Quotient of sum of exps

Average Error: 1.06% → 0.89%

Time: 8.8s

Precision: binary64

Cost: 25920

Math

\[\frac{e^{a}}{e^{a} + e^{b}} \]

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

↓

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp((a - log((exp(a) + exp(b)))))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

↓

public static double code(double a, double b) {
	return Math.exp((a - Math.log((Math.exp(a) + Math.exp(b)))));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_] := N[Exp[N[(a - N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Target

Original	1.06%
Target	0.02%
Herbie	0.89%

\[\frac{1}{1 + e^{b - a}} \]

Derivation

1. Initial program 1.06

   \[\frac{e^{a}}{e^{a} + e^{b}} \]

2. Applied egg-rr 0.89

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

3. Final simplification 0.89

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

Alternatives

Alternative 1
Error	1.06%
Cost	19520

\[\frac{e^{a}}{e^{a} + e^{b}} \]

Alternative 2
Error	1.4%
Cost	13252

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

Alternative 3
Error	20.73%
Cost	6860

\[\begin{array}{l} \mathbf{if}\;b \leq -0.96:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-255}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-221}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\right) + -1\\ \end{array} \]

Alternative 4
Error	3.14%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;b \leq -15500:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 140000000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\right) + -1\\ \end{array} \]

Alternative 5
Error	34.83%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;b \leq -15500:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\right) + -1\\ \end{array} \]

Alternative 6
Error	35.56%
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-226}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{b + 2} + -1\right)\\ \end{array} \]

Alternative 7
Error	35.54%
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-230}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 8
Error	47.11%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 2.1:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{b + -2}{b \cdot b}\\ \end{array} \]

Alternative 9
Error	60.11%
Cost	320

\[0.5 + a \cdot 0.25 \]

Alternative 10
Error	60.32%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023103 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))