symmetry log of sum of exp

Average Error: 30.2 → 1.1

Time: 15.1s

Precision: binary64

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

Math

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

↓

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

FPCore

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

↓

(FPCore (a b) :precision binary64 (+ (log1p (exp a)) (/ b (+ 1.0 (exp a)))))

C

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

↓

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

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) {
	return Math.log1p(Math.exp(a)) + (b / (1.0 + Math.exp(a)));
}

Python

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

↓

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

Julia

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

↓

function code(a, b)
	return Float64(log1p(exp(a)) + Float64(b / Float64(1.0 + exp(a))))
end

Wolfram

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

↓

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

Derivation

1. Initial program 30.2

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

2. Taylor expanded in b around 0 1.2

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

3. Simplified 1.1

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

4. Final simplification 1.1

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

Reproduce

herbie shell --seed 2022203 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))