Quotient of sum of exps

Percentage Accurate: 99.0% → 99.1%

Time: 8.7s

Alternatives: 17

Speedup: 1.0×

• could not determine a ground truth

  1. a = 1.3013793820117822e+33
  2. b = 1.474311795111808e+102

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: 99.0% 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: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)} \end{array} \]

FPCore

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

C

double code(double a, double b) {
	return exp(fma(log((exp(a) + exp(b))), -1.0, a));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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. associate-/r/ N/A

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

   4. inv-pow N/A

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

   5. pow-to-exp N/A

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

   6. lift-exp.f64 N/A

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

   7. prod-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-fma.f64 N/A

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]

   10. lower-log.f64 99.2

       \[\leadsto e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{a} + e^{b}\right)}, -1, a\right)} \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(e^{a} + e^{b}\right), -1, a\right)}} \]
5. Add Preprocessing

Alternative 2: 67.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{1}{b \cdot \mathsf{fma}\left(b, 0.5, \mathsf{fma}\left(b, b \cdot \left(0.16666666666666666 + \frac{2}{b \cdot \left(b \cdot b\right)}\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.0)
   (/
    1.0
    (*
     b
     (fma
      b
      0.5
      (fma b (* b (+ 0.16666666666666666 (/ 2.0 (* b (* b b))))) 1.0))))
   0.5))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.0) {
		tmp = 1.0 / (b * fma(b, 0.5, fma(b, (b * (0.16666666666666666 + (2.0 / (b * (b * b))))), 1.0)));
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.0)
		tmp = Float64(1.0 / Float64(b * fma(b, 0.5, fma(b, Float64(b * Float64(0.16666666666666666 + Float64(2.0 / Float64(b * Float64(b * b))))), 1.0))));
	else
		tmp = 0.5;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(b * N[(b * 0.5 + N[(b * N[(b * N[(0.16666666666666666 + N[(2.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.0

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-+.f64 N/A

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

      3. lower-exp.f64 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2 + \color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right)}, 2\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{{b}^{3} \cdot \left(\frac{1}{6} + \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b} + \left(\frac{1}{{b}^{2}} + 2 \cdot \frac{1}{{b}^{3}}\right)\right)}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 48.5%

       \[\leadsto \frac{1}{b \cdot \left(\mathsf{fma}\left(b, 0.5, 1\right) + \color{blue}{\left(\frac{2}{b \cdot \left(b \cdot b\right)} + 0.16666666666666666\right) \cdot \left(b \cdot b\right)}\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 67.3%

       \[\leadsto \frac{1}{\mathsf{fma}\left(b, 0.5, \mathsf{fma}\left(b, b \cdot \left(0.16666666666666666 + \frac{2}{b \cdot \left(b \cdot b\right)}\right), 1\right)\right) \cdot b} \]

   if 0.0 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.2%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-+.f64 N/A

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

      3. lower-exp.f64 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.2%

      \[\leadsto 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{1}{b \cdot \mathsf{fma}\left(b, 0.5, \mathsf{fma}\left(b, b \cdot \left(0.16666666666666666 + \frac{2}{b \cdot \left(b \cdot b\right)}\right), 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
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 2024222 
(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))))