Quotient of sum of exps

Average Accuracy: 98.8% → 100.0%

Time: 7.5s

Precision: binary64

Cost: 6848

Math

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

↓

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

FPCore

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))

↓

(FPCore (a b) :precision binary64 (/ 1.0 (+ 1.0 (exp (- b a)))))

C

double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}

↓

double code(double a, double b) {
	return 1.0 / (1.0 + exp((b - a)));
}

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 = 1.0d0 / (1.0d0 + exp((b - a)))
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 1.0 / (1.0 + Math.exp((b - a)));
}

Python

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

↓

def code(a, b):
	return 1.0 / (1.0 + math.exp((b - a)))

Julia

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

↓

function code(a, b)
	return Float64(1.0 / Float64(1.0 + exp(Float64(b - a))))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

↓

function tmp = code(a, b)
	tmp = 1.0 / (1.0 + exp((b - a)));
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[(1.0 / N[(1.0 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	98.8%
Target	100.0%
Herbie	100.0%

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

Derivation

1. Initial program 98.8%

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

2. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333}} \]
   Proof

   [Start] 98.8	\[ \frac{e^{a}}{e^{a} + e^{b}} \]
   add-cbrt-cube [=>] 98.2	\[ \color{blue}{\sqrt[3]{\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}}} \]
   pow1/3 [=>] 98.8	\[ \color{blue}{{\left(\left(\frac{e^{a}}{e^{a} + e^{b}} \cdot \frac{e^{a}}{e^{a} + e^{b}}\right) \cdot \frac{e^{a}}{e^{a} + e^{b}}\right)}^{0.3333333333333333}} \]
   pow3 [=>] 98.8	\[ {\color{blue}{\left({\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{3}\right)}}^{0.3333333333333333} \]
   clear-num [=>] 98.8	\[ {\left({\color{blue}{\left(\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}\right)}}^{3}\right)}^{0.3333333333333333} \]
   inv-pow [=>] 98.8	\[ {\left({\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-1}\right)}}^{3}\right)}^{0.3333333333333333} \]
   pow-pow [=>] 98.8	\[ {\color{blue}{\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{\left(-1 \cdot 3\right)}\right)}}^{0.3333333333333333} \]
   metadata-eval [=>] 98.8	\[ {\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{\color{blue}{-3}}\right)}^{0.3333333333333333} \]

3. Simplified 98.2%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{e^{b} + e^{a}}{e^{a}}\right)}^{-3}}} \]
   Proof

   [Start] 98.8	\[ {\left({\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-3}\right)}^{0.3333333333333333} \]
   unpow1/3 [=>] 98.2	\[ \color{blue}{\sqrt[3]{{\left(\frac{e^{a} + e^{b}}{e^{a}}\right)}^{-3}}} \]
   +-commutative [=>] 98.2	\[ \sqrt[3]{{\left(\frac{\color{blue}{e^{b} + e^{a}}}{e^{a}}\right)}^{-3}} \]

4. Taylor expanded in b around inf 98.8%

   \[\leadsto \color{blue}{\frac{e^{a}}{e^{a} + e^{b}}} \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   Proof

   [Start] 98.8	\[ \frac{e^{a}}{e^{a} + e^{b}} \]
   *-lft-identity [<=] 98.8	\[ \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]
   +-commutative [<=] 98.8	\[ \frac{1 \cdot e^{a}}{\color{blue}{e^{b} + e^{a}}} \]
   associate-/l* [=>] 98.8	\[ \color{blue}{\frac{1}{\frac{e^{b} + e^{a}}{e^{a}}}} \]
   *-lft-identity [<=] 98.8	\[ \frac{1}{\frac{\color{blue}{1 \cdot \left(e^{b} + e^{a}\right)}}{e^{a}}} \]
   associate-*l/ [<=] 98.8	\[ \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot \left(e^{b} + e^{a}\right)}} \]
   rec-exp [=>] 98.8	\[ \frac{1}{\color{blue}{e^{-a}} \cdot \left(e^{b} + e^{a}\right)} \]
   +-commutative [=>] 98.8	\[ \frac{1}{e^{-a} \cdot \color{blue}{\left(e^{a} + e^{b}\right)}} \]
   distribute-rgt-in [=>] 71.3	\[ \frac{1}{\color{blue}{e^{a} \cdot e^{-a} + e^{b} \cdot e^{-a}}} \]
   rec-exp [<=] 71.3	\[ \frac{1}{e^{a} \cdot \color{blue}{\frac{1}{e^{a}}} + e^{b} \cdot e^{-a}} \]
   rgt-mult-inverse [=>] 99.5	\[ \frac{1}{\color{blue}{1} + e^{b} \cdot e^{-a}} \]
   rec-exp [<=] 99.5	\[ \frac{1}{1 + e^{b} \cdot \color{blue}{\frac{1}{e^{a}}}} \]
   associate-*r/ [=>] 99.5	\[ \frac{1}{1 + \color{blue}{\frac{e^{b} \cdot 1}{e^{a}}}} \]
   *-rgt-identity [=>] 99.5	\[ \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]
   div-exp [=>] 100.0	\[ \frac{1}{1 + \color{blue}{e^{b - a}}} \]

6. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	98.6%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 2
Accuracy	76.6%
Cost	6860

\[\begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-45}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq -3.5 \cdot 10^{-294}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-288}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 3
Accuracy	64.0%
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 4
Accuracy	64.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;b \leq 240000:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{2}{b \cdot b}\right) + -1\\ \end{array} \]

Alternative 5
Accuracy	52.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 6
Accuracy	39.4%
Cost	320

\[0.5 + a \cdot 0.25 \]

Alternative 7
Accuracy	40.1%
Cost	320

\[\frac{1}{2 - a} \]

Alternative 8
Accuracy	39.3%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023146 
(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))))