Quotient of sum of exps

Average Accuracy: 98.8% → 98.8%

Time: 8.3s

Precision: binary64

Cost: 13513

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -38000 \lor \neg \left(b \leq 1.8 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + \left(b + 1\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (or (<= b -38000.0) (not (<= b 1.8e-5)))
   (/ 1.0 (+ (exp b) 1.0))
   (/ (exp a) (+ (exp a) (+ b 1.0)))))

C

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

↓

double code(double a, double b) {
	double tmp;
	if ((b <= -38000.0) || !(b <= 1.8e-5)) {
		tmp = 1.0 / (exp(b) + 1.0);
	} else {
		tmp = exp(a) / (exp(a) + (b + 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) :: tmp
    if ((b <= (-38000.0d0)) .or. (.not. (b <= 1.8d-5))) then
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    else
        tmp = exp(a) / (exp(a) + (b + 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 tmp;
	if ((b <= -38000.0) || !(b <= 1.8e-5)) {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	} else {
		tmp = Math.exp(a) / (Math.exp(a) + (b + 1.0));
	}
	return tmp;
}

Python

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

↓

def code(a, b):
	tmp = 0
	if (b <= -38000.0) or not (b <= 1.8e-5):
		tmp = 1.0 / (math.exp(b) + 1.0)
	else:
		tmp = math.exp(a) / (math.exp(a) + (b + 1.0))
	return tmp

Julia

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

↓

function code(a, b)
	tmp = 0.0
	if ((b <= -38000.0) || !(b <= 1.8e-5))
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	else
		tmp = Float64(exp(a) / Float64(exp(a) + Float64(b + 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)
	tmp = 0.0;
	if ((b <= -38000.0) || ~((b <= 1.8e-5)))
		tmp = 1.0 / (exp(b) + 1.0);
	else
		tmp = exp(a) / (exp(a) + (b + 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_] := If[Or[LessEqual[b, -38000.0], N[Not[LessEqual[b, 1.8e-5]], $MachinePrecision]], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[(b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	98.8%
Target	100.0%
Herbie	98.8%

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

Derivation

1. Split input into 2 regimes

2. if b < -38000 or 1.80000000000000005e-5 < b

   1. Initial program 98.5%

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

   2. Taylor expanded in a around 0 99.1%

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

   if -38000 < b < 1.80000000000000005e-5

   1. Initial program 99.2%

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

   2. Taylor expanded in b around 0 98.6%

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

   3. Simplified 98.6%

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

      [Start] 98.6	\[ \frac{e^{a}}{1 + \left(e^{a} + b\right)} \]
      associate-+r+ [=>] 98.6	\[ \frac{e^{a}}{\color{blue}{\left(1 + e^{a}\right) + b}} \]
      +-commutative [=>] 98.6	\[ \frac{e^{a}}{\color{blue}{\left(e^{a} + 1\right)} + b} \]
      associate-+l+ [=>] 98.6	\[ \frac{e^{a}}{\color{blue}{e^{a} + \left(1 + b\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -38000 \lor \neg \left(b \leq 1.8 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + \left(b + 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	39108

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

Alternative 2
Accuracy	98.2%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;b \leq -38000 \lor \neg \left(b \leq 1.35 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \end{array} \]

Alternative 3
Accuracy	98.2%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;b \leq -41000 \lor \neg \left(b \leq 0.0148\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2 + \left(a + b\right)}\\ \end{array} \]

Alternative 4
Accuracy	97.3%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -38000 \lor \neg \left(b \leq 1.35 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \end{array} \]

Alternative 5
Accuracy	81.2%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-17}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 6
Accuracy	64.7%
Cost	716

\[\begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{-73}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-54}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 0.29:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Accuracy	64.8%
Cost	716

\[\begin{array}{l} \mathbf{if}\;b \leq 3.3 \cdot 10^{-74}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-54}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 1.85:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8
Accuracy	79.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -350:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{b + 2}\right) + -1\\ \end{array} \]

Alternative 9
Accuracy	64.5%
Cost	460

\[\begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{-73}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-54}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 0.0008:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Accuracy	39.0%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023135 
(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))))