Quotient of sum of exps

Average Error: 0.6 → 0.6

Time: 4.3s

Precision: binary64

Cost: 26312

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{1 + e^{b}}\\ \mathbf{if}\;e^{b} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;e^{b} \leq 1.00002:\\ \;\;\;\;\frac{e^{a}}{e^{a} + \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp b)))))
   (if (<= (exp b) 2e-5)
     t_0
     (if (<= (exp b) 1.00002) (/ (exp a) (+ (exp a) (+ 1.0 b))) t_0))))

C

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

↓

double code(double a, double b) {
	double t_0 = 1.0 / (1.0 + exp(b));
	double tmp;
	if (exp(b) <= 2e-5) {
		tmp = t_0;
	} else if (exp(b) <= 1.00002) {
		tmp = exp(a) / (exp(a) + (1.0 + b));
	} else {
		tmp = t_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 = 1.0d0 / (1.0d0 + exp(b))
    if (exp(b) <= 2d-5) then
        tmp = t_0
    else if (exp(b) <= 1.00002d0) then
        tmp = exp(a) / (exp(a) + (1.0d0 + b))
    else
        tmp = t_0
    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 = 1.0 / (1.0 + Math.exp(b));
	double tmp;
	if (Math.exp(b) <= 2e-5) {
		tmp = t_0;
	} else if (Math.exp(b) <= 1.00002) {
		tmp = Math.exp(a) / (Math.exp(a) + (1.0 + b));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	t_0 = 1.0 / (1.0 + math.exp(b))
	tmp = 0
	if math.exp(b) <= 2e-5:
		tmp = t_0
	elif math.exp(b) <= 1.00002:
		tmp = math.exp(a) / (math.exp(a) + (1.0 + b))
	else:
		tmp = t_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(1.0 / Float64(1.0 + exp(b)))
	tmp = 0.0
	if (exp(b) <= 2e-5)
		tmp = t_0;
	elseif (exp(b) <= 1.00002)
		tmp = Float64(exp(a) / Float64(exp(a) + Float64(1.0 + b)));
	else
		tmp = t_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 = 1.0 / (1.0 + exp(b));
	tmp = 0.0;
	if (exp(b) <= 2e-5)
		tmp = t_0;
	elseif (exp(b) <= 1.00002)
		tmp = exp(a) / (exp(a) + (1.0 + b));
	else
		tmp = t_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[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[b], $MachinePrecision], 2e-5], t$95$0, If[LessEqual[N[Exp[b], $MachinePrecision], 1.00002], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	0.6
Target	0.0
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 2.00000000000000016e-5 or 1.00001999999999991 < (exp.f64 b)

   1. Initial program 0.8

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

   2. Taylor expanded in a around 0 0.5

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

   if 2.00000000000000016e-5 < (exp.f64 b) < 1.00001999999999991

   1. Initial program 0.4

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

   2. Taylor expanded in b around 0 0.7

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

   3. Simplified 0.7

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

      [Start] 0.7	\[ \frac{e^{a}}{1 + \left(e^{a} + b\right)} \]
      rational_best-simplify-115 [=>] 0.7	\[ \frac{e^{a}}{\color{blue}{e^{a} + \left(b + 1\right)}} \]
      rational_best-simplify-1 [=>] 0.7	\[ \frac{e^{a}}{e^{a} + \color{blue}{\left(1 + b\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{elif}\;e^{b} \leq 1.00002:\\ \;\;\;\;\frac{e^{a}}{e^{a} + \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	26184

\[\begin{array}{l} t_0 := \frac{1}{1 + e^{b}}\\ \mathbf{if}\;e^{b} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;e^{b} \leq 1.00002:\\ \;\;\;\;\frac{e^{a}}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.6
Cost	19520

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

Alternative 3
Error	21.5
Cost	6920

\[\begin{array}{l} \mathbf{if}\;b \leq 800:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{elif}\;b \leq 6.1 \cdot 10^{+90}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{b}\\ \end{array} \]

Alternative 4
Error	0.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;a \leq -700000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 5
Error	22.3
Cost	6592

\[\frac{e^{a}}{2} \]

Alternative 6
Error	38.8
Cost	320

\[0.5 + 0.25 \cdot a \]

Alternative 7
Error	38.9
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023104 
(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))))