symmetry log of sum of exp

Percentage Accurate: 61.6% → 94.1%

Time: 13.4s

Precision: binary64

Cost: 19788

• Unsound rule application detected in e-graph. Results may not be sound.

• 37.2% of points produce a very large (infinite) output. You may want to add a precondition.

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{b}{e^{a} + 1}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+79}:\\ \;\;\;\;{a}^{3} \cdot \left(b \cdot \left(--0.020833333333333332\right)\right)\\ \mathbf{elif}\;b \leq -24:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 720:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ b (+ (exp a) 1.0))))
   (if (<= b -1e+79)
     (* (pow a 3.0) (* b (- -0.020833333333333332)))
     (if (<= b -24.0)
       t_0
       (if (<= b 720.0)
         (log1p (+ (exp a) (expm1 b)))
         (if (<= b 2.4e+79) t_0 (log1p (exp b))))))))

C

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

↓

double code(double a, double b) {
	double t_0 = b / (exp(a) + 1.0);
	double tmp;
	if (b <= -1e+79) {
		tmp = pow(a, 3.0) * (b * -(-0.020833333333333332));
	} else if (b <= -24.0) {
		tmp = t_0;
	} else if (b <= 720.0) {
		tmp = log1p((exp(a) + expm1(b)));
	} else if (b <= 2.4e+79) {
		tmp = t_0;
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

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 = b / (Math.exp(a) + 1.0);
	double tmp;
	if (b <= -1e+79) {
		tmp = Math.pow(a, 3.0) * (b * -(-0.020833333333333332));
	} else if (b <= -24.0) {
		tmp = t_0;
	} else if (b <= 720.0) {
		tmp = Math.log1p((Math.exp(a) + Math.expm1(b)));
	} else if (b <= 2.4e+79) {
		tmp = t_0;
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	t_0 = b / (math.exp(a) + 1.0)
	tmp = 0
	if b <= -1e+79:
		tmp = math.pow(a, 3.0) * (b * -(-0.020833333333333332))
	elif b <= -24.0:
		tmp = t_0
	elif b <= 720.0:
		tmp = math.log1p((math.exp(a) + math.expm1(b)))
	elif b <= 2.4e+79:
		tmp = t_0
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

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

↓

function code(a, b)
	t_0 = Float64(b / Float64(exp(a) + 1.0))
	tmp = 0.0
	if (b <= -1e+79)
		tmp = Float64((a ^ 3.0) * Float64(b * Float64(-(-0.020833333333333332))));
	elseif (b <= -24.0)
		tmp = t_0;
	elseif (b <= 720.0)
		tmp = log1p(Float64(exp(a) + expm1(b)));
	elseif (b <= 2.4e+79)
		tmp = t_0;
	else
		tmp = log1p(exp(b));
	end
	return tmp
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[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1e+79], N[(N[Power[a, 3.0], $MachinePrecision] * N[(b * (--0.020833333333333332)), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -24.0], t$95$0, If[LessEqual[b, 720.0], N[Log[1 + N[(N[Exp[a], $MachinePrecision] + N[(Exp[b] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[b, 2.4e+79], t$95$0, N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -9.99999999999999967e78

   1. Initial program 17.2%

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

   2. Taylor expanded in b around 0 11.6%

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

   3. Simplified 11.6%

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

      [Start] 11.6%	\[ \log \left(1 + e^{a}\right) + \frac{b}{1 + e^{a}} \]
      log1p-def [=>] 11.6%	\[ \color{blue}{\mathsf{log1p}\left(e^{a}\right)} + \frac{b}{1 + e^{a}} \]

   4. Taylor expanded in a around 0 5.3%

      \[\leadsto \color{blue}{\left(0.5 - 0.25 \cdot b\right) \cdot a + \left(0.5 \cdot b + \left(-1 \cdot \left({a}^{3} \cdot \left(0.041666666666666664 \cdot b + \left(-0.0625 \cdot b + -0.5 \cdot \left(0.125 \cdot b + -0.125 \cdot b\right)\right)\right)\right) + \left({a}^{2} \cdot \left(0.125 - \left(0.125 \cdot b + -0.125 \cdot b\right)\right) + \log 2\right)\right)\right)} \]

   5. Taylor expanded in a around inf 5.3%

      \[\leadsto \left(0.5 - 0.25 \cdot b\right) \cdot a + \left(0.5 \cdot b + \color{blue}{-1 \cdot \left({a}^{3} \cdot \left(0.041666666666666664 \cdot b + \left(-0.0625 \cdot b + -0.5 \cdot \left(0.125 \cdot b + -0.125 \cdot b\right)\right)\right)\right)}\right) \]

