Quotient of sum of exps

Percentage Accurate: 99.1% → 99.1%

Time: 6.5s

Alternatives: 15

Speedup: 1.0×

• could not determine a ground truth

  1. a = 2.3000666984840257e+235
  2. b = -6.099102600639252e-31

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: 99.1% 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, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return 1.0 / ((exp(a) + exp(b)) / exp(a));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / ((exp(a) + exp(b)) / exp(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(1.0 / N[(N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision] / N[Exp[a], $MachinePrecision]), $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. frac-2neg N/A

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

   3. div-inv N/A

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

   4. associate-*r/ N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-neg-in N/A

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

   10. unsub-neg N/A

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

   11. lower--.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-neg.f64 99.2

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

   \[\leadsto \frac{1}{\frac{e^{a} + e^{b}}{e^{a}}} \]
6. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(e^{a} + e^{b}\right) \cdot e^{-a}} \end{array} \]

FPCore

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

C

double code(double a, double b) {
	return 1.0 / ((exp(a) + exp(b)) * exp(-a));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(1.0 / N[(N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision] * N[Exp[(-a)], $MachinePrecision]), $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. lower-/.f64 N/A

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

   4. div-inv N/A

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

   5. lower-*.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. rec-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 3: 99.1% 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]

Derivation

1. Initial program 99.2%

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

Alternative 4: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.2:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.2) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.2) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.2d0) then
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.2) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.2:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.2)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.2)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.2], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.20000000000000001

   1. Initial program 98.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. rec-exp N/A

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

      7. lft-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. lower-+.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 0.20000000000000001 < (exp.f64 a)

   1. Initial program 99.4%

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

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

      3. lower-exp.f64 98.2

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

   5. Applied rewrites 98.2%

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

Alternative 5: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.2:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.2) (/ (exp a) (+ 1.0 1.0)) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.2) {
		tmp = exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.2d0) then
        tmp = exp(a) / (1.0d0 + 1.0d0)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.2) {
		tmp = Math.exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.2:
		tmp = math.exp(a) / (1.0 + 1.0)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.2)
		tmp = Float64(exp(a) / Float64(1.0 + 1.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.2)
		tmp = exp(a) / (1.0 + 1.0);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.2], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.20000000000000001

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 97.8%

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

   if 0.20000000000000001 < (exp.f64 a)

   1. Initial program 99.4%

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

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

      3. lower-exp.f64 98.2

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

   5. Applied rewrites 98.2%

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

Alternative 6: 92.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.02e+103)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.02e+103) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.02e+103)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.02e+103], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.01999999999999991e103

   1. Initial program 98.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. rec-exp N/A

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

      7. lft-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. lower-+.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 100.0%

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

   if -1.01999999999999991e103 < a

   1. Initial program 99.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 91.8

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

   5. Applied rewrites 91.8%

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

Alternative 7: 73.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;b \leq 4.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(t\_0, b \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b \cdot b, b\right), -4\right)}{\mathsf{fma}\left(b, t\_0, -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b (fma b 0.16666666666666666 0.5) 1.0)))
   (if (<= b 4.2e+51)
     (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
     (if (<= b 1.05e+103)
       (/
        1.0
        (/
         (fma t_0 (* b (fma (fma b 0.16666666666666666 0.5) (* b b) b)) -4.0)
         (fma b t_0 -2.0)))
       (/ 1.0 (* b (* b (* b 0.16666666666666666))))))))

