Quotient of sum of exps

Percentage Accurate: 98.9% → 98.9%

Time: 9.1s

Alternatives: 15

Speedup: 1.0×

• could not determine a ground truth

  1. a = 2.8281361897846115e+84
  2. b = 1.3405517681964826e+123

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: 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]

Derivation

1. Initial program 99.2%

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

Alternative 2: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6000000:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6000000.0) (* (exp a) 0.5) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6000000.0) {
		tmp = exp(a) * 0.5;
	} 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 (a <= (-6000000.0d0)) then
        tmp = exp(a) * 0.5d0
    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 (a <= -6000000.0) {
		tmp = Math.exp(a) * 0.5;
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6000000.0:
		tmp = math.exp(a) * 0.5
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6000000.0)
		tmp = Float64(exp(a) * 0.5);
	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 (a <= -6000000.0)
		tmp = exp(a) * 0.5;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6000000.0], N[(N[Exp[a], $MachinePrecision] * 0.5), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6e6

   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. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 100.0%

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

      1. lift-exp.f64 N/A

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

      2. div-inv N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      3. metadata-eval N/A

         \[\leadsto e^{a} \cdot \color{blue}{\frac{1}{2}} \]

      4. lower-*.f64 100.0

         \[\leadsto \color{blue}{e^{a} \cdot 0.5} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{e^{a} \cdot 0.5} \]

   if -6e6 < a

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6000000:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(0.5, b \cdot b, b\right), -4\right)\\ \mathbf{if}\;b \leq -14.6:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+50}:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{t\_0}{t\_0 \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), -2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma b 0.5 1.0) (* b (fma 0.5 (* b b) b)) -4.0)))
   (if (<= b -14.6)
     (+ (exp b) 1.0)
     (if (<= b 5.8e+50)
       (* (exp a) 0.5)
       (if (<= b 1.6e+77)
         (/ t_0 (* t_0 (fma b (fma b 0.5 1.0) 2.0)))
         (if (<= b 2e+154)
           (/ (fma b (fma b 0.5 1.0) -2.0) t_0)
           (/ 2.0 (* b b))))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(b, 0.5, 1.0), (b * fma(0.5, (b * b), b)), -4.0);
	double tmp;
	if (b <= -14.6) {
		tmp = exp(b) + 1.0;
	} else if (b <= 5.8e+50) {
		tmp = exp(a) * 0.5;
	} else if (b <= 1.6e+77) {
		tmp = t_0 / (t_0 * fma(b, fma(b, 0.5, 1.0), 2.0));
	} else if (b <= 2e+154) {
		tmp = fma(b, fma(b, 0.5, 1.0), -2.0) / t_0;
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(b, 0.5, 1.0), Float64(b * fma(0.5, Float64(b * b), b)), -4.0)
	tmp = 0.0
	if (b <= -14.6)
		tmp = Float64(exp(b) + 1.0);
	elseif (b <= 5.8e+50)
		tmp = Float64(exp(a) * 0.5);
	elseif (b <= 1.6e+77)
		tmp = Float64(t_0 / Float64(t_0 * fma(b, fma(b, 0.5, 1.0), 2.0)));
	elseif (b <= 2e+154)
		tmp = Float64(fma(b, fma(b, 0.5, 1.0), -2.0) / t_0);
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * 0.5 + 1.0), $MachinePrecision] * N[(b * N[(0.5 * N[(b * b), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + -4.0), $MachinePrecision]}, If[LessEqual[b, -14.6], N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[b, 5.8e+50], N[(N[Exp[a], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[b, 1.6e+77], N[(t$95$0 / N[(t$95$0 * N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+154], N[(N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + -2.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -14.5999999999999996

   1. Initial program 96.4%

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

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

   5. Simplified 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -14.5999999999999996 < b < 5.8e50

   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. Simplified 96.9%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 95.1%

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

      1. lift-exp.f64 N/A

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

      2. div-inv N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      3. metadata-eval N/A

         \[\leadsto e^{a} \cdot \color{blue}{\frac{1}{2}} \]

      4. lower-*.f64 95.1

         \[\leadsto \color{blue}{e^{a} \cdot 0.5} \]

   10. Applied egg-rr 95.1%

       \[\leadsto \color{blue}{e^{a} \cdot 0.5} \]

   if 5.8e50 < b < 1.6000000000000001e77

   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. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 4.3

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

   8. Simplified 4.3%

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

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. associate-/r/ N/A

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

      4. flip-- N/A

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

      5. lift-fma.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

   10. Applied egg-rr 91.3%

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

   if 1.6000000000000001e77 < b < 2.00000000000000007e154

   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. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 6.1

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

   8. Simplified 6.1%

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

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   10. Applied egg-rr 100.0%

