symmetry log of sum of exp

Average Error: 29.9 → 1.1

Time: 25.1s

Precision: binary64

Cost: 26180

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0.999995:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right) - \frac{a}{-1 - e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{1 + e^{a}} + \mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 0.999995)
   (- (log1p (exp b)) (/ a (- -1.0 (exp b))))
   (+ (/ b (+ 1.0 (exp a))) (log1p (exp a)))))

C

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

↓

double code(double a, double b) {
	double tmp;
	if (exp(b) <= 0.999995) {
		tmp = log1p(exp(b)) - (a / (-1.0 - exp(b)));
	} else {
		tmp = (b / (1.0 + exp(a))) + log1p(exp(a));
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(b) <= 0.999995) {
		tmp = Math.log1p(Math.exp(b)) - (a / (-1.0 - Math.exp(b)));
	} else {
		tmp = (b / (1.0 + Math.exp(a))) + Math.log1p(Math.exp(a));
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	tmp = 0
	if math.exp(b) <= 0.999995:
		tmp = math.log1p(math.exp(b)) - (a / (-1.0 - math.exp(b)))
	else:
		tmp = (b / (1.0 + math.exp(a))) + math.log1p(math.exp(a))
	return tmp

Julia

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

↓

function code(a, b)
	tmp = 0.0
	if (exp(b) <= 0.999995)
		tmp = Float64(log1p(exp(b)) - Float64(a / Float64(-1.0 - exp(b))));
	else
		tmp = Float64(Float64(b / Float64(1.0 + exp(a))) + log1p(exp(a)));
	end
	return tmp
end

Wolfram

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

↓

code[a_, b_] := If[LessEqual[N[Exp[b], $MachinePrecision], 0.999995], N[(N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision] - N[(a / N[(-1.0 - N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 0.99999499999999997

   1. Initial program 56.7

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

   2. Taylor expanded in a around 0 1.0

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

   3. Applied egg-rr 0.9

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

   if 0.99999499999999997 < (exp.f64 b)

   1. Initial program 20.2

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

   2. Taylor expanded in b around 0 20.6

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

   3. Taylor expanded in b around 0 1.3

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

   4. Simplified 1.1

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

Alternatives

Alternative 1
Error	15.9
Cost	52036

\[\begin{array}{l} t_0 := \mathsf{log1p}\left(e^{a}\right)\\ t_1 := 0.5 \cdot t_0\\ \mathbf{if}\;e^{b} \leq 1:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right) - \frac{a}{-1 - e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{e^{t_0}} + t_1\right) + t_1\\ \end{array} \]

Alternative 2
Error	1.2
Cost	26180

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{1 + e^{a}} + \mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]

Alternative 3
Error	1.8
Cost	19524

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

Alternative 4
Error	17.9
Cost	13260

\[\begin{array}{l} t_0 := \mathsf{log1p}\left(e^{b}\right)\\ \mathbf{if}\;b \leq -260:\\ \;\;\;\;\frac{a}{1 + e^{b}}\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{-247}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	17.2
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + \left(a + b\right)\right)\\ \end{array} \]

Alternative 6
Error	17.2
Cost	13128

\[\begin{array}{l} \mathbf{if}\;b \leq -92:\\ \;\;\;\;\frac{a}{1 + e^{b}}\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(2 + b\right) + \left(0.5 \cdot b\right) \cdot b\right)\\ \end{array} \]

Alternative 7
Error	18.1
Cost	8008

\[\begin{array}{l} t_0 := \left(2 + b\right) + \left(0.5 \cdot b\right) \cdot b\\ t_1 := \log t_0\\ \mathbf{if}\;b \leq -50:\\ \;\;\;\;\frac{a}{1 + e^{b}}\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-246}:\\ \;\;\;\;\frac{a}{t_0} + t_1\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	18.1
Cost	7496

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4:\\ \;\;\;\;\frac{a}{1 + e^{b}}\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{-247}:\\ \;\;\;\;0.5 \cdot a + \left(\log 2 + \left(0.5 - 0.25 \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(2 + b\right) + \left(0.5 \cdot b\right) \cdot b\right)\\ \end{array} \]

Alternative 9
Error	18.3
Cost	7372

\[\begin{array}{l} t_0 := \log \left(\left(2 + b\right) + \left(0.5 \cdot b\right) \cdot b\right)\\ \mathbf{if}\;b \leq -39:\\ \;\;\;\;\frac{a}{1 + e^{b}}\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{-247}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	18.7
Cost	7244

\[\begin{array}{l} t_0 := \log \left(2 + a\right) + 0.5 \cdot b\\ \mathbf{if}\;b \leq -1.4:\\ \;\;\;\;\frac{a}{1 + e^{b}}\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{-247}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	18.7
Cost	7116

\[\begin{array}{l} t_0 := \log \left(2 + \left(a + b\right)\right)\\ \mathbf{if}\;b \leq -1:\\ \;\;\;\;\frac{a}{1 + e^{b}}\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{-247}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-300}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	30.6
Cost	6852

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

Alternative 13
Error	30.7
Cost	6596

\[\begin{array}{l} \mathbf{if}\;b \leq -85:\\ \;\;\;\;\frac{a}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 14
Error	56.9
Cost	324

\[\begin{array}{l} \mathbf{if}\;a \leq -2.46 \cdot 10^{-57}:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{2}\\ \end{array} \]

Alternative 15
Error	59.4
Cost	192

\[0.5 \cdot b \]

Reproduce

herbie shell --seed 2023010 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))