Average Error: 29.8 → 1.2
Time: 23.0s
Precision: binary64
\[ \begin{array}{c}[a, b] = \mathsf{sort}([a, b])\\ \end{array} \]
\[\log \left(e^{a} + e^{b}\right) \]
↓
\[\begin{array}{l} t_0 := \mathsf{log1p}\left(e^{a}\right)\\ t_1 := \sqrt[3]{\sqrt[3]{t_0}}\\ t_2 := \sqrt[3]{t_1}\\ t_0 + \frac{b}{{\left({\left(e^{{\left(t_2 \cdot \left({\left({t_2}^{3}\right)}^{2} \cdot {t_2}^{2}\right)\right)}^{2}}\right)}^{\left({t_1}^{2}\right)}\right)}^{t_1}} \end{array} \]
(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (log1p (exp a))) (t_1 (cbrt (cbrt t_0))) (t_2 (cbrt t_1)))
   (+
    t_0
    (/
     b
     (pow
      (pow
       (exp (pow (* t_2 (* (pow (pow t_2 3.0) 2.0) (pow t_2 2.0))) 2.0))
       (pow t_1 2.0))
      t_1)))))
double code(double a, double b) {
	return log((exp(a) + exp(b)));
}
↓
double code(double a, double b) {
	double t_0 = log1p(exp(a));
	double t_1 = cbrt(cbrt(t_0));
	double t_2 = cbrt(t_1);
	return t_0 + (b / pow(pow(exp(pow((t_2 * (pow(pow(t_2, 3.0), 2.0) * pow(t_2, 2.0))), 2.0)), pow(t_1, 2.0)), t_1));
}

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 t_0 = Math.log1p(Math.exp(a));
	double t_1 = Math.cbrt(Math.cbrt(t_0));
	double t_2 = Math.cbrt(t_1);
	return t_0 + (b / Math.pow(Math.pow(Math.exp(Math.pow((t_2 * (Math.pow(Math.pow(t_2, 3.0), 2.0) * Math.pow(t_2, 2.0))), 2.0)), Math.pow(t_1, 2.0)), t_1));
}

function code(a, b)
	return log(Float64(exp(a) + exp(b)))
end
↓
function code(a, b)
	t_0 = log1p(exp(a))
	t_1 = cbrt(cbrt(t_0))
	t_2 = cbrt(t_1)
	return Float64(t_0 + Float64(b / ((exp((Float64(t_2 * Float64(((t_2 ^ 3.0) ^ 2.0) * (t_2 ^ 2.0))) ^ 2.0)) ^ (t_1 ^ 2.0)) ^ t_1)))
end

code[a_, b_] := N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[a_, b_] := Block[{t$95$0 = N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 1/3], $MachinePrecision]}, N[(t$95$0 + N[(b / N[Power[N[Power[N[Exp[N[Power[N[(t$95$2 * N[(N[Power[N[Power[t$95$2, 3.0], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision], t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

\log \left(e^{a} + e^{b}\right)
↓
\begin{array}{l}
t_0 := \mathsf{log1p}\left(e^{a}\right)\\
t_1 := \sqrt[3]{\sqrt[3]{t_0}}\\
t_2 := \sqrt[3]{t_1}\\
t_0 + \frac{b}{{\left({\left(e^{{\left(t_2 \cdot \left({\left({t_2}^{3}\right)}^{2} \cdot {t_2}^{2}\right)\right)}^{2}}\right)}^{\left({t_1}^{2}\right)}\right)}^{t_1}}
\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.8

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

  2. Taylor expanded in b around 0 1.4

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

  3. Simplified1.2

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

  4. Applied egg-rr1.2

    \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{{\left({\left(e^{{\left(\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}\right)}^{2}}\right)}^{\left({\left(\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}\right)}^{2}\right)}\right)}^{\left(\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}\right)}}} \]

  5. Applied egg-rr1.2

    \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{{\left({\left(e^{{\color{blue}{\left(\left({\left(\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}}\right)}^{2}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}}\right)}}^{2}}\right)}^{\left({\left(\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}\right)}^{2}\right)}\right)}^{\left(\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}\right)}} \]

  6. Applied egg-rr1.2

    \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{{\left({\left(e^{{\left(\left({\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}}\right)}^{3}\right)}}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}}\right)}^{2}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}}\right)}^{2}}\right)}^{\left({\left(\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}\right)}^{2}\right)}\right)}^{\left(\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}\right)}} \]

  7. Final simplification1.2

    \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{{\left({\left(e^{{\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}} \cdot \left({\left({\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}}\right)}^{3}\right)}^{2} \cdot {\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}}\right)}^{2}\right)\right)}^{2}}\right)}^{\left({\left(\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}\right)}^{2}\right)}\right)}^{\left(\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(e^{a}\right)}}\right)}} \]

Reproduce

herbie shell --seed 2022209 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))