symmetry log of sum of exp

Average Error: 29.7 → 0.7

Time: 21.0s

Precision: binary64

Cost: 25924

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} - -1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ 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) <= 0.0) {
		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) <= 0.0d0) 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) <= 0.0) {
		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) <= 0.0:
		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) <= 0.0)
		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) <= 0.0)
		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], 0.0], 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) < 0.0

   1. Initial program 58.1

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

   2. Taylor expanded in b around 0 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

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

      [Start] 0	\[ \log \left(1 + e^{a}\right) + \left(0.5 \cdot \left(\left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right) \cdot {b}^{2}\right) + \frac{b}{1 + e^{a}}\right) \]
      rational.json-simplify-1 [=>] 0	\[ \log \left(1 + e^{a}\right) + \color{blue}{\left(\frac{b}{1 + e^{a}} + 0.5 \cdot \left(\left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right) \cdot {b}^{2}\right)\right)} \]
      rational.json-simplify-41 [=>] 0	\[ \color{blue}{\frac{b}{1 + e^{a}} + \left(0.5 \cdot \left(\left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right) \cdot {b}^{2}\right) + \log \left(1 + e^{a}\right)\right)} \]

   4. Taylor expanded in b around 0 0

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

   5. Taylor expanded in b around inf 0

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

   6. Simplified 0

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

      [Start] 0	\[ \frac{b}{1 + e^{a}} \]
      rational.json-simplify-17 [=>] 0	\[ \frac{b}{\color{blue}{e^{a} - -1}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 1.5

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} - -1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	46528

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

Alternative 2
Error	1.1
Cost	19776

\[\begin{array}{l} t_0 := 1 + e^{a}\\ \log t_0 + \frac{b}{t_0} \end{array} \]

Alternative 3
Error	1.4
Cost	19524

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} - -1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{b} - -1\right)\\ \end{array} \]

Alternative 4
Error	1.5
Cost	7364

\[\begin{array}{l} \mathbf{if}\;a \leq -1.35:\\ \;\;\;\;\frac{b}{e^{a} - -1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(0.5 - a \cdot 0.25\right) + \left(\log 2 + 0.5 \cdot a\right)\\ \end{array} \]

Alternative 5
Error	27.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -130:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 6
Error	1.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -25:\\ \;\;\;\;\frac{b}{e^{a} - -1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 7
Error	27.9
Cost	6724

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

Alternative 8
Error	27.8
Cost	6724

\[\begin{array}{l} \mathbf{if}\;a \leq -92:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + b\right)\\ \end{array} \]

Alternative 9
Error	28.1
Cost	6596

\[\begin{array}{l} \mathbf{if}\;a \leq -75:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 10
Error	56.3
Cost	192

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

Reproduce

herbie shell --seed 2023067 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))