   6. Taylor expanded in a around inf 17.8%

      \[\leadsto \color{blue}{-1 \cdot \left({a}^{3} \cdot \left(0.041666666666666664 \cdot b + \left(-0.0625 \cdot b + -0.5 \cdot \left(0.125 \cdot b + -0.125 \cdot b\right)\right)\right)\right)} \]

   7. Simplified 17.8%

      \[\leadsto \color{blue}{\left(b \cdot -0.020833333333333332\right) \cdot \left(-{a}^{3}\right)} \]
      Step-by-step derivation

      [Start] 17.8%	\[ -1 \cdot \left({a}^{3} \cdot \left(0.041666666666666664 \cdot b + \left(-0.0625 \cdot b + -0.5 \cdot \left(0.125 \cdot b + -0.125 \cdot b\right)\right)\right)\right) \]
      mul-1-neg [=>] 17.8%	\[ \color{blue}{-{a}^{3} \cdot \left(0.041666666666666664 \cdot b + \left(-0.0625 \cdot b + -0.5 \cdot \left(0.125 \cdot b + -0.125 \cdot b\right)\right)\right)} \]
      *-commutative [=>] 17.8%	\[ -\color{blue}{\left(0.041666666666666664 \cdot b + \left(-0.0625 \cdot b + -0.5 \cdot \left(0.125 \cdot b + -0.125 \cdot b\right)\right)\right) \cdot {a}^{3}} \]
      distribute-rgt-neg-in [=>] 17.8%	\[ \color{blue}{\left(0.041666666666666664 \cdot b + \left(-0.0625 \cdot b + -0.5 \cdot \left(0.125 \cdot b + -0.125 \cdot b\right)\right)\right) \cdot \left(-{a}^{3}\right)} \]
      distribute-rgt-out [=>] 17.8%	\[ \left(0.041666666666666664 \cdot b + \left(-0.0625 \cdot b + -0.5 \cdot \color{blue}{\left(b \cdot \left(0.125 + -0.125\right)\right)}\right)\right) \cdot \left(-{a}^{3}\right) \]
      metadata-eval [=>] 17.8%	\[ \left(0.041666666666666664 \cdot b + \left(-0.0625 \cdot b + -0.5 \cdot \left(b \cdot \color{blue}{0}\right)\right)\right) \cdot \left(-{a}^{3}\right) \]
      mul0-rgt [=>] 17.8%	\[ \left(0.041666666666666664 \cdot b + \left(-0.0625 \cdot b + -0.5 \cdot \color{blue}{0}\right)\right) \cdot \left(-{a}^{3}\right) \]
      metadata-eval [=>] 17.8%	\[ \left(0.041666666666666664 \cdot b + \left(-0.0625 \cdot b + \color{blue}{0}\right)\right) \cdot \left(-{a}^{3}\right) \]
      +-rgt-identity [=>] 17.8%	\[ \left(0.041666666666666664 \cdot b + \color{blue}{-0.0625 \cdot b}\right) \cdot \left(-{a}^{3}\right) \]
      distribute-rgt-out [=>] 17.8%	\[ \color{blue}{\left(b \cdot \left(0.041666666666666664 + -0.0625\right)\right)} \cdot \left(-{a}^{3}\right) \]
      metadata-eval [=>] 17.8%	\[ \left(b \cdot \color{blue}{-0.020833333333333332}\right) \cdot \left(-{a}^{3}\right) \]

   if -9.99999999999999967e78 < b < -24 or 720 < b < 2.39999999999999986e79

   1. Initial program 15.8%

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

   2. Taylor expanded in b around 0 40.3%

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

   3. Simplified 40.3%

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

      [Start] 40.3%	\[ \log \left(1 + e^{a}\right) + \frac{b}{1 + e^{a}} \]
      log1p-def [=>] 40.3%	\[ \color{blue}{\mathsf{log1p}\left(e^{a}\right)} + \frac{b}{1 + e^{a}} \]

