Quotient of sum of exps

Average Accuracy: 99.0% → 99.2%

Time: 9.1s

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	99.0%
Target	100.0%
Herbie	99.2%

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

Derivation

1. Initial program 99.0%

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

2. Applied egg-rr 99.2%

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

   [Start] 99.0	\[ \frac{e^{a}}{e^{a} + e^{b}} \]
   add-exp-log [=>] 99.0	\[ \frac{e^{a}}{\color{blue}{e^{\log \left(e^{a} + e^{b}\right)}}} \]
   div-exp [=>] 99.2	\[ \color{blue}{e^{a - \log \left(e^{a} + e^{b}\right)}} \]

3. Final simplification 99.2%

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

Alternatives

Alternative 1
Accuracy	67.4%
Cost	19652

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

Alternative 2
Accuracy	99.0%
Cost	19520

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

Alternative 3
Accuracy	98.6%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 4
Accuracy	97.9%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999999999995:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 5
Accuracy	63.1%
Cost	6596

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-125}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 6
Accuracy	39.7%
Cost	320

\[0.5 + a \cdot 0.25 \]

Alternative 7
Accuracy	39.6%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023142 
(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))))