?

Average Error: 29.5 → 1.6
Time: 15.1s
Precision: binary64
Cost: 13508

?

\[ \begin{array}{c}[a, b] = \mathsf{sort}([a, b])\\ \end{array} \]
\[\log \left(e^{a} + e^{b}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-108}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (if (<= a -1e-108) (log1p (+ (exp a) (+ b (* 0.5 (* b b))))) (log1p (exp b))))
double code(double a, double b) {
	return log((exp(a) + exp(b)));
}
↓
double code(double a, double b) {
	double tmp;
	if (a <= -1e-108) {
		tmp = log1p((exp(a) + (b + (0.5 * (b * b)))));
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

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 (a <= -1e-108) {
		tmp = Math.log1p((Math.exp(a) + (b + (0.5 * (b * b)))));
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

def code(a, b):
	return math.log((math.exp(a) + math.exp(b)))
↓
def code(a, b):
	tmp = 0
	if a <= -1e-108:
		tmp = math.log1p((math.exp(a) + (b + (0.5 * (b * b)))))
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

function code(a, b)
	return log(Float64(exp(a) + exp(b)))
end
↓
function code(a, b)
	tmp = 0.0
	if (a <= -1e-108)
		tmp = log1p(Float64(exp(a) + Float64(b + Float64(0.5 * Float64(b * b)))));
	else
		tmp = log1p(exp(b));
	end
	return tmp
end

code[a_, b_] := N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[a_, b_] := If[LessEqual[a, -1e-108], N[Log[1 + N[(N[Exp[a], $MachinePrecision] + N[(b + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

\log \left(e^{a} + e^{b}\right)
↓
\begin{array}{l}
\mathbf{if}\;a \leq -1 \cdot 10^{-108}:\\
\;\;\;\;\mathsf{log1p}\left(e^{a} + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\


\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 2 regimes

  2. if a < -1.00000000000000004e-108

    1. Initial program 44.0

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

    2. Applied egg-rr44.3

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

    3. Simplified1.1

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

      [Start]44.3

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

      log-prod [<=]44.4

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

      rem-square-sqrt [=>]44.0

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

      log1p-expm1 [<=]44.1

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

      expm1-def [<=]44.1

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

      rem-exp-log [=>]44.1

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

      associate--l+ [=>]44.0

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

      expm1-def [=>]1.1

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

    4. Taylor expanded in b around 0 1.8

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

    5. Simplified1.8

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

      [Start]1.8

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

      unpow2 [=>]1.8

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

    if -1.00000000000000004e-108 < a

    1. Initial program 1.4

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

    2. Taylor expanded in a around 0 1.4

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

    3. Simplified1.3

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

      [Start]1.4

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

      log1p-def [=>]1.3

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-108}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error1.2
Cost19392
\[\mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right) \]
Alternative 2
Error2.0
Cost13124
\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-108}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
Alternative 3
Error30.5
Cost12996
\[\begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
Alternative 4
Error31.2
Cost12864
\[\mathsf{log1p}\left(e^{a}\right) \]
Alternative 5
Error32.0
Cost6720
\[b \cdot 0.5 + \log 2 \]
Alternative 6
Error32.2
Cost6592
\[\log \left(b + 2\right) \]
Alternative 7
Error32.2
Cost6592
\[\mathsf{log1p}\left(b + 1\right) \]
Alternative 8
Error32.5
Cost6464
\[\log 2 \]
Alternative 9
Error62.3
Cost320
\[\frac{a}{b + 2} \]

Error

Reproduce?

herbie shell --seed 2023025 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))