Quotient of sum of exps

Average Error: 0.8 → 0.8

Time: 4.7s

Precision: binary64

Cost: 19520

Math

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

↓

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

FPCore

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

↓

(FPCore (a b) :precision binary64 (/ (exp a) (+ (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) / (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) / (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.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.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 Float64(exp(a) / 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) / (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[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.8
Target	0.0
Herbie	0.8

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

Derivation

1. Initial program 0.8

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

2. Final simplification 0.8

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

Alternatives

Alternative 1
Error	0.6
Cost	19849

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0.9999995 \lor \neg \left(e^{b} \leq 1.002\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \]

Alternative 2
Error	1.0
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	14.7
Cost	7124

\[\begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-13}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-106}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-187}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{-240}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 4
Error	22.1
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq 6.3 \cdot 10^{-59}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 5
Error	21.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq 135000000:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{2}{b \cdot b}\right) + -1\\ \end{array} \]

Alternative 6
Error	29.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 6.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 7
Error	38.6
Cost	320

\[0.5 + a \cdot 0.25 \]

Alternative 8
Error	38.1
Cost	320

\[\frac{1}{2 - a} \]

Alternative 9
Error	38.7
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023057 
(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))))