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(FPCore (x y) :precision binary64 (log (+ (exp x) (exp y))))

(FPCore (x y) :precision binary64 (log1p (+ (exp y) (expm1 x))))

double code(double x, double y) {
	return log((exp(x) + exp(y)));
}

double code(double x, double y) {
	return log1p((exp(y) + expm1(x)));
}

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

public static double code(double x, double y) {
	return Math.log1p((Math.exp(y) + Math.expm1(x)));
}

def code(x, y):
	return math.log((math.exp(x) + math.exp(y)))

def code(x, y):
	return math.log1p((math.exp(y) + math.expm1(x)))

function code(x, y)
	return log(Float64(exp(x) + exp(y)))
end

function code(x, y)
	return log1p(Float64(exp(y) + expm1(x)))
end

code[x_, y_] := N[Log[N[(N[Exp[x], $MachinePrecision] + N[Exp[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

code[x_, y_] := N[Log[1 + N[(N[Exp[y], $MachinePrecision] + N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\log \left(e^{x} + e^{y}\right)

\mathsf{log1p}\left(e^{y} + \mathsf{expm1}\left(x\right)\right)

Error?

Try it out?

Derivation?

1. Initial program 0.0

   \[\log \left(e^{x} + e^{y}\right) \]

2. Taylor expanded in x around inf 0.0

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

3. Simplified0.0

   \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{y} + \mathsf{expm1}\left(x\right)\right)} \]
   Proof

   [Start]0.0	\[ \log \left(e^{y} + e^{x}\right) \]
   +-commutative [=>]0.0	\[ \log \color{blue}{\left(e^{x} + e^{y}\right)} \]
   log1p-expm1 [<=]0.0	\[ \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(e^{x} + e^{y}\right)\right)\right)} \]
   expm1-def [<=]0.0	\[ \mathsf{log1p}\left(\color{blue}{e^{\log \left(e^{x} + e^{y}\right)} - 1}\right) \]
   rem-exp-log [=>]0.0	\[ \mathsf{log1p}\left(\color{blue}{\left(e^{x} + e^{y}\right)} - 1\right) \]
   +-commutative [<=]0.0	\[ \mathsf{log1p}\left(\color{blue}{\left(e^{y} + e^{x}\right)} - 1\right) \]
   associate--l+ [=>]0.0	\[ \mathsf{log1p}\left(\color{blue}{e^{y} + \left(e^{x} - 1\right)}\right) \]
   expm1-def [=>]0.0	\[ \mathsf{log1p}\left(e^{y} + \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

4. Final simplification0.0

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

Alternatives

Error

Reproduce?

herbie shell --seed 1 
(FPCore (x y)
  :name "log(exp(x)+exp(y))"
  :precision binary64
  :pre (and (and (<= 1.0 x) (<= x 500.0)) (and (<= 1.0 y) (<= y 500.0)))
  (log (+ (exp x) (exp y))))