Quotient of sum of exps

Average Error: 0.88% → 0.02%

Time: 7.1s

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	0.88%
Target	0.02%
Herbie	0.02%

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

Derivation

1. Initial program 0.88

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

2. Applied egg-rr 0.89

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

3. Taylor expanded in a around inf 0.88

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

4. Simplified 0.02

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

   [Start] 0.88	\[ \frac{e^{a}}{e^{a} + e^{b}} \]
   *-lft-identity [<=] 0.88	\[ \color{blue}{1 \cdot \frac{e^{a}}{e^{a} + e^{b}}} \]
   associate-*r/ [=>] 0.88	\[ \color{blue}{\frac{1 \cdot e^{a}}{e^{a} + e^{b}}} \]
   remove-double-neg [<=] 0.88	\[ \frac{1 \cdot e^{a}}{\color{blue}{\left(-\left(-e^{a}\right)\right)} + e^{b}} \]
   neg-sub0 [=>] 0.88	\[ \frac{1 \cdot e^{a}}{\color{blue}{\left(0 - \left(-e^{a}\right)\right)} + e^{b}} \]
   associate-+l- [=>] 0.88	\[ \frac{1 \cdot e^{a}}{\color{blue}{0 - \left(\left(-e^{a}\right) - e^{b}\right)}} \]
   neg-sub0 [<=] 0.88	\[ \frac{1 \cdot e^{a}}{\color{blue}{-\left(\left(-e^{a}\right) - e^{b}\right)}} \]
   neg-mul-1 [=>] 0.88	\[ \frac{1 \cdot e^{a}}{\color{blue}{-1 \cdot \left(\left(-e^{a}\right) - e^{b}\right)}} \]
   times-frac [=>] 0.88	\[ \color{blue}{\frac{1}{-1} \cdot \frac{e^{a}}{\left(-e^{a}\right) - e^{b}}} \]
   metadata-eval [=>] 0.88	\[ \color{blue}{-1} \cdot \frac{e^{a}}{\left(-e^{a}\right) - e^{b}} \]
   neg-mul-1 [<=] 0.88	\[ \color{blue}{-\frac{e^{a}}{\left(-e^{a}\right) - e^{b}}} \]
   distribute-frac-neg [<=] 0.88	\[ \color{blue}{\frac{-e^{a}}{\left(-e^{a}\right) - e^{b}}} \]
   neg-mul-1 [=>] 0.88	\[ \frac{\color{blue}{-1 \cdot e^{a}}}{\left(-e^{a}\right) - e^{b}} \]
   associate-/l* [=>] 0.89	\[ \color{blue}{\frac{-1}{\frac{\left(-e^{a}\right) - e^{b}}{e^{a}}}} \]
   div-sub [=>] 27.06	\[ \frac{-1}{\color{blue}{\frac{-e^{a}}{e^{a}} - \frac{e^{b}}{e^{a}}}} \]
   distribute-frac-neg [=>] 27.06	\[ \frac{-1}{\color{blue}{\left(-\frac{e^{a}}{e^{a}}\right)} - \frac{e^{b}}{e^{a}}} \]
   *-lft-identity [<=] 27.06	\[ \frac{-1}{\left(-\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}}\right) - \frac{e^{b}}{e^{a}}} \]
   associate-*l/ [<=] 27.06	\[ \frac{-1}{\left(-\color{blue}{\frac{1}{e^{a}} \cdot e^{a}}\right) - \frac{e^{b}}{e^{a}}} \]
   lft-mult-inverse [=>] 0.39	\[ \frac{-1}{\left(-\color{blue}{1}\right) - \frac{e^{b}}{e^{a}}} \]
   metadata-eval [=>] 0.39	\[ \frac{-1}{\color{blue}{-1} - \frac{e^{b}}{e^{a}}} \]
   div-exp [=>] 0.02	\[ \frac{-1}{-1 - \color{blue}{e^{b - a}}} \]

5. Final simplification 0.02

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

Alternatives

Alternative 1
Error	1.64%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;b \leq -14500000:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2
Error	1.5%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -580000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 3
Error	17.58%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	36.46%
Cost	716

\[\begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-228}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{-264}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-45}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	20.54%
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -1.76 \cdot 10^{-6}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{b + 2}\right)\\ \end{array} \]

Alternative 6
Error	36.5%
Cost	460

\[\begin{array}{l} \mathbf{if}\;a \leq -6.2 \cdot 10^{-44}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq -4.6 \cdot 10^{-92}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-125}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7
Error	60.07%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023089 
(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))))