Average Error: 0.6 → 0.7
Time: 3.3s
Precision: binary64
\[\frac{e^{a}}{e^{a} + e^{b}} \]
↓
\[\log \left({\left(e^{e^{a}}\right)}^{\left(\frac{1}{e^{a} + e^{b}}\right)}\right) \]
(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b)
 :precision binary64
 (log (pow (exp (exp a)) (/ 1.0 (+ (exp a) (exp b))))))
double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}
↓
double code(double a, double b) {
	return log(pow(exp(exp(a)), (1.0 / (exp(a) + exp(b)))));
}

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 = log((exp(exp(a)) ** (1.0d0 / (exp(a) + exp(b)))))
end function

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.log(Math.pow(Math.exp(Math.exp(a)), (1.0 / (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.log(math.pow(math.exp(math.exp(a)), (1.0 / (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 log((exp(exp(a)) ^ Float64(1.0 / Float64(exp(a) + exp(b)))))
end

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end
↓
function tmp = code(a, b)
	tmp = log((exp(exp(a)) ^ (1.0 / (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[Log[N[Power[N[Exp[N[Exp[a], $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

\frac{e^{a}}{e^{a} + e^{b}}
↓
\log \left({\left(e^{e^{a}}\right)}^{\left(\frac{1}{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

Original0.6
Target0.0
Herbie0.7
\[\frac{1}{1 + e^{b - a}} \]

Derivation

  1. Initial program 0.6

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

  2. Applied add-log-exp_binary640.7

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

  3. Applied div-inv_binary640.7

    \[\leadsto \log \left(e^{\color{blue}{e^{a} \cdot \frac{1}{e^{a} + e^{b}}}}\right) \]

  4. Applied exp-prod_binary640.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2022129 
(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))))