Quotient of sum of exps

Percentage Accurate: 98.9% → 98.3%

Time: 7.9s

Precision: binary64

Cost: 6852

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -620000000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (if (<= a -620000000000.0) 0.0 (/ 1.0 (+ (exp 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 (a <= -620000000000.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (exp(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 (a <= (-620000000000.0d0)) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 / (exp(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 (a <= -620000000000.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (Math.exp(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 a <= -620000000000.0:
		tmp = 0.0
	else:
		tmp = 1.0 / (math.exp(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 (a <= -620000000000.0)
		tmp = 0.0;
	else
		tmp = Float64(1.0 / Float64(exp(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 (a <= -620000000000.0)
		tmp = 0.0;
	else
		tmp = 1.0 / (exp(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[LessEqual[a, -620000000000.0], 0.0, N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Target

Original	98.9%
Target	100.0%
Herbie	98.3%

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

Derivation

1. Split input into 2 regimes

2. if a < -6.2e11

   1. Initial program 96.7%

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

   2. Taylor expanded in a around 0 25.4%

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

   3. Taylor expanded in b around 0 3.7%

      \[\leadsto \frac{1}{\color{blue}{2 + b}} \]

   4. Simplified 3.7%

      \[\leadsto \frac{1}{\color{blue}{b + 2}} \]
      Step-by-step derivation

      [Start] 3.7	\[ \frac{1}{2 + b} \]
      +-commutative [=>] 3.7	\[ \frac{1}{\color{blue}{b + 2}} \]

   5. Applied egg-rr 23.8%

      \[\leadsto \color{blue}{\left(1 + \frac{1}{b + 2}\right) - 1} \]
      Step-by-step derivation

      [Start] 3.7	\[ \frac{1}{b + 2} \]
      expm1-log1p-u [=>] 3.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{b + 2}\right)\right)} \]
      expm1-udef [=>] 23.8	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{b + 2}\right)} - 1} \]
      log1p-udef [=>] 23.8	\[ e^{\color{blue}{\log \left(1 + \frac{1}{b + 2}\right)}} - 1 \]
      add-exp-log [<=] 23.8	\[ \color{blue}{\left(1 + \frac{1}{b + 2}\right)} - 1 \]

   6. Simplified 23.8%

      \[\leadsto \color{blue}{1 + \left(\frac{1}{b + 2} - 1\right)} \]
      Step-by-step derivation

      [Start] 23.8	\[ \left(1 + \frac{1}{b + 2}\right) - 1 \]
      associate--l+ [=>] 23.8	\[ \color{blue}{1 + \left(\frac{1}{b + 2} - 1\right)} \]

   7. Taylor expanded in b around inf 100.0%

      \[\leadsto 1 + \color{blue}{-1} \]

   if -6.2e11 < a

   1. Initial program 97.9%

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

   2. Taylor expanded in a around 0 97.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -620000000000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	80.8%
Cost	26185

\[\begin{array}{l} \mathbf{if}\;e^{b} \leq 1 \lor \neg \left(e^{b} \leq 1.0000005\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	25920

\[e^{a - \log \left(e^{a} + e^{b}\right)} \]

Alternative 3
Accuracy	98.9%
Cost	19520

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

Alternative 4
Accuracy	76.1%
Cost	7124

\[\begin{array}{l} t_0 := 0.5 + a \cdot 0.25\\ \mathbf{if}\;b \leq -2.1 \cdot 10^{-72}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq -3.7 \cdot 10^{-164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-193}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq -1.02 \cdot 10^{-240}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-275}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Accuracy	79.6%
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq -3000:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{b + 2} + -1\right)\\ \end{array} \]

Alternative 6
Accuracy	63.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;a \leq -0.37:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-57}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7
Accuracy	63.5%
Cost	460

\[\begin{array}{l} \mathbf{if}\;a \leq -0.53:\\ \;\;\;\;0\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-57}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8
Accuracy	39.2%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023159 
(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))))