symmetry log of sum of exp

?

Percentage Accurate: 53.4% → 97.8%
Time: 15.3s
Precision: binary64
Cost: 19392

?

\[\log \left(e^{a} + e^{b}\right) \]
↓
\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \end{array} \mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right) \]
(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))
↓
;; Ensure these are sorted, for example in Racket, do
(match-define (list a b) (sort (a b) <))

(FPCore (a b) :precision binary64 (log1p (+ (exp a) (expm1 b))))
double code(double a, double b) {
	return log((exp(a) + exp(b)));
}
↓
// Ensure these are sorted
assert(a < b);

double code(double a, double b) {
	return log1p((exp(a) + expm1(b)));
}

public static double code(double a, double b) {
	return Math.log((Math.exp(a) + Math.exp(b)));
}
↓
// Ensure these are sorted
assert a < b;

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

def code(a, b):
	return math.log((math.exp(a) + math.exp(b)))
↓
[a, b] = sort([a, b])

def code(a, b):
	return math.log1p((math.exp(a) + math.expm1(b)))

function code(a, b)
	return log(Float64(exp(a) + exp(b)))
end
↓
a, b = sort([a, b])

function code(a, b)
	return log1p(Float64(exp(a) + expm1(b)))
end

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

\log \left(e^{a} + e^{b}\right)
↓
\begin{array}{l}
[a, b] = \mathsf{sort}([a, b])\\
\end{array}
\mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right)

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 56.9%

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

  2. Applied egg-rr56.1%

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

    [Start]56.9

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

    add-sqr-sqrt [=>]55.7

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

    log-prod [=>]56.1

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

  3. Simplified75.8%

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

    [Start]56.1

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

    log-prod [<=]55.7

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

    rem-square-sqrt [=>]56.9

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

    log1p-expm1 [<=]56.5

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

    expm1-def [<=]56.5

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

    rem-exp-log [=>]56.5

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

    associate--l+ [=>]56.6

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

    expm1-def [=>]75.8

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

  4. Final simplification75.8%

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

Alternatives

Alternative 1
Accuracy51.9%
Cost12996
\[\begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
Alternative 2
Accuracy95.7%
Cost12992
\[\mathsf{log1p}\left(e^{a} + b\right) \]
Alternative 3
Accuracy50.2%
Cost12864
\[\mathsf{log1p}\left(e^{a}\right) \]
Alternative 4
Accuracy49.2%
Cost6720
\[b \cdot 0.5 + \log 2 \]
Alternative 5
Accuracy48.9%
Cost6592
\[\log \left(b + 2\right) \]
Alternative 6
Accuracy48.4%
Cost6464
\[\log 2 \]
Alternative 7
Accuracy2.6%
Cost320
\[\frac{a}{b + 2} \]

Error

Reproduce?

herbie shell --seed 2023160 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))