Quotient of sum of exps

Percentage Accurate: 98.8% → 99.1%

Time: 5.5s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. a = 2.1951458659181632e+42
  2. b = 3.780291627567537e-58

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: 98.8% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.99997 \lor \neg \left(e^{b} \leq 1.0002\right):\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-a} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= (exp b) 0.99997) (not (<= (exp b) 1.0002)))
   (pow (+ (exp b) 1.0) -1.0)
   (pow (+ (exp (- a)) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(b) <= 0.99997) || !(exp(b) <= 1.0002)) {
		tmp = pow((exp(b) + 1.0), -1.0);
	} else {
		tmp = pow((exp(-a) + 1.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((exp(b) <= 0.99997d0) .or. (.not. (exp(b) <= 1.0002d0))) then
        tmp = (exp(b) + 1.0d0) ** (-1.0d0)
    else
        tmp = (exp(-a) + 1.0d0) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.exp(b) <= 0.99997) || !(Math.exp(b) <= 1.0002)) {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	} else {
		tmp = Math.pow((Math.exp(-a) + 1.0), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.exp(b) <= 0.99997) or not (math.exp(b) <= 1.0002):
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	else:
		tmp = math.pow((math.exp(-a) + 1.0), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((exp(b) <= 0.99997) || !(exp(b) <= 1.0002))
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	else
		tmp = Float64(exp(Float64(-a)) + 1.0) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((exp(b) <= 0.99997) || ~((exp(b) <= 1.0002)))
		tmp = (exp(b) + 1.0) ^ -1.0;
	else
		tmp = (exp(-a) + 1.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[N[Exp[b], $MachinePrecision], 0.99997], N[Not[LessEqual[N[Exp[b], $MachinePrecision], 1.0002]], $MachinePrecision]], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[Exp[(-a)], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 0.99997000000000003 or 1.0002 < (exp.f64 b)

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

   if 0.99997000000000003 < (exp.f64 b) < 1.0002

   1. Initial program 99.2%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. inv-pow N/A

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

      5. unpow-1 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. remove-double-div N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. lift-exp.f64 N/A

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

      12. exp-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-exp.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-exp.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 99.6

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

   9. Applied rewrites 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0.99997 \lor \neg \left(e^{b} \leq 1.0002\right):\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{-a} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. inv-pow N/A

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

   5. unpow-1 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. rec-exp N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. exp-neg N/A

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

   6. lift-exp.f64 N/A

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

   7. remove-double-div N/A

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

   8. clear-num N/A

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

   9. lower-/.f64 N/A

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

   10. div-inv N/A

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

   11. lift-exp.f64 N/A

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

   12. exp-neg N/A

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

   13. lift-neg.f64 N/A

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

   14. lift-exp.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

6. Applied rewrites 99.2%

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

7. Final simplification 99.2%

   \[\leadsto {\left(\mathsf{fma}\left(e^{-a}, e^{b}, 1\right)\right)}^{-1} \]
8. Add Preprocessing

Alternative 3: 53.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 1.0002:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 1.0002)
   (pow (- 2.0 a) -1.0)
   (pow (fma (fma 0.5 b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (exp(b) <= 1.0002) {
		tmp = pow((2.0 - a), -1.0);
	} else {
		tmp = pow(fma(fma(0.5, b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (exp(b) <= 1.0002)
		tmp = Float64(2.0 - a) ^ -1.0;
	else
		tmp = fma(fma(0.5, b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[b], $MachinePrecision], 1.0002], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 1.0002

   1. Initial program 98.4%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. inv-pow N/A

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

      5. unpow-1 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. remove-double-div N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. lift-exp.f64 N/A

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

      12. exp-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-exp.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-exp.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 78.5

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

   9. Applied rewrites 78.5%

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

   10. Taylor expanded in a around 0

       \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 51.0%

       \[\leadsto \frac{1}{2 - \color{blue}{a}} \]

   if 1.0002 < (exp.f64 b)

