Quotient of sum of exps

Percentage Accurate: 98.9% → 100.0%

Time: 5.6s

Alternatives: 10

Speedup: 2.7×

• could not determine a ground truth

  1. a = 8.724553103329932e+61
  2. b = 3.5361157278208164e+120

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   5. +-commutative N/A

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

   6. div-add N/A

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

   7. *-inverses N/A

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

   8. lower-+.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. div-exp N/A

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

   12. lower-exp.f64 N/A

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

   13. lower--.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 56.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.6:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.6)
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))
   0.5))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.6) {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.6)
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	else
		tmp = 0.5;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.6], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.599999999999999978

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 74.5

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

   5. Applied rewrites 74.5%

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

   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 63.5%

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

   if 0.599999999999999978 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 95.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 96.6

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 18.8%

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

Alternative 3: 53.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.6:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.6)
   (/ 1.0 (fma (fma 0.5 b 1.0) b 2.0))
   0.5))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.6) {
		tmp = 1.0 / fma(fma(0.5, b, 1.0), b, 2.0);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.6)
		tmp = Float64(1.0 / fma(fma(0.5, b, 1.0), b, 2.0));
	else
		tmp = 0.5;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.6], N[(1.0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.599999999999999978

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 74.5

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

   5. Applied rewrites 74.5%

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

   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 60.0%

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

   if 0.599999999999999978 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 95.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 96.6

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 18.8%

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

Alternative 4: 52.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.4998:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5 \cdot b, b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.4998)
   (/ 1.0 (fma (* 0.5 b) b 2.0))
   0.5))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.4998) {
		tmp = 1.0 / fma((0.5 * b), b, 2.0);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.4998)
		tmp = Float64(1.0 / fma(Float64(0.5 * b), b, 2.0));
	else
		tmp = 0.5;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.4998], N[(1.0 / N[(N[(0.5 * b), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.49980000000000002

   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 59.7

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

   5. Applied rewrites 59.7%

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

   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 35.9%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 35.5%

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

   if 0.49980000000000002 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (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 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 67.3%

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

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{e^{b} + 1}\\ \mathbf{if}\;e^{b} \leq 0.9999999:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;e^{b} \leq 1.00000000002:\\ \;\;\;\;\frac{1}{e^{-a} + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ (exp b) 1.0))))
   (if (<= (exp b) 0.9999999)
     t_0
     (if (<= (exp b) 1.00000000002) (/ 1.0 (+ (exp (- a)) 1.0)) t_0))))

C

double code(double a, double b) {
	double t_0 = 1.0 / (exp(b) + 1.0);
	double tmp;
	if (exp(b) <= 0.9999999) {
		tmp = t_0;
	} else if (exp(b) <= 1.00000000002) {
		tmp = 1.0 / (exp(-a) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (exp(b) + 1.0d0)
    if (exp(b) <= 0.9999999d0) then
        tmp = t_0
    else if (exp(b) <= 1.00000000002d0) then
        tmp = 1.0d0 / (exp(-a) + 1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 1.0 / (Math.exp(b) + 1.0);
	double tmp;
	if (Math.exp(b) <= 0.9999999) {
		tmp = t_0;
	} else if (Math.exp(b) <= 1.00000000002) {
		tmp = 1.0 / (Math.exp(-a) + 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 1.0 / (math.exp(b) + 1.0)
	tmp = 0
	if math.exp(b) <= 0.9999999:
		tmp = t_0
	elif math.exp(b) <= 1.00000000002:
		tmp = 1.0 / (math.exp(-a) + 1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(1.0 / Float64(exp(b) + 1.0))
	tmp = 0.0
	if (exp(b) <= 0.9999999)
		tmp = t_0;
	elseif (exp(b) <= 1.00000000002)
		tmp = Float64(1.0 / Float64(exp(Float64(-a)) + 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 1.0 / (exp(b) + 1.0);
	tmp = 0.0;
	if (exp(b) <= 0.9999999)
		tmp = t_0;
	elseif (exp(b) <= 1.00000000002)
		tmp = 1.0 / (exp(-a) + 1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Exp[b], $MachinePrecision], 0.9999999], t$95$0, If[LessEqual[N[Exp[b], $MachinePrecision], 1.00000000002], N[(1.0 / N[(N[Exp[(-a)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 0.999999900000000053 or 1.00000000002 < (exp.f64 b)

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 0.999999900000000053 < (exp.f64 b) < 1.00000000002

   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. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. *-inverses N/A

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

      8. lower-+.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. div-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. neg-mul-1 N/A

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

      2. lower-neg.f64 99.6

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

   7. Applied rewrites 99.6%

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

Alternative 6: 92.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b + 1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right), a, -1\right), a, 1\right), 1\right)}\\ \mathbf{elif}\;a \leq -880000000:\\ \;\;\;\;{b}^{3} \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9.6e+41)
   (/
    1.0
    (fma
     (+ b 1.0)
     (fma (fma (fma -0.16666666666666666 a 0.5) a -1.0) a 1.0)
     1.0))
   (if (<= a -880000000.0)
     (* (pow b 3.0) 0.020833333333333332)
     (/ 1.0 (+ (exp b) 1.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9.6e+41) {
		tmp = 1.0 / fma((b + 1.0), fma(fma(fma(-0.16666666666666666, a, 0.5), a, -1.0), a, 1.0), 1.0);
	} else if (a <= -880000000.0) {
		tmp = pow(b, 3.0) * 0.020833333333333332;
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9.6e+41)
		tmp = Float64(1.0 / fma(Float64(b + 1.0), fma(fma(fma(-0.16666666666666666, a, 0.5), a, -1.0), a, 1.0), 1.0));
	elseif (a <= -880000000.0)
		tmp = Float64((b ^ 3.0) * 0.020833333333333332);
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9.6e+41], N[(1.0 / N[(N[(b + 1.0), $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * a + 0.5), $MachinePrecision] * a + -1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -880000000.0], N[(N[Power[b, 3.0], $MachinePrecision] * 0.020833333333333332), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.6000000000000007e41

