taylor expanded in b around 0 96.4%

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

      1. log1p-def 96.4%

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

      2. +-commutative 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in a around 0 18.2%

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

   8. Taylor expanded in b around inf 18.2%

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

      1. *-commutative 18.2%

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

   10. Simplified 18.2%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0 69.4%

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

      1. log1p-def 69.4%

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

   4. Simplified 69.4%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternative 4: 97.8% 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 60.2%

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

   1. add-cbrt-cube 60.1%

      \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)}\right)} \]

   2. pow1/3 60.2%

      \[\leadsto \log \color{blue}{\left({\left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)}^{0.3333333333333333}\right)} \]

   3. log-pow 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)} \]

   4. pow3 59.4%

      \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(e^{a} + e^{b}\right)}^{3}\right)} \]

   5. log-pow 59.4%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

3. Applied egg-rr 59.4%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* 60.2%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \log \left(e^{a} + e^{b}\right)} \]

   2. metadata-eval 60.2%

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

   3. *-un-lft-identity 60.2%

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

   4. log1p-expm1-u 59.9%

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

   5. log1p-udef 59.9%

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

   6. expm1-udef 59.9%

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

   7. add-exp-log 59.9%

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

5. Applied egg-rr 59.9%

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

   1. log1p-def 59.9%

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

   2. associate--l+ 60.0%

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

   3. expm1-def 77.2%

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

7. Simplified 77.2%

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

8. Final simplification 77.2%

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

Alternative 5: 58.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 11.6%

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

      1. add-cbrt-cube 11.6%

         \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)}\right)} \]

      2. pow1/3 11.6%

         \[\leadsto \log \color{blue}{\left({\left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)}^{0.3333333333333333}\right)} \]

      3. log-pow 11.6%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)} \]

      4. pow3 11.6%

         \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(e^{a} + e^{b}\right)}^{3}\right)} \]

      5. log-pow 11.6%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

   3. Applied egg-rr 11.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

   4. Taylor expanded in b around 0 96.4%

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

      1. log1p-def 96.4%

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

      2. +-commutative 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in a around 0 18.2%

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

   8. Taylor expanded in b around inf 18.2%

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

      1. *-commutative 18.2%

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

   10. Simplified 18.2%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 72.5%

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

      1. add-cbrt-cube 72.5%

         \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)}\right)} \]

      2. pow1/3 72.5%

         \[\leadsto \log \color{blue}{\left({\left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)}^{0.3333333333333333}\right)} \]

      3. log-pow 71.5%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)} \]

      4. pow3 71.5%

         \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(e^{a} + e^{b}\right)}^{3}\right)} \]

      5. log-pow 71.5%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

   3. Applied egg-rr 71.5%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

   4. Taylor expanded in b around 0 69.6%

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

      1. log1p-def 69.6%

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

      2. +-commutative 69.6%

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

   6. Simplified 69.6%

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

   7. Taylor expanded in a around 0 69.6%

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

   8. Taylor expanded in a around 0 68.8%

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

      1. distribute-lft-out 68.8%

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

      2. +-commutative 68.8%

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

   10. Simplified 68.8%

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

4. Final simplification 58.5%

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

Alternative 6: 96.0% 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 60.2%

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

2. Taylor expanded in b around 0 57.2%

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

   1. associate-+r+ 57.2%

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

4. Simplified 57.2%

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

5. Taylor expanded in b around 0 56.2%

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

   1. +-commutative 56.2%

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

   2. associate-+r+ 56.2%

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

7. Simplified 56.2%

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

   1. associate-+l+ 56.2%

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

   2. log1p-def 73.4%

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

9. Applied egg-rr 73.4%

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

10. Final simplification 73.4%

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

Alternative 7: 57.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.36d0)) then
        tmp = b * 0.5d0
    else
        tmp = log(2.0d0) + (a * 0.5d0)
    end if
    code = tmp
end function

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.36:
		tmp = b * 0.5
	else:
		tmp = math.log(2.0) + (a * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.36)
		tmp = Float64(b * 0.5);
	else
		tmp = Float64(log(2.0) + Float64(a * 0.5));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.36)
		tmp = b * 0.5;
	else
		tmp = log(2.0) + (a * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.36], N[(b * 0.5), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3600000000000001

   1. Initial program 11.6%

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

      1. add-cbrt-cube 11.6%

         \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)}\right)} \]

      2. pow1/3 11.6%

         \[\leadsto \log \color{blue}{\left({\left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)}^{0.3333333333333333}\right)} \]

      3. log-pow 11.6%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)} \]

      4. pow3 11.6%

         \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(e^{a} + e^{b}\right)}^{3}\right)} \]

      5. log-pow 11.6%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

   3. Applied egg-rr 11.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

   4. Taylor expanded in b around 0 96.4%

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

      1. log1p-def 96.4%

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

      2. +-commutative 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in a around 0 18.2%

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

   8. Taylor expanded in b around inf 18.2%

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

      1. *-commutative 18.2%

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

   10. Simplified 18.2%

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

   if -1.3600000000000001 < a

   1. Initial program 72.5%

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

   2. Taylor expanded in b around 0 69.7%

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

      1. log1p-def 69.7%

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

   4. Simplified 69.7%

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

   5. Taylor expanded in a around 0 68.5%

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

      1. *-commutative 68.5%

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

   7. Simplified 68.5%

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

4. Final simplification 58.3%

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

Alternative 8: 57.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -102:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -102.0) (* b 0.5) (+ (* b 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -102.0) {
		tmp = b * 0.5;
	} 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 (a <= (-102.0d0)) then
        tmp = b * 0.5d0
    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 (a <= -102.0) {
		tmp = b * 0.5;
	} else {
		tmp = (b * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -102.0:
		tmp = b * 0.5
	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 (a <= -102.0)
		tmp = Float64(b * 0.5);
	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 (a <= -102.0)
		tmp = b * 0.5;
	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[a, -102.0], N[(b * 0.5), $MachinePrecision], N[(N[(b * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -102

