Quotient of sum of exps

Percentage Accurate: 99.0% → 99.0%

Time: 8.5s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

  1. a = 1.2613206104298157e+23
  2. b = -2.110762997923037e+247

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

Alternative 2: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ (exp a) (+ 1.0 1.0)) (/ 1.0 (+ (exp b) 1.0))))

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = exp(a) / (1.0d0 + 1.0d0)
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = Math.exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = math.exp(a) / (1.0 + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 100.0%

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

   if 0.0 < (exp.f64 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 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.25 \cdot 10^{+79}:\\ \;\;\;\;\frac{a + 1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.25e+79)
   (/
    (+ a 1.0)
    (+ 1.0 (fma a (fma a (fma a 0.16666666666666666 0.5) 1.0) 1.0)))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.25e+79) {
		tmp = (a + 1.0) / (1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0));
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.25e+79)
		tmp = Float64(Float64(a + 1.0) / Float64(1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0)));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.25e+79], N[(N[(a + 1.0), $MachinePrecision] / N[(1.0 + N[(a * N[(a * N[(a * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.25000000000000003e79

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 100.0%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 2.1

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

   11. Applied rewrites 2.1%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{a + 1}{\color{blue}{\left(a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) + 1\right)} + 1} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{a + 1}{\color{blue}{\mathsf{fma}\left(a, 1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right), 1\right)} + 1} \]

       3. +-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right) + 1}, 1\right) + 1} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + \frac{1}{6} \cdot a, 1\right)}, 1\right) + 1} \]

       5. +-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\frac{1}{6} \cdot a + \frac{1}{2}}, 1\right), 1\right) + 1} \]

       6. *-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{a \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) + 1} \]

       7. lower-fma.f64 83.5

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

   14. Applied rewrites 83.5%

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

   if -5.25000000000000003e79 < 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 93.8

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

   5. Applied rewrites 93.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.25 \cdot 10^{+79}:\\ \;\;\;\;\frac{a + 1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;b \leq -14.5:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{a + 1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot a\right)}{1 + 1}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(t\_0, b \cdot \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b\right), -4\right)}{\mathsf{fma}\left(b, t\_0, -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b (fma b 0.16666666666666666 0.5) 1.0)))
   (if (<= b -14.5)
     (+ (exp b) 1.0)
     (if (<= b 1.6e+30)
       (/
        (+ a 1.0)
        (+ 1.0 (fma a (fma a (fma a 0.16666666666666666 0.5) 1.0) 1.0)))
       (if (<= b 4.4e+51)
         (/ (* 0.5 (* a a)) (+ 1.0 1.0))
         (if (<= b 1.02e+103)
           (/
            1.0
            (/
             (fma
              t_0
              (* b (fma b (* b (fma b 0.16666666666666666 0.5)) 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 <= -14.5) {
		tmp = exp(b) + 1.0;
	} else if (b <= 1.6e+30) {
		tmp = (a + 1.0) / (1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0));
	} else if (b <= 4.4e+51) {
		tmp = (0.5 * (a * a)) / (1.0 + 1.0);
	} else if (b <= 1.02e+103) {
		tmp = 1.0 / (fma(t_0, (b * fma(b, (b * fma(b, 0.16666666666666666, 0.5)), 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 <= -14.5)
		tmp = Float64(exp(b) + 1.0);
	elseif (b <= 1.6e+30)
		tmp = Float64(Float64(a + 1.0) / Float64(1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0)));
	elseif (b <= 4.4e+51)
		tmp = Float64(Float64(0.5 * Float64(a * a)) / Float64(1.0 + 1.0));
	elseif (b <= 1.02e+103)
		tmp = Float64(1.0 / Float64(fma(t_0, Float64(b * fma(b, Float64(b * fma(b, 0.16666666666666666, 0.5)), 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, -14.5], N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[b, 1.6e+30], N[(N[(a + 1.0), $MachinePrecision] / N[(1.0 + N[(a * N[(a * N[(a * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e+51], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.02e+103], N[(1.0 / N[(N[(t$95$0 * N[(b * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $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 5 regimes

2. if b < -14.5

   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%

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

   7. Applied rewrites 100.0%

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

   if -14.5 < b < 1.59999999999999986e30

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 97.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 96.6%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 61.3

