symmetry log of sum of exp

Average Error: 29.7 → 0.9

Time: 17.5s

Precision: binary64

Cost: 46208

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

Math

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

↓

\[\begin{array}{l} t_0 := 1 + e^{a}\\ \frac{b}{t_0} + \mathsf{fma}\left(b \cdot b, \frac{0.5}{t_0} + \frac{-0.5}{{t_0}^{2}}, \mathsf{log1p}\left(e^{a}\right)\right) \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp a))))
   (+
    (/ b t_0)
    (fma (* b b) (+ (/ 0.5 t_0) (/ -0.5 (pow t_0 2.0))) (log1p (exp a))))))

C

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

↓

double code(double a, double b) {
	double t_0 = 1.0 + exp(a);
	return (b / t_0) + fma((b * b), ((0.5 / t_0) + (-0.5 / pow(t_0, 2.0))), log1p(exp(a)));
}

Julia

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

↓

function code(a, b)
	t_0 = Float64(1.0 + exp(a))
	return Float64(Float64(b / t_0) + fma(Float64(b * b), Float64(Float64(0.5 / t_0) + Float64(-0.5 / (t_0 ^ 2.0))), log1p(exp(a))))
end

Wolfram

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

↓

code[a_, b_] := Block[{t$95$0 = N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]}, N[(N[(b / t$95$0), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(N[(0.5 / t$95$0), $MachinePrecision] + N[(-0.5 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 29.7

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

2. Taylor expanded in b around 0 1.0

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

3. Simplified 0.9

   \[\leadsto \color{blue}{\frac{b}{1 + e^{a}} + \mathsf{fma}\left(b \cdot b, \frac{0.5}{1 + e^{a}} + \frac{-0.5}{{\left(1 + e^{a}\right)}^{2}}, \mathsf{log1p}\left(e^{a}\right)\right)} \]
   Proof

   [Start] 1.0	\[ \log \left(1 + e^{a}\right) + \left(0.5 \cdot \left(\left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right) \cdot {b}^{2}\right) + \frac{b}{1 + e^{a}}\right) \]
   associate-+r+ [=>] 1.0	\[ \color{blue}{\left(\log \left(1 + e^{a}\right) + 0.5 \cdot \left(\left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right) \cdot {b}^{2}\right)\right) + \frac{b}{1 + e^{a}}} \]
   +-commutative [=>] 1.0	\[ \color{blue}{\frac{b}{1 + e^{a}} + \left(\log \left(1 + e^{a}\right) + 0.5 \cdot \left(\left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right) \cdot {b}^{2}\right)\right)} \]
   +-commutative [=>] 1.0	\[ \frac{b}{1 + e^{a}} + \color{blue}{\left(0.5 \cdot \left(\left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right) \cdot {b}^{2}\right) + \log \left(1 + e^{a}\right)\right)} \]
   associate-*r* [=>] 1.0	\[ \frac{b}{1 + e^{a}} + \left(\color{blue}{\left(0.5 \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) \cdot {b}^{2}} + \log \left(1 + e^{a}\right)\right) \]
   *-commutative [=>] 1.0	\[ \frac{b}{1 + e^{a}} + \left(\color{blue}{{b}^{2} \cdot \left(0.5 \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right)} + \log \left(1 + e^{a}\right)\right) \]
   fma-def [=>] 1.0	\[ \frac{b}{1 + e^{a}} + \color{blue}{\mathsf{fma}\left({b}^{2}, 0.5 \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right), \log \left(1 + e^{a}\right)\right)} \]

4. Final simplification 0.9

   \[\leadsto \frac{b}{1 + e^{a}} + \mathsf{fma}\left(b \cdot b, \frac{0.5}{1 + e^{a}} + \frac{-0.5}{{\left(1 + e^{a}\right)}^{2}}, \mathsf{log1p}\left(e^{a}\right)\right) \]

Alternatives

Alternative 1
Error	1.0
Cost	19648

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

Alternative 2
Error	1.4
Cost	19392

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

Alternative 3
Error	2.6
Cost	12992

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

Alternative 4
Error	31.6
Cost	12864

\[\mathsf{log1p}\left(e^{a}\right) \]

Alternative 5
Error	32.4
Cost	6720

\[b \cdot 0.5 + \log 2 \]

Alternative 6
Error	32.5
Cost	6592

\[\log \left(b + 2\right) \]

Alternative 7
Error	32.9
Cost	6464

\[\log 2 \]

Alternative 8
Error	62.3
Cost	192

\[a \cdot 0.5 \]

Alternative 9
Error	61.2
Cost	192

\[\frac{2}{b} \]

Reproduce

herbie shell --seed 2023016 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))