symmetry log of sum of exp

Average Error: 30.0 → 0.6

Time: 1.3min

Precision: binary64

Cost: 58824

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

Math

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

↓

\[\begin{array}{l} t_0 := e^{a} + e^{b}\\ t_1 := 1 + e^{a}\\ t_2 := \log t_1 + \frac{b}{t_1}\\ \mathbf{if}\;t_0 \leq 1.5:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\log \left(e^{b} \cdot 1.5 - \left(\left(e^{b} - e^{a}\right) + \frac{e^{b}}{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (exp a) (exp b)))
        (t_1 (+ 1.0 (exp a)))
        (t_2 (+ (log t_1) (/ b t_1))))
   (if (<= t_0 1.5)
     t_2
     (if (<= t_0 5e+102)
       (log (- (* (exp b) 1.5) (+ (- (exp b) (exp a)) (/ (exp b) -2.0))))
       t_2))))

C

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

↓

double code(double a, double b) {
	double t_0 = exp(a) + exp(b);
	double t_1 = 1.0 + exp(a);
	double t_2 = log(t_1) + (b / t_1);
	double tmp;
	if (t_0 <= 1.5) {
		tmp = t_2;
	} else if (t_0 <= 5e+102) {
		tmp = log(((exp(b) * 1.5) - ((exp(b) - exp(a)) + (exp(b) / -2.0))));
	} else {
		tmp = t_2;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = exp(a) + exp(b)
    t_1 = 1.0d0 + exp(a)
    t_2 = log(t_1) + (b / t_1)
    if (t_0 <= 1.5d0) then
        tmp = t_2
    else if (t_0 <= 5d+102) then
        tmp = log(((exp(b) * 1.5d0) - ((exp(b) - exp(a)) + (exp(b) / (-2.0d0)))))
    else
        tmp = t_2
    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 t_0 = Math.exp(a) + Math.exp(b);
	double t_1 = 1.0 + Math.exp(a);
	double t_2 = Math.log(t_1) + (b / t_1);
	double tmp;
	if (t_0 <= 1.5) {
		tmp = t_2;
	} else if (t_0 <= 5e+102) {
		tmp = Math.log(((Math.exp(b) * 1.5) - ((Math.exp(b) - Math.exp(a)) + (Math.exp(b) / -2.0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	t_0 = math.exp(a) + math.exp(b)
	t_1 = 1.0 + math.exp(a)
	t_2 = math.log(t_1) + (b / t_1)
	tmp = 0
	if t_0 <= 1.5:
		tmp = t_2
	elif t_0 <= 5e+102:
		tmp = math.log(((math.exp(b) * 1.5) - ((math.exp(b) - math.exp(a)) + (math.exp(b) / -2.0))))
	else:
		tmp = t_2
	return tmp

Julia

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

↓

function code(a, b)
	t_0 = Float64(exp(a) + exp(b))
	t_1 = Float64(1.0 + exp(a))
	t_2 = Float64(log(t_1) + Float64(b / t_1))
	tmp = 0.0
	if (t_0 <= 1.5)
		tmp = t_2;
	elseif (t_0 <= 5e+102)
		tmp = log(Float64(Float64(exp(b) * 1.5) - Float64(Float64(exp(b) - exp(a)) + Float64(exp(b) / -2.0))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b)
	t_0 = exp(a) + exp(b);
	t_1 = 1.0 + exp(a);
	t_2 = log(t_1) + (b / t_1);
	tmp = 0.0;
	if (t_0 <= 1.5)
		tmp = t_2;
	elseif (t_0 <= 5e+102)
		tmp = log(((exp(b) * 1.5) - ((exp(b) - exp(a)) + (exp(b) / -2.0))));
	else
		tmp = t_2;
	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_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t$95$1], $MachinePrecision] + N[(b / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1.5], t$95$2, If[LessEqual[t$95$0, 5e+102], N[Log[N[(N[(N[Exp[b], $MachinePrecision] * 1.5), $MachinePrecision] - N[(N[(N[Exp[b], $MachinePrecision] - N[Exp[a], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[b], $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 a) (exp.f64 b)) < 1.5 or 5e102 < (+.f64 (exp.f64 a) (exp.f64 b))

