Average Error: 0.7 → 0.7
Time: 2.1s
Precision: binary64
\[\frac{e^{a}}{e^{a} + e^{b}} \]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right) \]
(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (expm1 (log1p (/ (exp a) (+ (exp a) (exp b))))))
double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}
↓
double code(double a, double b) {
	return expm1(log1p((exp(a) / (exp(a) + exp(b)))));
}

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.expm1(Math.log1p((Math.exp(a) / (Math.exp(a) + Math.exp(b)))));
}

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))
↓
def code(a, b):
	return math.expm1(math.log1p((math.exp(a) / (math.exp(a) + math.exp(b)))))

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end
↓
function code(a, b)
	return expm1(log1p(Float64(exp(a) / Float64(exp(a) + exp(b)))))
end

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[Log[1 + N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]

\frac{e^{a}}{e^{a} + e^{b}}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.7
Target0.0
Herbie0.7
\[\frac{1}{1 + e^{b - a}} \]

Derivation

  1. Initial program 0.7

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

  2. Applied egg-rr0.7

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)} \]

  3. Final simplification0.7

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right) \]

Reproduce

herbie shell --seed 2022166 
(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))))