symmetry log of sum of exp

Average Error: 30.1 → 0.6

Time: 20.0s

Precision: binary64

Cost: 25924

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-123) (/ 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-123) {
		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-123) 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-123) {
		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-123:
		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-123)
		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-123)
		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-123], 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) < 2.0000000000000001e-123

   1. Initial program 58.2

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

   2. Taylor expanded in b around 0 0.0

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

   3. Simplified 0

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]
      Proof
      (+.f64 (log1p.f64 (exp.f64 a)) (/.f64 b (+.f64 1 (exp.f64 a)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (exp.f64 a)))) (/.f64 b (+.f64 1 (exp.f64 a)))): 2 points increase in error, 0 points decrease in error

   4. Taylor expanded in b around inf 0.0

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

   if 2.0000000000000001e-123 < (exp.f64 a)

   1. Initial program 1.1

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	19648

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

Alternative 2
Error	1.1
Cost	19396

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-123}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

Alternative 3
Error	27.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -282.81711097496895:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 4
Error	1.6
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -282.81711097496895:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 5
Error	28.0
Cost	6724

\[\begin{array}{l} \mathbf{if}\;a \leq -282.81711097496895:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]

Alternative 6
Error	28.4
Cost	6596

\[\begin{array}{l} \mathbf{if}\;a \leq -282.81711097496895:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 7
Error	56.2
Cost	192

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

Reproduce

herbie shell --seed 2022313 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))