   1. Initial program 58.2

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

   2. Taylor expanded in b around 0 1.2

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

   if 1.5 < (+.f64 (exp.f64 a) (exp.f64 b)) < 5e102

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

   3. Simplified 0.0

      \[\leadsto \log \color{blue}{\left(e^{b} \cdot 1.5 - \left(\left(e^{b} - e^{a}\right) + \frac{e^{b}}{-2}\right)\right)} \]
      Proof

      [Start] 0.0	\[ \log \left(3 \cdot \frac{e^{b}}{2} - \left(\left(e^{b} - e^{a}\right) + \frac{e^{b}}{-2}\right)\right) \]
      rational_best-simplify-55 [=>] 0.0	\[ \log \left(\color{blue}{e^{b} \cdot \frac{3}{2}} - \left(\left(e^{b} - e^{a}\right) + \frac{e^{b}}{-2}\right)\right) \]
      metadata-eval [=>] 0.0	\[ \log \left(e^{b} \cdot \color{blue}{1.5} - \left(\left(e^{b} - e^{a}\right) + \frac{e^{b}}{-2}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} + e^{b} \leq 1.5:\\ \;\;\;\;\log \left(1 + e^{a}\right) + \frac{b}{1 + e^{a}}\\ \mathbf{elif}\;e^{a} + e^{b} \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\log \left(e^{b} \cdot 1.5 - \left(\left(e^{b} - e^{a}\right) + \frac{e^{b}}{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{a}\right) + \frac{b}{1 + e^{a}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	46528

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

Alternative 2
Error	0.7
Cost	45896

\[\begin{array}{l} t_0 := e^{a} + e^{b}\\ t_1 := 1 + e^{a}\\ t_2 := \log t_1 + \frac{b}{t_1}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\log \left(e^{b} \cdot 1.5 - \left(-1 + 0.5 \cdot e^{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	1.9
Cost	32452

\[\begin{array}{l} t_0 := e^{a} + e^{b}\\ \mathbf{if}\;t_0 \leq 1.0000002:\\ \;\;\;\;\frac{b}{1 + e^{a}} + 0.125 \cdot {b}^{2}\\ \mathbf{else}:\\ \;\;\;\;\log t_0\\ \end{array} \]

Alternative 4
Error	3.0
Cost	19972

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}} + 0.125 \cdot {b}^{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{b}\right)\\ \end{array} \]

Alternative 5
Error	26.9
Cost	19780

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 1.00004:\\ \;\;\;\;\log \left(1 + e^{a}\right) + \frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{b}\right)\\ \end{array} \]

Alternative 6
Error	30.6
Cost	13252

\[\begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\log \left(e^{a} + \left(1 + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{b}\right)\\ \end{array} \]

Alternative 7
Error	31.1
Cost	13124

\[\begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{-25}:\\ \;\;\;\;\log \left(1 + e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{b}\right)\\ \end{array} \]

Alternative 8
Error	32.1
Cost	12992

\[\log \left(1 + e^{a}\right) \]

Alternative 9
Error	31.9
Cost	7496

\[\begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;0.125 \cdot {b}^{2}\\ \mathbf{elif}\;a \leq -8.4 \cdot 10^{-16}:\\ \;\;\;\;\left(\frac{a}{8} - \left(a \cdot -0.25 - \log 2\right)\right) + \frac{a}{8}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot b + \log 2\\ \end{array} \]

Alternative 10
Error	31.9
Cost	6984

\[\begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;0.125 \cdot {b}^{2}\\ \mathbf{elif}\;a \leq -8.4 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot a + \log 2\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + b\right)\\ \end{array} \]

Alternative 11
Error	31.9
Cost	6984

\[\begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;0.125 \cdot {b}^{2}\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot a + \log 2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot b + \log 2\\ \end{array} \]

Alternative 12
Error	31.9
Cost	6856

\[\begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;0.125 \cdot {b}^{2}\\ \mathbf{elif}\;a \leq -3.8 \cdot 10^{-16}:\\ \;\;\;\;\log \left(2 + a\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + b\right)\\ \end{array} \]

Alternative 13
Error	32.9
Cost	6592

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

Alternative 14
Error	33.2
Cost	6464

\[\log 2 \]

Alternative 15
Error	62.3
Cost	192

\[0.5 \cdot a \]

Reproduce

herbie shell --seed 2023099 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))