Quotient of sum of exps

Percentage Accurate: 98.9% → 98.9%

Time: 6.3s

Alternatives: 15

Speedup: 1.0×

• could not determine a ground truth

  1. a = 7.818227542460876e+71
  2. b = 3.4615475814980363e+164

Specification

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

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

C

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

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

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

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

C

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

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

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{b} + e^{a}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(b) + exp(a))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(b) + Math.exp(a));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing

3. Final simplification 98.8%

   \[\leadsto \frac{e^{a}}{e^{b} + e^{a}} \]
4. Add Preprocessing

Alternative 2: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999999:\\ \;\;\;\;\frac{1}{e^{-a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999999999)
   (/ 1.0 (+ (exp (- a)) 1.0))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999999999) {
		tmp = 1.0 / (exp(-a) + 1.0);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.999999999d0) then
        tmp = 1.0d0 / (exp(-a) + 1.0d0)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.999999999) {
		tmp = 1.0 / (Math.exp(-a) + 1.0);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.999999999:
		tmp = 1.0 / (math.exp(-a) + 1.0)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.999999999)
		tmp = Float64(1.0 / Float64(exp(Float64(-a)) + 1.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.999999999)
		tmp = 1.0 / (exp(-a) + 1.0);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.999999999], N[(1.0 / N[(N[Exp[(-a)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.999999999000000028

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 98.4

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{neg}\left(a\right)} \cdot \left(1 + e^{a}\right)}} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-+.f64 N/A

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

      6. neg-mul-1 N/A

         \[\leadsto \frac{1}{e^{\color{blue}{-1 \cdot a}} + 1} \]

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{e^{-1 \cdot a}} + 1} \]

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 97.1

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

   7. Applied rewrites 97.1%

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

   if 0.999999999000000028 < (exp.f64 a)

   1. Initial program 99.1%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999999:\\ \;\;\;\;\frac{1}{e^{-a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{b - a}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 1.0 (exp (- b a)))))

C

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 = 1.0d0 / (1.0d0 + exp((b - a)))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (1.0 + Math.exp((b - a)));
}

Python

def code(a, b):
	return 1.0 / (1.0 + math.exp((b - a)))

Julia

function code(a, b)
	return Float64(1.0 / Float64(1.0 + exp(Float64(b - a))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (1.0 + exp((b - a)));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(1.0 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024229 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ 1 (exp (- b a)))))

  (/ (exp a) (+ (exp a) (exp b))))