   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. +-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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 55.9%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), \color{blue}{b}, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 1.0002:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -45000000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -45000000.0) (/ (exp a) 2.0) (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -45000000.0) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = pow((exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-45000000.0d0)) then
        tmp = exp(a) / 2.0d0
    else
        tmp = (exp(b) + 1.0d0) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -45000000.0) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -45000000.0:
		tmp = math.exp(a) / 2.0
	else:
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -45000000.0)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -45000000.0)
		tmp = exp(a) / 2.0;
	else
		tmp = (exp(b) + 1.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -45000000.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5e7

   1. Initial program 98.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{e^{a}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

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

   if -4.5e7 < a

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -45000000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+56}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right), a, -1\right), a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.4e+56)
   (pow (fma (fma (fma -0.16666666666666666 a 0.5) a -1.0) a 2.0) -1.0)
   (pow (fma (fma (* 0.16666666666666666 b) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.4e+56) {
		tmp = pow(fma(fma(fma(-0.16666666666666666, a, 0.5), a, -1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(fma(fma((0.16666666666666666 * b), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.4e+56)
		tmp = fma(fma(fma(-0.16666666666666666, a, 0.5), a, -1.0), a, 2.0) ^ -1.0;
	else
		tmp = fma(fma(Float64(0.16666666666666666 * b), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.4e+56], N[Power[N[(N[(N[(-0.16666666666666666 * a + 0.5), $MachinePrecision] * a + -1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.40000000000000004e56

   1. Initial program 98.4%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. inv-pow N/A

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

      5. unpow-1 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. remove-double-div N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. lift-exp.f64 N/A

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

      12. exp-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-exp.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-exp.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 77.8

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

   9. Applied rewrites 77.8%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 70.1%

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

   if 1.40000000000000004e56 < b

   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. +-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 100.0

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

   5. Applied rewrites 100.0%

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

   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 86.6%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot b, b, 1\right), b, 2\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 86.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+56}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right), a, -1\right), a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.76 \cdot 10^{+61}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.76e+61)
   (/ (exp a) 2.0)
   (pow (fma (fma (* 0.16666666666666666 b) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.76e+61) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = pow(fma(fma((0.16666666666666666 * b), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.76e+61)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = fma(fma(Float64(0.16666666666666666 * b), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.76e+61], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.75999999999999997e61

   1. Initial program 98.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 76.6

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

   5. Applied rewrites 76.6%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{e^{a}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 72.9%

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

   if 1.75999999999999997e61 < b

   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. +-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 100.0

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

   5. Applied rewrites 100.0%

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

   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 90.7%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot b, b, 1\right), b, 2\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 90.7%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.76 \cdot 10^{+61}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+56}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 1\right) + 1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.4e+56)
   (pow (+ (fma (fma 0.5 a -1.0) a 1.0) 1.0) -1.0)
   (pow (fma (fma (* 0.16666666666666666 b) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.4e+56) {
		tmp = pow((fma(fma(0.5, a, -1.0), a, 1.0) + 1.0), -1.0);
	} else {
		tmp = pow(fma(fma((0.16666666666666666 * b), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.4e+56)
		tmp = Float64(fma(fma(0.5, a, -1.0), a, 1.0) + 1.0) ^ -1.0;
	else
		tmp = fma(fma(Float64(0.16666666666666666 * b), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.4e+56], N[Power[N[(N[(N[(0.5 * a + -1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.40000000000000004e56

   1. Initial program 98.4%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. inv-pow N/A

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

      5. unpow-1 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. remove-double-div N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. lift-exp.f64 N/A

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

      12. exp-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-exp.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-exp.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 77.8

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

   9. Applied rewrites 77.8%

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

   10. Taylor expanded in a around 0

       \[\leadsto \frac{1}{\left(1 + a \cdot \left(\frac{1}{2} \cdot a - 1\right)\right) + 1} \]
   11. Step-by-step derivation

   12. Applied rewrites 67.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 1\right) + 1} \]

   if 1.40000000000000004e56 < b

   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. +-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 100.0