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. 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. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. *-inverses N/A

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

      8. lower-+.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. div-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-exp.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 88.3%

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

   if -9.6000000000000007e41 < a < -8.8e8

   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 19.3

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

   5. Applied rewrites 19.3%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.020833333333333332, -0.25\right), \color{blue}{b}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{48} \cdot {b}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 67.7%

       \[\leadsto {b}^{3} \cdot 0.020833333333333332 \]

   if -8.8e8 < a

   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 97.1

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

   5. Applied rewrites 97.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b + 1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right), a, -1\right), a, 1\right), 1\right)}\\ \mathbf{elif}\;a \leq -880000000:\\ \;\;\;\;{b}^{3} \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2300:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b + 1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right), a, -1\right), a, 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -2300.0)
   0.5
   (if (<= b 9.2e+102)
     (/
      1.0
      (fma
       (+ b 1.0)
       (fma (fma (fma -0.16666666666666666 a 0.5) a -1.0) a 1.0)
       1.0))
     (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2300.0) {
		tmp = 0.5;
	} else if (b <= 9.2e+102) {
		tmp = 1.0 / fma((b + 1.0), fma(fma(fma(-0.16666666666666666, a, 0.5), a, -1.0), a, 1.0), 1.0);
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -2300.0)
		tmp = 0.5;
	elseif (b <= 9.2e+102)
		tmp = Float64(1.0 / fma(Float64(b + 1.0), fma(fma(fma(-0.16666666666666666, a, 0.5), a, -1.0), a, 1.0), 1.0));
	else
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -2300.0], 0.5, If[LessEqual[b, 9.2e+102], N[(1.0 / N[(N[(b + 1.0), $MachinePrecision] * N[(N[(N[(-0.16666666666666666 * a + 0.5), $MachinePrecision] * a + -1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2300

   1. Initial program 97.8%

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

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -2300 < b < 9.1999999999999995e102

   1. Initial program 99.3%

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

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. *-inverses N/A

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

      8. lower-+.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. div-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-exp.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 90.6

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 77.2%

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

   if 9.1999999999999995e102 < 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 100.0%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2300:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b + 1, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right), a, -1\right), a, 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2300:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b + 1, \mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -2300.0)
   0.5
   (if (<= b 5.2e+102)
     (/ 1.0 (fma (+ b 1.0) (fma (fma 0.5 a -1.0) a 1.0) 1.0))
     (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2300.0) {
		tmp = 0.5;
	} else if (b <= 5.2e+102) {
		tmp = 1.0 / fma((b + 1.0), fma(fma(0.5, a, -1.0), a, 1.0), 1.0);
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -2300.0)
		tmp = 0.5;
	elseif (b <= 5.2e+102)
		tmp = Float64(1.0 / fma(Float64(b + 1.0), fma(fma(0.5, a, -1.0), a, 1.0), 1.0));
	else
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -2300.0], 0.5, If[LessEqual[b, 5.2e+102], N[(1.0 / N[(N[(b + 1.0), $MachinePrecision] * N[(N[(0.5 * a + -1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2300

   1. Initial program 97.8%

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

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -2300 < b < 5.20000000000000013e102

   1. Initial program 99.3%

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

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. *-inverses N/A

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

      8. lower-+.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. div-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-exp.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 90.6

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

   7. Applied rewrites 90.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 70.9%

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

   if 5.20000000000000013e102 < 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 100.0%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2300:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b + 1, \mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b + 1, 1 - a, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.4)
   0.5
   (if (<= b 2.5e+118)
     (/ 1.0 (fma (+ b 1.0) (- 1.0 a) 1.0))
     (/ 1.0 (fma (fma 0.5 b 1.0) b 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.4) {
		tmp = 0.5;
	} else if (b <= 2.5e+118) {
		tmp = 1.0 / fma((b + 1.0), (1.0 - a), 1.0);
	} else {
		tmp = 1.0 / fma(fma(0.5, b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.4)
		tmp = 0.5;
	elseif (b <= 2.5e+118)
		tmp = Float64(1.0 / fma(Float64(b + 1.0), Float64(1.0 - a), 1.0));
	else
		tmp = Float64(1.0 / fma(fma(0.5, b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.4], 0.5, If[LessEqual[b, 2.5e+118], N[(1.0 / N[(N[(b + 1.0), $MachinePrecision] * N[(1.0 - a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.3999999999999999

   1. Initial program 97.8%

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

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -1.3999999999999999 < b < 2.49999999999999986e118

   1. Initial program 99.3%

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

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

      5. +-commutative N/A

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

      6. div-add N/A

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

      7. *-inverses N/A

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

      8. lower-+.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. div-exp N/A

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

      12. lower-exp.f64 N/A

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

      13. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt1-in N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-exp.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 89.6

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

   7. Applied rewrites 89.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 53.9%

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

   if 2.49999999999999986e118 < 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 93.8%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b + 1, 1 - a, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.2% 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 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. +-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 78.7

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

5. Applied rewrites 78.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 35.9%

   \[\leadsto 0.5 \]
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 2024312 
(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))))