symmetry log of sum of exp

Average Error: 29.4 → 1.1

Time: 21.2s

Precision: binary64

Cost: 19776

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

Math

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

↓

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

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp a)))) (+ (log t_0) (/ b t_0))))

C

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

↓

double code(double a, double b) {
	double t_0 = 1.0 + exp(a);
	return log(t_0) + (b / t_0);
}

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
    t_0 = 1.0d0 + exp(a)
    code = log(t_0) + (b / t_0)
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 = 1.0 + Math.exp(a);
	return Math.log(t_0) + (b / t_0);
}

Python

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

↓

def code(a, b):
	t_0 = 1.0 + math.exp(a)
	return math.log(t_0) + (b / t_0)

Julia

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

↓

function code(a, b)
	t_0 = Float64(1.0 + exp(a))
	return Float64(log(t_0) + Float64(b / t_0))
end

MATLAB

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

↓

function tmp = code(a, b)
	t_0 = 1.0 + exp(a);
	tmp = log(t_0) + (b / t_0);
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[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]}, N[(N[Log[t$95$0], $MachinePrecision] + N[(b / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 29.4

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

2. Taylor expanded in b around 0 1.1

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

3. Final simplification 1.1

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

Alternatives

Alternative 1
Error	1.6
Cost	19524

\[\begin{array}{l} t_0 := 1 + e^{a}\\ \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\log t_0\\ \end{array} \]

Alternative 2
Error	1.2
Cost	13252

\[\begin{array}{l} \mathbf{if}\;a \leq -37:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \left(b - -1\right)\right)\\ \end{array} \]

Alternative 3
Error	27.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -118:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot b + \log 2\\ \end{array} \]

Alternative 4
Error	1.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -37:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot b + \log 2\\ \end{array} \]

Alternative 5
Error	27.5
Cost	6724

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

Alternative 6
Error	27.5
Cost	6724

\[\begin{array}{l} \mathbf{if}\;a \leq -145:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + b\right)\\ \end{array} \]

Alternative 7
Error	27.8
Cost	6596

\[\begin{array}{l} \mathbf{if}\;a \leq -160:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 8
Error	56.4
Cost	192

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

Reproduce

herbie shell --seed 2023089 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))