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

   11. Applied rewrites 61.3%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{a + 1}{\color{blue}{\left(a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) + 1\right)} + 1} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{a + 1}{\color{blue}{\mathsf{fma}\left(a, 1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right), 1\right)} + 1} \]

       3. +-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right) + 1}, 1\right) + 1} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + \frac{1}{6} \cdot a, 1\right)}, 1\right) + 1} \]

       5. +-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\frac{1}{6} \cdot a + \frac{1}{2}}, 1\right), 1\right) + 1} \]

       6. *-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{a \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) + 1} \]

       7. lower-fma.f64 82.8

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

   14. Applied rewrites 82.8%

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

   if 1.59999999999999986e30 < b < 4.39999999999999984e51

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 3.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{a \cdot \left(1 + \frac{1}{2} \cdot a\right) + 1}}{1 + 1} \]

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 3.1

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

   11. Applied rewrites 3.1%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 83.9%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\left(a \cdot a\right)}}{1 + 1} \]

   if 4.39999999999999984e51 < b < 1.01999999999999991e103

   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 6.8%

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

   10. Applied rewrites 100.0%

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

   if 1.01999999999999991e103 < b

   1. Initial program 98.1%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. 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 5 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -14.5:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{a + 1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot a\right)}{1 + 1}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), b \cdot \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b\right), -4\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;b \leq 1.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{a + 1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot a\right)}{1 + 1}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(t\_0, b \cdot \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b\right), -4\right)}{\mathsf{fma}\left(b, t\_0, -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b (fma b 0.16666666666666666 0.5) 1.0)))
   (if (<= b 1.6e+30)
     (/
      (+ a 1.0)
      (+ 1.0 (fma a (fma a (fma a 0.16666666666666666 0.5) 1.0) 1.0)))
     (if (<= b 4.4e+51)
       (/ (* 0.5 (* a a)) (+ 1.0 1.0))
       (if (<= b 1.02e+103)
         (/
          1.0
          (/
           (fma t_0 (* b (fma b (* b (fma b 0.16666666666666666 0.5)) 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 <= 1.6e+30) {
		tmp = (a + 1.0) / (1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0));
	} else if (b <= 4.4e+51) {
		tmp = (0.5 * (a * a)) / (1.0 + 1.0);
	} else if (b <= 1.02e+103) {
		tmp = 1.0 / (fma(t_0, (b * fma(b, (b * fma(b, 0.16666666666666666, 0.5)), 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 <= 1.6e+30)
		tmp = Float64(Float64(a + 1.0) / Float64(1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0)));
	elseif (b <= 4.4e+51)
		tmp = Float64(Float64(0.5 * Float64(a * a)) / Float64(1.0 + 1.0));
	elseif (b <= 1.02e+103)
		tmp = Float64(1.0 / Float64(fma(t_0, Float64(b * fma(b, Float64(b * fma(b, 0.16666666666666666, 0.5)), 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, 1.6e+30], N[(N[(a + 1.0), $MachinePrecision] / N[(1.0 + N[(a * N[(a * N[(a * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e+51], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.02e+103], N[(1.0 / N[(N[(t$95$0 * N[(b * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $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 4 regimes

2. if b < 1.59999999999999986e30

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 74.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.5%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 48.7

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

   11. Applied rewrites 48.7%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{a + 1}{\color{blue}{\left(a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) + 1\right)} + 1} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{a + 1}{\color{blue}{\mathsf{fma}\left(a, 1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right), 1\right)} + 1} \]

       3. +-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right) + 1}, 1\right) + 1} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + \frac{1}{6} \cdot a, 1\right)}, 1\right) + 1} \]

       5. +-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\frac{1}{6} \cdot a + \frac{1}{2}}, 1\right), 1\right) + 1} \]

       6. *-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{a \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) + 1} \]

       7. lower-fma.f64 63.8

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

   14. Applied rewrites 63.8%

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

   if 1.59999999999999986e30 < b < 4.39999999999999984e51

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 3.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{a \cdot \left(1 + \frac{1}{2} \cdot a\right) + 1}}{1 + 1} \]

