Quotient of sum of exps

Average Error: 0.6 → 0.6

Time: 9.6s

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	12.1
Cost	19848

\[\begin{array}{l} t_0 := \frac{1}{1 + e^{b}}\\ \mathbf{if}\;e^{b} \leq 1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;e^{b} \leq 2:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.3
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9998:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 3
Error	12.1
Cost	772

\[\begin{array}{l} \mathbf{if}\;b \leq 28000000000:\\ \;\;\;\;-1 + \left(1 - \frac{-1}{\left(-a\right) + 2}\right)\\ \mathbf{else}:\\ \;\;\;\;-2 - \left(\frac{-1}{b + 2} + -2\right)\\ \end{array} \]

Alternative 4
Error	22.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{\left(-a\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{-1}{b + 2}\right)\\ \end{array} \]

Alternative 5
Error	22.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 28000000000:\\ \;\;\;\;\frac{1}{\left(-a\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - \frac{-1}{b}\right)\\ \end{array} \]

Alternative 6
Error	38.3
Cost	384

\[\frac{1}{\left(-a\right) + 2} \]

Alternative 7
Error	38.7
Cost	320

\[0.5 + 0.25 \cdot a \]

Alternative 8
Error	38.9
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023067 
(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))))