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

   5. Applied rewrites 100.0%

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

   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 86.6%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot b, b, 1\right), b, 2\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 86.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+56}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 1\right) + 1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+56}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 1\right) + 1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot \left(b \cdot b\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.4e+56)
   (pow (+ (fma (fma 0.5 a -1.0) a 1.0) 1.0) -1.0)
   (pow (* (fma 0.16666666666666666 b 0.5) (* b b)) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.4e+56) {
		tmp = pow((fma(fma(0.5, a, -1.0), a, 1.0) + 1.0), -1.0);
	} else {
		tmp = pow((fma(0.16666666666666666, b, 0.5) * (b * b)), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.4e+56)
		tmp = Float64(fma(fma(0.5, a, -1.0), a, 1.0) + 1.0) ^ -1.0;
	else
		tmp = Float64(fma(0.16666666666666666, b, 0.5) * Float64(b * b)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.4e+56], N[Power[N[(N[(N[(0.5 * a + -1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.40000000000000004e56

   1. Initial program 98.4%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. inv-pow N/A

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

      5. unpow-1 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. remove-double-div N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. lift-exp.f64 N/A

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

      12. exp-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-exp.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-exp.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 77.8

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

   9. Applied rewrites 77.8%

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

   10. Taylor expanded in a around 0

       \[\leadsto \frac{1}{\left(1 + a \cdot \left(\frac{1}{2} \cdot a - 1\right)\right) + 1} \]
   11. Step-by-step derivation

   12. Applied rewrites 67.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 1\right) + 1} \]

   if 1.40000000000000004e56 < b

   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. +-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 100.0

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

   5. Applied rewrites 100.0%

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

   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 86.6%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 86.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot \left(b \cdot \color{blue}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+56}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 1\right) + 1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot \left(b \cdot b\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+154}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 1\right) + 1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.15e+154)
   (pow (+ (fma (fma 0.5 a -1.0) a 1.0) 1.0) -1.0)
   (pow (* (* 0.5 b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.15e+154) {
		tmp = pow((fma(fma(0.5, a, -1.0), a, 1.0) + 1.0), -1.0);
	} else {
		tmp = pow(((0.5 * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.15e+154)
		tmp = Float64(fma(fma(0.5, a, -1.0), a, 1.0) + 1.0) ^ -1.0;
	else
		tmp = Float64(Float64(0.5 * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.15e+154], N[Power[N[(N[(N[(0.5 * a + -1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.5 * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15e154

   1. Initial program 98.6%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. inv-pow N/A

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

      5. unpow-1 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. remove-double-div N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. lift-exp.f64 N/A

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

      12. exp-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-exp.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-exp.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 72.8

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

   9. Applied rewrites 72.8%

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

   10. Taylor expanded in a around 0

       \[\leadsto \frac{1}{\left(1 + a \cdot \left(\frac{1}{2} \cdot a - 1\right)\right) + 1} \]
   11. Step-by-step derivation

   12. Applied rewrites 61.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 1\right) + 1} \]

   if 1.15e154 < b

   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. +-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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\frac{1}{2} \cdot {b}^{\color{blue}{2}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\left(0.5 \cdot b\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+154}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 1\right) + 1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+154}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.15e+154)
   (pow (fma (fma 0.5 a -1.0) a 2.0) -1.0)
   (pow (* (* 0.5 b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.15e+154) {
		tmp = pow(fma(fma(0.5, a, -1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(((0.5 * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.15e+154)
		tmp = fma(fma(0.5, a, -1.0), a, 2.0) ^ -1.0;
	else
		tmp = Float64(Float64(0.5 * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.15e+154], N[Power[N[(N[(0.5 * a + -1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.5 * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15e154

   1. Initial program 98.6%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. inv-pow N/A

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

      5. unpow-1 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. remove-double-div N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. lift-exp.f64 N/A

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

      12. exp-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-exp.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-exp.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 72.8

