Quotient of sum of exps

Average Error: 0.6 → 0.4

Time: 6.3s

Precision: binary64

Cost: 39108

Math

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

↓

\[\begin{array}{l} t_0 := \frac{e^{a}}{e^{a} + e^{b}}\\ \mathbf{if}\;t_0 \leq 0.6:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{e^{b} + 1}\right) + -1\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (exp a) (+ (exp a) (exp b)))))
   (if (<= t_0 0.6) t_0 (+ (+ 1.0 (/ 1.0 (+ (exp b) 1.0))) -1.0))))

C

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

↓

double code(double a, double b) {
	double t_0 = exp(a) / (exp(a) + exp(b));
	double tmp;
	if (t_0 <= 0.6) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (1.0 / (exp(b) + 1.0))) + -1.0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(a) / (exp(a) + exp(b))
    if (t_0 <= 0.6d0) then
        tmp = t_0
    else
        tmp = (1.0d0 + (1.0d0 / (exp(b) + 1.0d0))) + (-1.0d0)
    end if
    code = tmp
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) {
	double t_0 = Math.exp(a) / (Math.exp(a) + Math.exp(b));
	double tmp;
	if (t_0 <= 0.6) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (1.0 / (Math.exp(b) + 1.0))) + -1.0;
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	t_0 = math.exp(a) / (math.exp(a) + math.exp(b))
	tmp = 0
	if t_0 <= 0.6:
		tmp = t_0
	else:
		tmp = (1.0 + (1.0 / (math.exp(b) + 1.0))) + -1.0
	return tmp

Julia

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

↓

function code(a, b)
	t_0 = Float64(exp(a) / Float64(exp(a) + exp(b)))
	tmp = 0.0
	if (t_0 <= 0.6)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(1.0 / Float64(exp(b) + 1.0))) + -1.0);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b)
	t_0 = exp(a) / (exp(a) + exp(b));
	tmp = 0.0;
	if (t_0 <= 0.6)
		tmp = t_0;
	else
		tmp = (1.0 + (1.0 / (exp(b) + 1.0))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.6], t$95$0, N[(N[(1.0 + N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Target

Original	0.6
Target	0.0
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.599999999999999978

   1. Initial program 0.0

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

   if 0.599999999999999978 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 3.3

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

   2. Applied egg-rr 3.3

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

   3. Taylor expanded in a around 0 2.0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{1 + e^{b}}\right)} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.6:\\ \;\;\;\;\frac{e^{a}}{e^{a} + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{e^{b} + 1}\right) + -1\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	26312

\[\begin{array}{l} t_0 := \frac{1}{e^{b} + 1}\\ \mathbf{if}\;e^{b} \leq 0.6:\\ \;\;\;\;\left(1 + t_0\right) + -1\\ \mathbf{elif}\;e^{b} \leq 1.000002:\\ \;\;\;\;\frac{e^{a}}{e^{a} + \left(b + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.8
Cost	26184

\[\begin{array}{l} t_0 := \frac{1}{e^{b} + 1}\\ \mathbf{if}\;e^{b} \leq 0.6:\\ \;\;\;\;\left(1 + t_0\right) + -1\\ \mathbf{elif}\;e^{b} \leq 1.000000005:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -13111326.671473132:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 4
Error	25.2
Cost	1376

\[\begin{array}{l} \mathbf{if}\;a \leq -29.865202000468962:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq -9.676835304439497 \cdot 10^{-222}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;a \leq -1.2204010151244642 \cdot 10^{-257}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq 2.082802020257218 \cdot 10^{-263}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;a \leq 7.933027014547591 \cdot 10^{-215}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq 1.788982764330052 \cdot 10^{-154}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;a \leq 7.780091078854733 \cdot 10^{-99}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq 5.142742043405975 \cdot 10^{-51}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	25.3
Cost	1120

\[\begin{array}{l} \mathbf{if}\;a \leq -1134.2530607568654:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq -9.676835304439497 \cdot 10^{-222}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;a \leq -1.2204010151244642 \cdot 10^{-257}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq 2.082802020257218 \cdot 10^{-263}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;a \leq 7.933027014547591 \cdot 10^{-215}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq 1.788982764330052 \cdot 10^{-154}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;a \leq 1.4994810814086305 \cdot 10^{-109}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq 5.142742043405975 \cdot 10^{-51}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	25.1
Cost	1120

\[\begin{array}{l} \mathbf{if}\;a \leq -29.865202000468962:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq -9.676835304439497 \cdot 10^{-222}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;a \leq -1.2204010151244642 \cdot 10^{-257}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq 2.082802020257218 \cdot 10^{-263}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;a \leq 7.933027014547591 \cdot 10^{-215}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq 1.788982764330052 \cdot 10^{-154}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;a \leq 1.4994810814086305 \cdot 10^{-109}:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq 5.142742043405975 \cdot 10^{-51}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	12.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -1.1127435327369119 \cdot 10^{-5}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 8
Error	38.6
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022289 
(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))))