symmetry log of sum of exp

Average Accuracy: 52.3% → 98.5%

Time: 18.1s

Precision: binary64

Cost: 25924

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-322) (/ b (+ (exp a) 1.0)) (log (+ (exp a) (exp b)))))

C

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

↓

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-322) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((exp(a) + exp(b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = log((exp(a) + exp(b)))
end function

↓

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 2d-322) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((exp(a) + exp(b)))
    end if
    code = tmp
end function

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 tmp;
	if (Math.exp(a) <= 2e-322) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((Math.exp(a) + Math.exp(b)));
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	tmp = 0
	if math.exp(a) <= 2e-322:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((math.exp(a) + math.exp(b)))
	return tmp

Julia

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

↓

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-322)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(exp(a) + exp(b)));
	end
	return tmp
end

MATLAB

function tmp = code(a, b)
	tmp = log((exp(a) + exp(b)));
end

↓

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 2e-322)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((exp(a) + exp(b)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 2e-322], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1.97626e-322

   1. Initial program 8.9%

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

   2. Taylor expanded in b around 0 7.2%

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

   3. Simplified 7.2%

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

      [Start] 7.2	\[ \log \left(1 + \left(e^{a} + b\right)\right) \]
      associate-+r+ [=>] 7.2	\[ \log \color{blue}{\left(\left(1 + e^{a}\right) + b\right)} \]
      +-commutative [=>] 7.2	\[ \log \left(\color{blue}{\left(e^{a} + 1\right)} + b\right) \]
      associate-+l+ [=>] 7.2	\[ \log \color{blue}{\left(e^{a} + \left(1 + b\right)\right)} \]

   4. Taylor expanded in b around 0 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{2 + \mathsf{expm1}\left(a\right)}} \]
      Proof

      [Start] 100.0	\[ \log \left(1 + e^{a}\right) + \frac{b}{1 + e^{a}} \]
      log1p-def [=>] 100.0	\[ \color{blue}{\mathsf{log1p}\left(e^{a}\right)} + \frac{b}{1 + e^{a}} \]
      +-commutative [<=] 100.0	\[ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{e^{a} + 1}} \]
      +-commutative [=>] 100.0	\[ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{1 + e^{a}}} \]
      metadata-eval [<=] 100.0	\[ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{\left(2 - 1\right)} + e^{a}} \]
      associate--r- [<=] 100.0	\[ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{2 - \left(1 - e^{a}\right)}} \]
      metadata-eval [<=] 100.0	\[ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{\left(2 + 0\right)} - \left(1 - e^{a}\right)} \]
      associate-+r- [<=] 100.0	\[ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{2 + \left(0 - \left(1 - e^{a}\right)\right)}} \]
      associate--r- [=>] 100.0	\[ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{2 + \color{blue}{\left(\left(0 - 1\right) + e^{a}\right)}} \]
      metadata-eval [=>] 100.0	\[ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{2 + \left(\color{blue}{-1} + e^{a}\right)} \]
      +-commutative [<=] 100.0	\[ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{2 + \color{blue}{\left(e^{a} + -1\right)}} \]
      metadata-eval [<=] 100.0	\[ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{2 + \left(e^{a} + \color{blue}{\left(-1\right)}\right)} \]
      sub-neg [<=] 100.0	\[ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{2 + \color{blue}{\left(e^{a} - 1\right)}} \]
      expm1-def [=>] 100.0	\[ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{2 + \color{blue}{\mathsf{expm1}\left(a\right)}} \]

   6. Taylor expanded in b around inf 100.0%

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

   if 1.97626e-322 < (exp.f64 a)

   1. Initial program 96.9%

      \[\log \left(e^{a} + e^{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.2%
Cost	26056

\[\begin{array}{l} t_0 := e^{a} + 1\\ \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{b}{t_0}\\ \mathbf{elif}\;e^{a} \leq 1:\\ \;\;\;\;\log t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternative 2
Accuracy	97.5%
Cost	25928

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{elif}\;e^{a} \leq 1:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternative 3
Accuracy	97.5%
Cost	19396

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]

Alternative 4
Accuracy	97.6%
Cost	19392

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

Alternative 5
Accuracy	97.2%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 0.5 + \left(b \cdot 0.5 + \log 2\right)\\ \end{array} \]

Alternative 6
Accuracy	96.5%
Cost	13376

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

Alternative 7
Accuracy	97.1%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-322}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + \left(a + 2\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	95.8%
Cost	12992

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

Alternative 9
Accuracy	49.1%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;b \cdot \left(0.5 + a \cdot -0.25\right)\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\log \left(a + 2\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]

Alternative 10
Accuracy	49.1%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;b \cdot \left(0.5 + a \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + \left(a + 2\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	48.5%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;b \cdot \left(0.5 + a \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(a + 2\right)\\ \end{array} \]

Alternative 12
Accuracy	48.2%
Cost	6720

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

Alternative 13
Accuracy	47.3%
Cost	6464

\[\log 2 \]

Alternative 14
Accuracy	5.4%
Cost	448

\[b \cdot \left(0.5 + a \cdot -0.25\right) \]

Alternative 15
Accuracy	2.6%
Cost	192

\[a \cdot 0.5 \]

Reproduce

herbie shell --seed 2023135 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))