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

   9. Applied rewrites 72.8%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 61.6%

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

   if 1.15e154 < b

   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. +-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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\frac{1}{2} \cdot {b}^{\color{blue}{2}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\left(0.5 \cdot b\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+154}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{+53}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.4e+53) (pow (- 2.0 a) -1.0) (pow (* (* 0.5 b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.4e+53) {
		tmp = pow((2.0 - a), -1.0);
	} else {
		tmp = pow(((0.5 * b) * b), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.4d+53) then
        tmp = (2.0d0 - a) ** (-1.0d0)
    else
        tmp = ((0.5d0 * b) * b) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.4e+53) {
		tmp = Math.pow((2.0 - a), -1.0);
	} else {
		tmp = Math.pow(((0.5 * b) * b), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.4e+53:
		tmp = math.pow((2.0 - a), -1.0)
	else:
		tmp = math.pow(((0.5 * b) * b), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.4e+53)
		tmp = Float64(2.0 - a) ^ -1.0;
	else
		tmp = Float64(Float64(0.5 * b) * b) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.4e+53)
		tmp = (2.0 - a) ^ -1.0;
	else
		tmp = ((0.5 * b) * b) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.4e+53], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.5 * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.4e53

   1. Initial program 98.4%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. inv-pow N/A

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

      5. unpow-1 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. remove-double-div N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. div-inv N/A

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

      11. lift-exp.f64 N/A

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

      12. exp-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-exp.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-lft-in N/A

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-exp.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 77.8

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

   9. Applied rewrites 77.8%

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

   10. Taylor expanded in a around 0

       \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot a}} \]
   11. Step-by-step derivation

   12. Applied rewrites 48.8%

       \[\leadsto \frac{1}{2 - \color{blue}{a}} \]

   if 2.4e53 < b

   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. +-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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 63.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\frac{1}{2} \cdot {b}^{\color{blue}{2}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.0%

       \[\leadsto \frac{1}{\left(0.5 \cdot b\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{+53}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ {\left(2 - a\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (pow (- 2.0 a) -1.0))

C

double code(double a, double b) {
	return pow((2.0 - a), -1.0);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (2.0d0 - a) ** (-1.0d0)
end function

Java

public static double code(double a, double b) {
	return Math.pow((2.0 - a), -1.0);
}

Python

def code(a, b):
	return math.pow((2.0 - a), -1.0)

Julia

function code(a, b)
	return Float64(2.0 - a) ^ -1.0
end

MATLAB

function tmp = code(a, b)
	tmp = (2.0 - a) ^ -1.0;
end

Wolfram

code[a_, b_] := N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 98.8%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. inv-pow N/A

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

   5. unpow-1 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. rec-exp N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 98.8

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

4. Applied rewrites 98.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. exp-neg N/A

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

   6. lift-exp.f64 N/A

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

   7. remove-double-div N/A

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

   8. clear-num N/A

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

   9. lower-/.f64 N/A

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

   10. div-inv N/A

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

   11. lift-exp.f64 N/A

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

   12. exp-neg N/A

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

   13. lift-neg.f64 N/A

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

   14. lift-exp.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. distribute-lft-in N/A

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

6. Applied rewrites 99.2%

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

7. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. neg-mul-1 N/A

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

   4. lower-exp.f64 N/A

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

   5. neg-mul-1 N/A

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

   6. lower-neg.f64 65.8

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

9. Applied rewrites 65.8%

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

10. Taylor expanded in a around 0

    \[\leadsto \frac{1}{2 + \color{blue}{-1 \cdot a}} \]
11. Step-by-step derivation

12. Applied rewrites 37.7%

    \[\leadsto \frac{1}{2 - \color{blue}{a}} \]

13. Final simplification 37.7%

    \[\leadsto {\left(2 - a\right)}^{-1} \]
14. Add Preprocessing

Alternative 13: 40.2% accurate, 315.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.5)

C

double code(double a, double b) {
	return 0.5;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0
end function

Java

public static double code(double a, double b) {
	return 0.5;
}

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5;
end

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 79.9

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

5. Applied rewrites 79.9%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 36.8%

   \[\leadsto 0.5 \]
9. 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 2024322 
(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))))