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 3.1

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

   11. Applied rewrites 3.1%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 83.9%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\left(a \cdot a\right)}}{1 + 1} \]

   if 4.39999999999999984e51 < b < 1.01999999999999991e103

   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 6.8%

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

   10. Applied rewrites 100.0%

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

   if 1.01999999999999991e103 < b

   1. Initial program 98.1%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. 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 4 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{a + 1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot a\right)}{1 + 1}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), b \cdot \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b\right), -4\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), -2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{a + 1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+77}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot a\right)}{1 + 1}\\ \mathbf{elif}\;b \leq 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b \cdot \left(b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)\right), -1\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), -1\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.6e+30)
   (/
    (+ a 1.0)
    (+ 1.0 (fma a (fma a (fma a 0.16666666666666666 0.5) 1.0) 1.0)))
   (if (<= b 2.9e+77)
     (/ (* 0.5 (* a a)) (+ 1.0 1.0))
     (if (<= b 1e+154)
       (/
        1.0
        (fma
         b
         (/
          (fma
           (fma b 0.16666666666666666 0.5)
           (* b (* b (fma b 0.16666666666666666 0.5)))
           -1.0)
          (fma b (fma b 0.16666666666666666 0.5) -1.0))
         2.0))
       (/ 1.0 (fma b (fma b 0.5 1.0) 2.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.6e+30) {
		tmp = (a + 1.0) / (1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0));
	} else if (b <= 2.9e+77) {
		tmp = (0.5 * (a * a)) / (1.0 + 1.0);
	} else if (b <= 1e+154) {
		tmp = 1.0 / fma(b, (fma(fma(b, 0.16666666666666666, 0.5), (b * (b * fma(b, 0.16666666666666666, 0.5))), -1.0) / fma(b, fma(b, 0.16666666666666666, 0.5), -1.0)), 2.0);
	} else {
		tmp = 1.0 / fma(b, fma(b, 0.5, 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.6e+30)
		tmp = Float64(Float64(a + 1.0) / Float64(1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0)));
	elseif (b <= 2.9e+77)
		tmp = Float64(Float64(0.5 * Float64(a * a)) / Float64(1.0 + 1.0));
	elseif (b <= 1e+154)
		tmp = Float64(1.0 / fma(b, Float64(fma(fma(b, 0.16666666666666666, 0.5), Float64(b * Float64(b * fma(b, 0.16666666666666666, 0.5))), -1.0) / fma(b, fma(b, 0.16666666666666666, 0.5), -1.0)), 2.0));
	else
		tmp = Float64(1.0 / fma(b, fma(b, 0.5, 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.6e+30], N[(N[(a + 1.0), $MachinePrecision] / N[(1.0 + N[(a * N[(a * N[(a * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.9e+77], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+154], N[(1.0 / N[(b * N[(N[(N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < 1.59999999999999986e30

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 74.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.5%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 48.7

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

   11. Applied rewrites 48.7%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{a + 1}{\color{blue}{\left(a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) + 1\right)} + 1} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{a + 1}{\color{blue}{\mathsf{fma}\left(a, 1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right), 1\right)} + 1} \]

       3. +-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right) + 1}, 1\right) + 1} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + \frac{1}{6} \cdot a, 1\right)}, 1\right) + 1} \]

       5. +-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\frac{1}{6} \cdot a + \frac{1}{2}}, 1\right), 1\right) + 1} \]

       6. *-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{a \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) + 1} \]

       7. lower-fma.f64 63.8

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

   14. Applied rewrites 63.8%

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

   if 1.59999999999999986e30 < b < 2.9000000000000002e77

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 10.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 10.0%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{a \cdot \left(1 + \frac{1}{2} \cdot a\right) + 1}}{1 + 1} \]

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 3.0

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

   11. Applied rewrites 3.0%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 52.5%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\left(a \cdot a\right)}}{1 + 1} \]