   4. Taylor expanded in b around inf 27.9%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   if -24 < b < 720

   1. Initial program 69.9%

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

   2. Applied egg-rr 69.9%

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

      [Start] 69.9%	\[ \log \left(e^{a} + e^{b}\right) \]
      log1p-expm1-u [=>] 69.9%	\[ \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(e^{a} + e^{b}\right)\right)\right)} \]
      expm1-udef [=>] 69.9%	\[ \mathsf{log1p}\left(\color{blue}{e^{\log \left(e^{a} + e^{b}\right)} - 1}\right) \]
      add-exp-log [<=] 69.9%	\[ \mathsf{log1p}\left(\color{blue}{\left(e^{a} + e^{b}\right)} - 1\right) \]

   3. Simplified 91.2%

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

      [Start] 69.9%	\[ \mathsf{log1p}\left(\left(e^{a} + e^{b}\right) - 1\right) \]
      associate--l+ [=>] 69.9%	\[ \mathsf{log1p}\left(\color{blue}{e^{a} + \left(e^{b} - 1\right)}\right) \]
      expm1-def [=>] 91.2%	\[ \mathsf{log1p}\left(e^{a} + \color{blue}{\mathsf{expm1}\left(b\right)}\right) \]

   if 2.39999999999999986e79 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 100.0%

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

   3. Simplified 100.0%

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

      [Start] 100.0%	\[ \log \left(1 + e^{b}\right) \]
      log1p-def [=>] 100.0%	\[ \color{blue}{\mathsf{log1p}\left(e^{b}\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+79}:\\ \;\;\;\;{a}^{3} \cdot \left(b \cdot \left(--0.020833333333333332\right)\right)\\ \mathbf{elif}\;b \leq -24:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;b \leq 720:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+79}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	94.1%
Cost	19788

\[\begin{array}{l} t_0 := \frac{b}{e^{a} + 1}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+79}:\\ \;\;\;\;{a}^{3} \cdot \left(b \cdot \left(--0.020833333333333332\right)\right)\\ \mathbf{elif}\;b \leq -24:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 720:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternative 2
Accuracy	86.5%
Cost	32452

\[\begin{array}{l} t_0 := e^{a} + e^{b}\\ \mathbf{if}\;t_0 \leq 1.5:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log t_0\\ \end{array} \]

Alternative 3
Accuracy	93.7%
Cost	19912

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+79}:\\ \;\;\;\;{a}^{3} \cdot \left(b \cdot \left(--0.020833333333333332\right)\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right) + \frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternative 4
Accuracy	65.3%
Cost	13260

\[\begin{array}{l} \mathbf{if}\;a \leq -1850:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-245}:\\ \;\;\;\;a \cdot \left(0.5 - b \cdot 0.25\right) + \left(b \cdot 0.5 + \log 2\right)\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-44}:\\ \;\;\;\;b \cdot \left(\frac{1}{1 - e^{a \cdot 2}} - \frac{-0.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]

Alternative 5
Accuracy	80.6%
Cost	12996

\[\begin{array}{l} \mathbf{if}\;a \leq -1850:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternative 6
Accuracy	64.8%
Cost	7628

\[\begin{array}{l} \mathbf{if}\;a \leq -11:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-245}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \mathbf{elif}\;a \leq 2.2:\\ \;\;\;\;b \cdot \left(\frac{1}{1 - e^{a \cdot 2}} - \frac{-0.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(b \cdot \left(--0.020833333333333332\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	65.0%
Cost	7628

\[\begin{array}{l} \mathbf{if}\;a \leq -1850:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-245}:\\ \;\;\;\;a \cdot \left(0.5 - b \cdot 0.25\right) + \left(b \cdot 0.5 + \log 2\right)\\ \mathbf{elif}\;a \leq 215:\\ \;\;\;\;b \cdot \left(\frac{1}{1 - e^{a \cdot 2}} - \frac{-0.5}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(b \cdot \left(--0.020833333333333332\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	62.0%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;a \leq -8:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-22}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.5 + \left({a}^{3} \cdot 0.020833333333333332 + a \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	62.0%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;a \leq -17:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;a \leq 1000:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot \left(b \cdot \left(--0.020833333333333332\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	31.6%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \]

Alternative 11
Accuracy	33.1%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -1850:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 12
Accuracy	57.9%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -9.5:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 13
Accuracy	31.3%
Cost	6596

\[\begin{array}{l} \mathbf{if}\;a \leq -12.5:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 14
Accuracy	15.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{+83}:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.5 + a \cdot -0.25\right)\\ \end{array} \]

Alternative 15
Accuracy	10.4%
Cost	192

\[b \cdot 0.5 \]

Reproduce

herbie shell --seed 2023255 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))