Quotient of sum of exps

?

Percentage Accurate: 99.0% → 99.2%
Time: 7.3s
Precision: binary64
Cost: 25920

?

\[\frac{e^{a}}{e^{a} + e^{b}} \]
↓
\[\begin{array}{l} \\ e^{a - \log \left(e^{a} + e^{b}\right)} \end{array} \]
(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))
↓
(FPCore (a b) :precision binary64 (exp (- a (log (+ (exp a) (exp b))))))
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)))));
}

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

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)))));
}

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)))))

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

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

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]

\frac{e^{a}}{e^{a} + e^{b}}
↓
\begin{array}{l}

\\
e^{a - \log \left(e^{a} + e^{b}\right)}
\end{array}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 13 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original99.0%
Target100.0%
Herbie99.2%
\[\frac{1}{1 + e^{b - a}} \]

Derivation?

  1. Initial program 99.2%

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

  2. Applied egg-rr99.6%

    \[\leadsto \color{blue}{e^{a - \log \left(e^{a} + e^{b}\right)}} \]
    Step-by-step derivation

    [Start]99.2%

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

    add-exp-log [=>]99.2%

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

    div-exp [=>]99.6%

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

  3. Final simplification99.6%

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

Alternatives

Alternative 1
Accuracy99.2%
Cost25920
\[e^{a - \log \left(e^{a} + e^{b}\right)} \]
Alternative 2
Accuracy99.0%
Cost19520
\[\frac{e^{a}}{e^{a} + e^{b}} \]
Alternative 3
Accuracy98.5%
Cost13252
\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
Alternative 4
Accuracy98.3%
Cost13248
\[\frac{e^{a}}{1 + \left(a + e^{b}\right)} \]
Alternative 5
Accuracy71.7%
Cost6724
\[\begin{array}{l} \mathbf{if}\;a \leq -490:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-42}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
Alternative 6
Accuracy72.6%
Cost1356
\[\begin{array}{l} t_0 := b \cdot \left(-b\right)\\ \mathbf{if}\;b \leq -1.45:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{t_0 - b \cdot -2}{b \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]
Alternative 7
Accuracy68.7%
Cost836
\[\begin{array}{l} \mathbf{if}\;b \leq -0.94:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
Alternative 8
Accuracy68.2%
Cost708
\[\begin{array}{l} \mathbf{if}\;b \leq -0.065:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \]
Alternative 9
Accuracy68.7%
Cost584
\[\begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.9:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{b \cdot b}\\ \end{array} \]
Alternative 10
Accuracy55.2%
Cost452
\[\begin{array}{l} \mathbf{if}\;b \leq -0.98:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
Alternative 11
Accuracy55.1%
Cost452
\[\begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \]
Alternative 12
Accuracy54.9%
Cost196
\[\begin{array}{l} \mathbf{if}\;b \leq -0.48:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
Alternative 13
Accuracy40.0%
Cost64
\[0.5 \]

Reproduce?

herbie shell --seed 2023167 
(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))))