   if 2.9000000000000002e77 < b < 1.00000000000000004e154

   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 78.2%

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

   10. Applied rewrites 100.0%

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

   if 1.00000000000000004e154 < b

   1. Initial program 97.2%

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

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{a + 1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+77}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot a\right)}{1 + 1}\\ \mathbf{elif}\;b \leq 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b \cdot \left(b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)\right), -1\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), -1\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{a + 1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+95}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot a\right)}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.6e+30)
   (/
    (+ a 1.0)
    (+ 1.0 (fma a (fma a (fma a 0.16666666666666666 0.5) 1.0) 1.0)))
   (if (<= b 4.4e+95)
     (/ (* 0.5 (* a a)) (+ 1.0 1.0))
     (/ 1.0 (* b (* b (fma b 0.16666666666666666 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.6e+30) {
		tmp = (a + 1.0) / (1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0));
	} else if (b <= 4.4e+95) {
		tmp = (0.5 * (a * a)) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (b * (b * fma(b, 0.16666666666666666, 0.5)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.6e+30)
		tmp = Float64(Float64(a + 1.0) / Float64(1.0 + fma(a, fma(a, fma(a, 0.16666666666666666, 0.5), 1.0), 1.0)));
	elseif (b <= 4.4e+95)
		tmp = Float64(Float64(0.5 * Float64(a * a)) / Float64(1.0 + 1.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * fma(b, 0.16666666666666666, 0.5))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.6e+30], N[(N[(a + 1.0), $MachinePrecision] / N[(1.0 + N[(a * N[(a * N[(a * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e+95], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.59999999999999986e30

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 74.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.5%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 48.7

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

   11. Applied rewrites 48.7%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{a + 1}{\color{blue}{\left(a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) + 1\right)} + 1} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{a + 1}{\color{blue}{\mathsf{fma}\left(a, 1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right), 1\right)} + 1} \]

       3. +-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right) + 1}, 1\right) + 1} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + \frac{1}{6} \cdot a, 1\right)}, 1\right) + 1} \]

       5. +-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{\frac{1}{6} \cdot a + \frac{1}{2}}, 1\right), 1\right) + 1} \]

       6. *-commutative N/A

          \[\leadsto \frac{a + 1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \color{blue}{a \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) + 1} \]

       7. lower-fma.f64 63.8

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

   14. Applied rewrites 63.8%

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

   if 1.59999999999999986e30 < b < 4.3999999999999998e95

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 15.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 15.2%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{a \cdot \left(1 + \frac{1}{2} \cdot a\right) + 1}}{1 + 1} \]

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 3.0

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

   11. Applied rewrites 3.0%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 49.1%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\left(a \cdot a\right)}}{1 + 1} \]

   if 4.3999999999999998e95 < b

   1. Initial program 98.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 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 95.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 95.1%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{a + 1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.16666666666666666, 0.5\right), 1\right), 1\right)}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+95}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot a\right)}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.1% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{a + 1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right)}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+95}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot a\right)}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.6e+30)
   (/ (+ a 1.0) (+ 1.0 (fma a (fma a 0.5 1.0) 1.0)))
   (if (<= b 4.4e+95)
     (/ (* 0.5 (* a a)) (+ 1.0 1.0))
     (/ 1.0 (* b (* b (fma b 0.16666666666666666 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.6e+30) {
		tmp = (a + 1.0) / (1.0 + fma(a, fma(a, 0.5, 1.0), 1.0));
	} else if (b <= 4.4e+95) {
		tmp = (0.5 * (a * a)) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (b * (b * fma(b, 0.16666666666666666, 0.5)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.6e+30)
		tmp = Float64(Float64(a + 1.0) / Float64(1.0 + fma(a, fma(a, 0.5, 1.0), 1.0)));
	elseif (b <= 4.4e+95)
		tmp = Float64(Float64(0.5 * Float64(a * a)) / Float64(1.0 + 1.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * fma(b, 0.16666666666666666, 0.5))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.6e+30], N[(N[(a + 1.0), $MachinePrecision] / N[(1.0 + N[(a * N[(a * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e+95], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.59999999999999986e30

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 74.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.5%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 48.7

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

   11. Applied rewrites 48.7%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{a + 1}{\color{blue}{\left(a \cdot \left(1 + \frac{1}{2} \cdot a\right) + 1\right)} + 1} \]

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 60.6

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

   14. Applied rewrites 60.6%

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

   if 1.59999999999999986e30 < b < 4.3999999999999998e95

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 15.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 15.2%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{a \cdot \left(1 + \frac{1}{2} \cdot a\right) + 1}}{1 + 1} \]

