Quotient of sum of exps

Average Accuracy: 99.1% → 99.1%

Time: 5.2s

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	99.1%
Target	100.0%
Herbie	99.1%

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

Derivation

1. Initial program 99.2%

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

2. Final simplification 99.2%

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

Alternatives

Alternative 1
Accuracy	98.6%
Cost	26185

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

Alternative 2
Accuracy	98.6%
Cost	26184

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 0.5:\\ \;\;\;\;\frac{\mathsf{expm1}\left(b\right)}{\mathsf{expm1}\left(b \cdot 2\right)}\\ \mathbf{elif}\;e^{b} \leq 2:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 3
Accuracy	98.6%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 4
Accuracy	98.3%
Cost	13248

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

Alternative 5
Accuracy	62.5%
Cost	6920

\[\begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-203}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{elif}\;a \leq -3.1 \cdot 10^{-261}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 6
Accuracy	64.4%
Cost	6592

\[\frac{e^{a}}{2} \]

Alternative 7
Accuracy	38.8%
Cost	320

\[0.5 + a \cdot 0.25 \]

Alternative 8
Accuracy	38.6%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023157 -o generate:proofs
(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))))