Quotient of sum of exps

Average Error: 0.6 → 0.6

Time: 7.7s

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	0.6
Target	0.0
Herbie	0.6

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

Derivation

1. Initial program 0.6

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

2. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.7
Cost	19913

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0 \lor \neg \left(e^{b} \leq 1\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-1 + \frac{-1}{e^{a}}}\\ \end{array} \]

Alternative 2
Error	0.7
Cost	19849

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0 \lor \neg \left(e^{b} \leq 1\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \]

Alternative 3
Error	1.0
Cost	13252

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

Alternative 4
Error	2.1
Cost	6856

\[\begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 350:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	0.0
Cost	6848

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

Alternative 6
Error	7.2
Cost	6596

\[\begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\left(a + -0.5 \cdot \left(a \cdot a\right)\right) + -2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	13.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -100:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \frac{1}{b + 2}\right)\\ \end{array} \]

Alternative 8
Error	22.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq -0.000205:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 9
Error	22.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-5}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 10
Error	39.1
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023059 
(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))))