C

double code(double a, double b) {
	double t_0 = fma(b, fma(b, 0.16666666666666666, 0.5), 1.0);
	double tmp;
	if (b <= 4.2e+51) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else if (b <= 1.05e+103) {
		tmp = 1.0 / (fma(t_0, (b * fma(fma(b, 0.16666666666666666, 0.5), (b * b), b)), -4.0) / fma(b, t_0, -2.0));
	} else {
		tmp = 1.0 / (b * (b * (b * 0.16666666666666666)));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, fma(b, 0.16666666666666666, 0.5), 1.0)
	tmp = 0.0
	if (b <= 4.2e+51)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	elseif (b <= 1.05e+103)
		tmp = Float64(1.0 / Float64(fma(t_0, Float64(b * fma(fma(b, 0.16666666666666666, 0.5), Float64(b * b), b)), -4.0) / fma(b, t_0, -2.0)));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * Float64(b * 0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[b, 4.2e+51], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e+103], N[(1.0 / N[(N[(t$95$0 * N[(b * N[(N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + -4.0), $MachinePrecision] / N[(b * t$95$0 + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.2000000000000002e51

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. rec-exp N/A

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

      7. lft-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. lower-+.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lower-neg.f64 75.4

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 66.7%

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

   if 4.2000000000000002e51 < b < 1.0500000000000001e103

   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 100.0

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

   5. Applied rewrites 100.0%

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

   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 6.5%

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), b \cdot \mathsf{fma}\left(\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b \cdot b, b\right), -4\right)}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right)}, -2\right)}} \]

   if 1.0500000000000001e103 < 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. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 100.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\frac{1}{6} \cdot {b}^{\color{blue}{3}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \frac{1}{b \cdot \left(b \cdot \color{blue}{\left(b \cdot 0.16666666666666666\right)}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 71.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+55}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right) \cdot \left(b \cdot b\right), -1\right)}{\mathsf{fma}\left(b, 0.5, -1\right)}, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.3e+55)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (/
    1.0
    (fma
     b
     (/
      (fma
       (fma b 0.16666666666666666 0.5)
       (* (fma b 0.16666666666666666 0.5) (* b b))
       -1.0)
      (fma b 0.5 -1.0))
     2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.3e+55) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else {
		tmp = 1.0 / fma(b, (fma(fma(b, 0.16666666666666666, 0.5), (fma(b, 0.16666666666666666, 0.5) * (b * b)), -1.0) / fma(b, 0.5, -1.0)), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.3e+55)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	else
		tmp = Float64(1.0 / fma(b, Float64(fma(fma(b, 0.16666666666666666, 0.5), Float64(fma(b, 0.16666666666666666, 0.5) * Float64(b * b)), -1.0) / fma(b, 0.5, -1.0)), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.3e+55], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(b * 0.5 + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3e55

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. rec-exp N/A

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

      7. lft-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. lower-+.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lower-neg.f64 75.4

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 66.7%

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

   if 1.3e55 < 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. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 81.6%

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

   10. Applied rewrites 31.9%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 87.9%

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

Alternative 9: 70.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, -0.16666666666666666, 0.5\right), -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.8e+99)
   (/ 1.0 (fma a (fma a (fma a -0.16666666666666666 0.5) -1.0) 2.0))
   (/ 1.0 (* b (* b (* b 0.16666666666666666))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.8e+99) {
		tmp = 1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0);
	} else {
		tmp = 1.0 / (b * (b * (b * 0.16666666666666666)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.8e+99)
		tmp = Float64(1.0 / fma(a, fma(a, fma(a, -0.16666666666666666, 0.5), -1.0), 2.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * Float64(b * 0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.8e+99], N[(1.0 / N[(a * N[(a * N[(a * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.8000000000000004e99

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. rec-exp N/A

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

      7. lft-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. lower-+.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lower-neg.f64 72.0

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

   7. Applied rewrites 72.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 63.8%

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

   if 5.8000000000000004e99 < 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. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 98.2%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\frac{1}{6} \cdot {b}^{\color{blue}{3}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.2%

       \[\leadsto \frac{1}{b \cdot \left(b \cdot \color{blue}{\left(b \cdot 0.16666666666666666\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 66.3% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+55}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.3e+55)
   (/ 1.0 (fma a (fma a 0.5 -1.0) 2.0))
   (/ 1.0 (* b (* b (* b 0.16666666666666666))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.3e+55) {
		tmp = 1.0 / fma(a, fma(a, 0.5, -1.0), 2.0);
	} else {
		tmp = 1.0 / (b * (b * (b * 0.16666666666666666)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.3e+55)
		tmp = Float64(1.0 / fma(a, fma(a, 0.5, -1.0), 2.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * Float64(b * 0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.3e+55], N[(1.0 / N[(a * N[(a * 0.5 + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3e55