   1. Initial program 11.6%

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

      1. add-cbrt-cube 11.6%

         \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)}\right)} \]

      2. pow1/3 11.6%

         \[\leadsto \log \color{blue}{\left({\left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)}^{0.3333333333333333}\right)} \]

      3. log-pow 11.6%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)} \]

      4. pow3 11.6%

         \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(e^{a} + e^{b}\right)}^{3}\right)} \]

      5. log-pow 11.6%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

   3. Applied egg-rr 11.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

   4. Taylor expanded in b around 0 96.4%

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

      1. log1p-def 96.4%

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

      2. +-commutative 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in a around 0 18.2%

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

   8. Taylor expanded in b around inf 18.2%

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

      1. *-commutative 18.2%

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

   10. Simplified 18.2%

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

   if -102 < a

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0 69.4%

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

   3. Taylor expanded in b around 0 67.8%

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

      1. *-commutative 67.8%

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

   5. Simplified 67.8%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -102:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 9: 57.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -85:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -85.0) (* b 0.5) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -85.0) {
		tmp = b * 0.5;
	} else {
		tmp = log((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 <= (-85.0d0)) then
        tmp = b * 0.5d0
    else
        tmp = log((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 <= -85.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -85.0:
		tmp = b * 0.5
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -85.0)
		tmp = Float64(b * 0.5);
	else
		tmp = log(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 <= -85.0)
		tmp = b * 0.5;
	else
		tmp = log((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, -85.0], N[(b * 0.5), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -85

   1. Initial program 11.6%

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

      1. add-cbrt-cube 11.6%

         \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)}\right)} \]

      2. pow1/3 11.6%

         \[\leadsto \log \color{blue}{\left({\left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)}^{0.3333333333333333}\right)} \]

      3. log-pow 11.6%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)} \]

      4. pow3 11.6%

         \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(e^{a} + e^{b}\right)}^{3}\right)} \]

      5. log-pow 11.6%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

   3. Applied egg-rr 11.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

   4. Taylor expanded in b around 0 96.4%

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

      1. log1p-def 96.4%

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

      2. +-commutative 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in a around 0 18.2%

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

   8. Taylor expanded in b around inf 18.2%

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

      1. *-commutative 18.2%

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

   10. Simplified 18.2%

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

   if -85 < a

   1. Initial program 72.5%

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

   2. Taylor expanded in a around 0 69.4%

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

   3. Taylor expanded in b around 0 67.0%

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

      1. +-commutative 67.0%

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

   5. Simplified 67.0%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -85:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]

Alternative 10: 56.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -98:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -98.0) (* b 0.5) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -98.0) {
		tmp = b * 0.5;
	} else {
		tmp = log(2.0);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-98.0d0)) then
        tmp = b * 0.5d0
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -98.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -98.0:
		tmp = b * 0.5
	else:
		tmp = math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -98.0)
		tmp = Float64(b * 0.5);
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -98.0)
		tmp = b * 0.5;
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -98.0], N[(b * 0.5), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -98

   1. Initial program 11.6%

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

      1. add-cbrt-cube 11.6%

         \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)}\right)} \]

      2. pow1/3 11.6%

         \[\leadsto \log \color{blue}{\left({\left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)}^{0.3333333333333333}\right)} \]

      3. log-pow 11.6%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)} \]

      4. pow3 11.6%

         \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(e^{a} + e^{b}\right)}^{3}\right)} \]

      5. log-pow 11.6%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

   3. Applied egg-rr 11.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

   4. Taylor expanded in b around 0 96.4%

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

      1. log1p-def 96.4%

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

      2. +-commutative 96.4%

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

   6. Simplified 96.4%

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

   7. Taylor expanded in a around 0 18.2%

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

   8. Taylor expanded in b around inf 18.2%

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

      1. *-commutative 18.2%

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

   10. Simplified 18.2%

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

   if -98 < a

   1. Initial program 72.5%

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

   2. Taylor expanded in b around 0 69.7%

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

      1. log1p-def 69.7%

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

   4. Simplified 69.7%

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

   5. Taylor expanded in a around 0 67.5%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -98:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 11: 11.9% accurate, 101.0× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return 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 = b * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

   1. add-cbrt-cube 60.1%

      \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)}\right)} \]

   2. pow1/3 60.2%

      \[\leadsto \log \color{blue}{\left({\left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)}^{0.3333333333333333}\right)} \]

   3. log-pow 59.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(e^{a} + e^{b}\right) \cdot \left(e^{a} + e^{b}\right)\right) \cdot \left(e^{a} + e^{b}\right)\right)} \]

   4. pow3 59.4%

      \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(e^{a} + e^{b}\right)}^{3}\right)} \]

   5. log-pow 59.4%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

3. Applied egg-rr 59.4%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(e^{a} + e^{b}\right)\right)} \]

4. Taylor expanded in b around 0 75.0%

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

   1. log1p-def 75.0%

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

   2. +-commutative 75.0%

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

6. Simplified 75.0%

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

7. Taylor expanded in a around 0 59.2%

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

8. Taylor expanded in b around inf 6.6%

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

   1. *-commutative 6.6%

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

10. Simplified 6.6%

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

11. Final simplification 6.6%

    \[\leadsto b \cdot 0.5 \]

Reproduce

herbie shell --seed 2023333 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))