Average Error: 0.7 → 0.7
Time: 2.2s
Precision: binary64
\[\frac{e^{a}}{e^{a} + e^{b}} \]
↓
\[\begin{array}{l} t_0 := \sqrt{e^{b}}\\ \frac{e^{a}}{\mathsf{fma}\left(t_0, t_0, e^{a}\right)} \end{array} \]
(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (sqrt (exp b)))) (/ (exp a) (fma t_0 t_0 (exp a)))))
double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}
↓
double code(double a, double b) {
	double t_0 = sqrt(exp(b));
	return exp(a) / fma(t_0, t_0, exp(a));
}

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end
↓
function code(a, b)
	t_0 = sqrt(exp(b))
	return Float64(exp(a) / fma(t_0, t_0, exp(a)))
end

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[a_, b_] := Block[{t$95$0 = N[Sqrt[N[Exp[b], $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[a], $MachinePrecision] / N[(t$95$0 * t$95$0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{e^{a}}{e^{a} + e^{b}}
↓
\begin{array}{l}
t_0 := \sqrt{e^{b}}\\
\frac{e^{a}}{\mathsf{fma}\left(t_0, t_0, e^{a}\right)}
\end{array}

Error

Bits error versus a

Bits error versus b

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

    \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\sqrt{e^{b}}, \sqrt{e^{b}}, e^{a}\right)}} \]

  3. Final simplification0.7

    \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\sqrt{e^{b}}, \sqrt{e^{b}}, e^{a}\right)} \]

Reproduce

herbie shell --seed 2022134 
(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))))