?

Average Error: 29.5 → 0.7
Time: 17.1s
Precision: binary64
Cost: 25924

?

\[ \begin{array}{c}[a, b] = \mathsf{sort}([a, b])\\ \end{array} \]
\[\log \left(e^{a} + e^{b}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]
(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-38) (/ b (+ 1.0 (exp a))) (log (+ (exp a) (exp b)))))
double code(double a, double b) {
	return log((exp(a) + exp(b)));
}
↓
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-38) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((exp(a) + exp(b)));
	}
	return tmp;
}

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) <= 5d-38) then
        tmp = b / (1.0d0 + exp(a))
    else
        tmp = log((exp(a) + exp(b)))
    end if
    code = tmp
end function

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) <= 5e-38) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log((Math.exp(a) + Math.exp(b)));
	}
	return tmp;
}

def code(a, b):
	return math.log((math.exp(a) + math.exp(b)))
↓
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-38:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log((math.exp(a) + math.exp(b)))
	return tmp

function code(a, b)
	return log(Float64(exp(a) + exp(b)))
end
↓
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-38)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(exp(a) + exp(b)));
	end
	return tmp
end

function tmp = code(a, b)
	tmp = log((exp(a) + exp(b)));
end
↓
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-38)
		tmp = b / (1.0 + exp(a));
	else
		tmp = log((exp(a) + exp(b)));
	end
	tmp_2 = tmp;
end

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], 5e-38], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

\log \left(e^{a} + e^{b}\right)
↓
\begin{array}{l}
\mathbf{if}\;e^{a} \leq 5 \cdot 10^{-38}:\\
\;\;\;\;\frac{b}{1 + e^{a}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{a} + e^{b}\right)\\


\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 2 regimes

  2. if (exp.f64 a) < 5.00000000000000033e-38

    1. Initial program 58.3

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

    2. Taylor expanded in b around 0 0.1

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

    3. Taylor expanded in b around inf 0.1

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

    if 5.00000000000000033e-38 < (exp.f64 a)

    1. Initial program 1.3

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error1.1
Cost19776
\[\begin{array}{l} t_0 := 1 + e^{a}\\ \log t_0 + \frac{b}{t_0} \end{array} \]
Alternative 2
Error1.2
Cost19652
\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \left(e^{a} + b\right)\right)\\ \end{array} \]
Alternative 3
Error1.4
Cost19524
\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{b}\right)\\ \end{array} \]
Alternative 4
Error1.6
Cost13380
\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \left(b + \left(1 + a\right)\right)\right)\\ \end{array} \]
Alternative 5
Error1.5
Cost13380
\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + 0.5 \cdot \left(b + a\right)\\ \end{array} \]
Alternative 6
Error1.8
Cost13252
\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot b + \log 2\\ \end{array} \]
Alternative 7
Error27.5
Cost6852
\[\begin{array}{l} \mathbf{if}\;a \leq -175:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot b + \log 2\\ \end{array} \]
Alternative 8
Error27.7
Cost6724
\[\begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + a\right)\\ \end{array} \]
Alternative 9
Error27.6
Cost6724
\[\begin{array}{l} \mathbf{if}\;a \leq -150:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + b\right)\\ \end{array} \]
Alternative 10
Error27.9
Cost6596
\[\begin{array}{l} \mathbf{if}\;a \leq -125:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
Alternative 11
Error56.4
Cost192
\[0.5 \cdot b \]

Error

Reproduce?

herbie shell --seed 2023090 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))