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 3.0

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

   11. Applied rewrites 3.0%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 49.1%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\left(a \cdot a\right)}}{1 + 1} \]

   if 4.3999999999999998e95 < b

   1. Initial program 98.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 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 95.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 95.1%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{a + 1}{1 + \mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, 1\right), 1\right)}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+95}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot a\right)}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.5% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+95}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot a\right)}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.95e+15)
   (fma a 0.25 0.5)
   (if (<= b 4.4e+95)
     (/ (* 0.5 (* a a)) (+ 1.0 1.0))
     (/ 1.0 (* b (* b (fma b 0.16666666666666666 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.95e+15) {
		tmp = fma(a, 0.25, 0.5);
	} else if (b <= 4.4e+95) {
		tmp = (0.5 * (a * a)) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (b * (b * fma(b, 0.16666666666666666, 0.5)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.95e+15)
		tmp = fma(a, 0.25, 0.5);
	elseif (b <= 4.4e+95)
		tmp = Float64(Float64(0.5 * Float64(a * a)) / Float64(1.0 + 1.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * fma(b, 0.16666666666666666, 0.5))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.95e+15], N[(a * 0.25 + 0.5), $MachinePrecision], If[LessEqual[b, 4.4e+95], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.95e15

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(b \cdot e^{a}\right)}{{\left(1 + e^{a}\right)}^{2}}} + \frac{e^{a}}{1 + e^{a}} \]

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 46.4%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(b, -0.125, 0.5\right) + \mathsf{fma}\left(b, 0.125, -0.25\right)}, \mathsf{fma}\left(b, -0.25, 0.5\right)\right) \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \frac{1}{4} \cdot \color{blue}{a} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.5%

       \[\leadsto \mathsf{fma}\left(a, 0.25, 0.5\right) \]

   if 1.95e15 < b < 4.3999999999999998e95

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 22.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 22.5%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{a \cdot \left(1 + \frac{1}{2} \cdot a\right) + 1}}{1 + 1} \]

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 2.9

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

   11. Applied rewrites 2.9%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 40.0%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\left(a \cdot a\right)}}{1 + 1} \]

   if 4.3999999999999998e95 < b

   1. Initial program 98.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 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 95.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 95.1%

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

Alternative 10: 60.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+95}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot a\right)}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(b \cdot \left(b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.95e+15)
   (fma a 0.25 0.5)
   (if (<= b 4.4e+95)
     (/ (* 0.5 (* a a)) (+ 1.0 1.0))
     (/ 1.0 (* b (* b (* b 0.16666666666666666)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.95e+15) {
		tmp = fma(a, 0.25, 0.5);
	} else if (b <= 4.4e+95) {
		tmp = (0.5 * (a * a)) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (b * (b * (b * 0.16666666666666666)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.95e+15)
		tmp = fma(a, 0.25, 0.5);
	elseif (b <= 4.4e+95)
		tmp = Float64(Float64(0.5 * Float64(a * a)) / Float64(1.0 + 1.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(b * Float64(b * 0.16666666666666666))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.95e+15], N[(a * 0.25 + 0.5), $MachinePrecision], If[LessEqual[b, 4.4e+95], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(1.0 + 1.0), $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 < 1.95e15

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(b \cdot e^{a}\right)}{{\left(1 + e^{a}\right)}^{2}}} + \frac{e^{a}}{1 + e^{a}} \]

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 46.4%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(b, -0.125, 0.5\right) + \mathsf{fma}\left(b, 0.125, -0.25\right)}, \mathsf{fma}\left(b, -0.25, 0.5\right)\right) \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \frac{1}{4} \cdot \color{blue}{a} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.5%

       \[\leadsto \mathsf{fma}\left(a, 0.25, 0.5\right) \]

   if 1.95e15 < b < 4.3999999999999998e95

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 22.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 22.5%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{a \cdot \left(1 + \frac{1}{2} \cdot a\right) + 1}}{1 + 1} \]

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 2.9

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

   11. Applied rewrites 2.9%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 40.0%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\left(a \cdot a\right)}}{1 + 1} \]

   if 4.3999999999999998e95 < b

   1. Initial program 98.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 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 95.1%