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. rec-exp N/A

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

      7. lft-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. lower-+.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lower-neg.f64 75.4

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 61.6%

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

   if 1.3e55 < 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. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   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 81.6%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\frac{1}{6} \cdot {b}^{\color{blue}{3}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 81.6%

       \[\leadsto \frac{1}{b \cdot \left(b \cdot \color{blue}{\left(b \cdot 0.16666666666666666\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 63.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, -1\right), 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.65e+154)
   (/ 1.0 (fma a (fma a 0.5 -1.0) 2.0))
   (/ 1.0 (* b (* b 0.5)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.65e+154) {
		tmp = 1.0 / fma(a, fma(a, 0.5, -1.0), 2.0);
	} else {
		tmp = 1.0 / (b * (b * 0.5));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.65e+154)
		tmp = Float64(1.0 / fma(a, fma(a, 0.5, -1.0), 2.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.65e+154], N[(1.0 / N[(a * N[(a * 0.5 + -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.65e154

   1. Initial program 99.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. rec-exp N/A

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

      7. lft-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. lower-+.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lower-neg.f64 68.8

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

   7. Applied rewrites 68.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 55.0%

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

   if 1.65e154 < 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. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

   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%

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

Alternative 12: 52.5% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, b \cdot 0.5, b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9e+52) (/ 1.0 (- 2.0 a)) (/ 1.0 (fma b (* b 0.5) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9e+52) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 1.0 / fma(b, (b * 0.5), b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9e+52)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(1.0 / fma(b, Float64(b * 0.5), b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9e+52], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.9999999999999999e52

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. rec-exp N/A

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

      7. lft-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. lower-+.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lower-neg.f64 75.4

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 45.9%

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

   if 8.9999999999999999e52 < 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. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 58.7%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 58.7%

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

Alternative 13: 52.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9e+52) (/ 1.0 (- 2.0 a)) (/ 1.0 (* b (* b 0.5)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9e+52) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 1.0 / (b * (b * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 9d+52) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = 1.0d0 / (b * (b * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 9e+52) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = 1.0 / (b * (b * 0.5));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 9e+52:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = 1.0 / (b * (b * 0.5))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9e+52)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9e+52)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = 1.0 / (b * (b * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9e+52], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.9999999999999999e52

   1. Initial program 99.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. rec-exp N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. rec-exp N/A

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

      7. lft-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. lower-+.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lower-neg.f64 75.4

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 45.9%

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

   if 8.9999999999999999e52 < 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. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 58.7%

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

   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 58.7%

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

Alternative 14: 39.2% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 1.0 / (2.0 - a)

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(1.0 / N[(2.0 - 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. frac-2neg N/A

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

   3. div-inv N/A

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

   4. associate-*r/ N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-neg-in N/A

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

   10. unsub-neg N/A

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

   11. lower--.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-neg.f64 99.2

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in b around 0

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

   1. *-lft-identity N/A

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

   2. associate-*l/ N/A

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

   3. rec-exp N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. rec-exp N/A

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

   7. lft-mult-inverse N/A

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

   8. *-rgt-identity N/A

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

   9. lower-+.f64 N/A

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

   10. neg-mul-1 N/A

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

   11. lower-exp.f64 N/A

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

   12. neg-mul-1 N/A

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

   13. lower-neg.f64 64.2

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

7. Applied rewrites 64.2%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 35.8%

    \[\leadsto \frac{1}{2 - \color{blue}{a}} \]
11. Add Preprocessing

Alternative 15: 38.3% 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 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 79.2

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 35.1%

   \[\leadsto 0.5 \]
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 2024237 
(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))))