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

   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 95.1%

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

Alternative 11: 57.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.5 \cdot \left(a \cdot a\right)}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.95e+15)
   (fma a 0.25 0.5)
   (if (<= b 1.9e+154)
     (/ (* 0.5 (* a a)) (+ 1.0 1.0))
     (/ 1.0 (fma b (fma b 0.5 1.0) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.95e+15) {
		tmp = fma(a, 0.25, 0.5);
	} else if (b <= 1.9e+154) {
		tmp = (0.5 * (a * a)) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / fma(b, fma(b, 0.5, 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.95e+15)
		tmp = fma(a, 0.25, 0.5);
	elseif (b <= 1.9e+154)
		tmp = Float64(Float64(0.5 * Float64(a * a)) / Float64(1.0 + 1.0));
	else
		tmp = Float64(1.0 / fma(b, fma(b, 0.5, 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.95e+15], N[(a * 0.25 + 0.5), $MachinePrecision], If[LessEqual[b, 1.9e+154], N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.95e15

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(b \cdot e^{a}\right)}{{\left(1 + e^{a}\right)}^{2}}} + \frac{e^{a}}{1 + e^{a}} \]

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 46.4%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(b, -0.125, 0.5\right) + \mathsf{fma}\left(b, 0.125, -0.25\right)}, \mathsf{fma}\left(b, -0.25, 0.5\right)\right) \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \frac{1}{4} \cdot \color{blue}{a} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.5%

       \[\leadsto \mathsf{fma}\left(a, 0.25, 0.5\right) \]

   if 1.95e15 < b < 1.8999999999999999e154

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 25.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 25.5%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{a \cdot \left(1 + \frac{1}{2} \cdot a\right) + 1}}{1 + 1} \]

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 2.9

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

   11. Applied rewrites 2.9%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 35.1%

       \[\leadsto \frac{0.5 \cdot \color{blue}{\left(a \cdot a\right)}}{1 + 1} \]

   if 1.8999999999999999e154 < b

   1. Initial program 97.2%

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

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

Alternative 12: 53.8% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -2.6e-190) (fma a 0.25 0.5) (/ 1.0 (fma b (fma b 0.5 1.0) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.6e-190) {
		tmp = fma(a, 0.25, 0.5);
	} else {
		tmp = 1.0 / fma(b, fma(b, 0.5, 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -2.6e-190)
		tmp = fma(a, 0.25, 0.5);
	else
		tmp = Float64(1.0 / fma(b, fma(b, 0.5, 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -2.6e-190], N[(a * 0.25 + 0.5), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.5999999999999998e-190

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(b \cdot e^{a}\right)}{{\left(1 + e^{a}\right)}^{2}}} + \frac{e^{a}}{1 + e^{a}} \]

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 30.5%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(b, -0.125, 0.5\right) + \mathsf{fma}\left(b, 0.125, -0.25\right)}, \mathsf{fma}\left(b, -0.25, 0.5\right)\right) \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \frac{1}{4} \cdot \color{blue}{a} \]
   10. Step-by-step derivation

   11. Applied rewrites 38.8%

       \[\leadsto \mathsf{fma}\left(a, 0.25, 0.5\right) \]

   if -2.5999999999999998e-190 < b

   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 80.2

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

   5. Applied rewrites 80.2%

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

   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 57.1%

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

Alternative 13: 39.4% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, 0.25, 0.5\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma a 0.25 0.5))

C

double code(double a, double b) {
	return fma(a, 0.25, 0.5);
}

Julia

function code(a, b)
	return fma(a, 0.25, 0.5)
end

Wolfram

code[a_, b_] := N[(a * 0.25 + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(b \cdot e^{a}\right)}{{\left(1 + e^{a}\right)}^{2}}} + \frac{e^{a}}{1 + e^{a}} \]

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. times-frac N/A

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

   5. distribute-lft1-in N/A

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

   6. lower-*.f64 N/A

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

5. Applied rewrites 57.9%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 33.5%

   \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(b, -0.125, 0.5\right) + \mathsf{fma}\left(b, 0.125, -0.25\right)}, \mathsf{fma}\left(b, -0.25, 0.5\right)\right) \]

9. Taylor expanded in b around 0

   \[\leadsto \frac{1}{2} + \frac{1}{4} \cdot \color{blue}{a} \]
10. Step-by-step derivation

11. Applied rewrites 36.5%

    \[\leadsto \mathsf{fma}\left(a, 0.25, 0.5\right) \]
12. Add Preprocessing

Alternative 14: 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 99.6%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 82.8

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

5. Applied rewrites 82.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 36.2%

   \[\leadsto 0.5 \]
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 2024232 
(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))))