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

   if 2.00000000000000007e154 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

       3. lower-*.f64 100.0

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{2}{b \cdot b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -14.6:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+50}:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(0.5, b \cdot b, b\right), -4\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(0.5, b \cdot b, b\right), -4\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(0.5, b \cdot b, b\right), -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(0.5, b \cdot b, b\right), -4\right)\\ \mathbf{if}\;b \leq -1.6:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{t\_0}{t\_0 \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), -2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma b 0.5 1.0) (* b (fma 0.5 (* b b) b)) -4.0)))
   (if (<= b -1.6)
     (+ (exp b) 1.0)
     (if (<= b 1.6e+77)
       (/ t_0 (* t_0 (fma b (fma b 0.5 1.0) 2.0)))
       (if (<= b 2e+154)
         (/ (fma b (fma b 0.5 1.0) -2.0) t_0)
         (/ 2.0 (* b b)))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(b, 0.5, 1.0), (b * fma(0.5, (b * b), b)), -4.0);
	double tmp;
	if (b <= -1.6) {
		tmp = exp(b) + 1.0;
	} else if (b <= 1.6e+77) {
		tmp = t_0 / (t_0 * fma(b, fma(b, 0.5, 1.0), 2.0));
	} else if (b <= 2e+154) {
		tmp = fma(b, fma(b, 0.5, 1.0), -2.0) / t_0;
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(b, 0.5, 1.0), Float64(b * fma(0.5, Float64(b * b), b)), -4.0)
	tmp = 0.0
	if (b <= -1.6)
		tmp = Float64(exp(b) + 1.0);
	elseif (b <= 1.6e+77)
		tmp = Float64(t_0 / Float64(t_0 * fma(b, fma(b, 0.5, 1.0), 2.0)));
	elseif (b <= 2e+154)
		tmp = Float64(fma(b, fma(b, 0.5, 1.0), -2.0) / t_0);
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * 0.5 + 1.0), $MachinePrecision] * N[(b * N[(0.5 * N[(b * b), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + -4.0), $MachinePrecision]}, If[LessEqual[b, -1.6], N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[b, 1.6e+77], N[(t$95$0 / N[(t$95$0 * N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+154], N[(N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + -2.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.6000000000000001

   1. Initial program 96.4%

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

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

   5. Simplified 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -1.6000000000000001 < b < 1.6000000000000001e77

   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 67.7

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

   5. Simplified 67.7%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 56.5

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

   8. Simplified 56.5%

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

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. associate-/r/ N/A

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

      4. flip-- N/A

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

      5. lift-fma.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

   10. Applied egg-rr 62.7%

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

   if 1.6000000000000001e77 < b < 2.00000000000000007e154

   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. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 6.1

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

   8. Simplified 6.1%

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

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   10. Applied egg-rr 100.0%

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

   if 2.00000000000000007e154 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

       3. lower-*.f64 100.0

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{2}{b \cdot b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(0.5, b \cdot b, b\right), -4\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(0.5, b \cdot b, b\right), -4\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(0.5, b \cdot b, b\right), -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(0.5, b \cdot b, b\right), -4\right)\\ \mathbf{if}\;b \leq 9.6 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a \cdot 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{t\_0}{t\_0 \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), -2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma b 0.5 1.0) (* b (fma 0.5 (* b b) b)) -4.0)))
   (if (<= b 9.6e-65)
     (fma
      a
      (fma
       (* a a)
       (fma a (* a 0.0020833333333333333) -0.020833333333333332)
       0.25)
      0.5)
     (if (<= b 1.6e+77)
       (/ t_0 (* t_0 (fma b (fma b 0.5 1.0) 2.0)))
       (if (<= b 2e+154)
         (/ (fma b (fma b 0.5 1.0) -2.0) t_0)
         (/ 2.0 (* b b)))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(b, 0.5, 1.0), (b * fma(0.5, (b * b), b)), -4.0);
	double tmp;
	if (b <= 9.6e-65) {
		tmp = fma(a, fma((a * a), fma(a, (a * 0.0020833333333333333), -0.020833333333333332), 0.25), 0.5);
	} else if (b <= 1.6e+77) {
		tmp = t_0 / (t_0 * fma(b, fma(b, 0.5, 1.0), 2.0));
	} else if (b <= 2e+154) {
		tmp = fma(b, fma(b, 0.5, 1.0), -2.0) / t_0;
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(b, 0.5, 1.0), Float64(b * fma(0.5, Float64(b * b), b)), -4.0)
	tmp = 0.0
	if (b <= 9.6e-65)
		tmp = fma(a, fma(Float64(a * a), fma(a, Float64(a * 0.0020833333333333333), -0.020833333333333332), 0.25), 0.5);
	elseif (b <= 1.6e+77)
		tmp = Float64(t_0 / Float64(t_0 * fma(b, fma(b, 0.5, 1.0), 2.0)));
	elseif (b <= 2e+154)
		tmp = Float64(fma(b, fma(b, 0.5, 1.0), -2.0) / t_0);
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(b * 0.5 + 1.0), $MachinePrecision] * N[(b * N[(0.5 * N[(b * b), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + -4.0), $MachinePrecision]}, If[LessEqual[b, 9.6e-65], N[(a * N[(N[(a * a), $MachinePrecision] * N[(a * N[(a * 0.0020833333333333333), $MachinePrecision] + -0.020833333333333332), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision], If[LessEqual[b, 1.6e+77], N[(t$95$0 / N[(t$95$0 * N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+154], N[(N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + -2.0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 9.6000000000000006e-65

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 74.3%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right), \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right) + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left({a}^{2}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\frac{1}{480} \cdot {a}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{{a}^{2} \cdot \frac{1}{480}} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{480} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot \left(a \cdot \frac{1}{480}\right)} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, a \cdot \left(a \cdot \frac{1}{480}\right) + \color{blue}{\frac{-1}{48}}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\mathsf{fma}\left(a, a \cdot \frac{1}{480}, \frac{-1}{48}\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      13. lower-*.f64 52.3

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, \color{blue}{a \cdot 0.0020833333333333333}, -0.020833333333333332\right), 0.25\right), 0.5\right) \]

   8. Simplified 52.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a \cdot 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)} \]

   if 9.6000000000000006e-65 < b < 1.6000000000000001e77

   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 72.8

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

   5. Simplified 72.8%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 18.8

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

   8. Simplified 18.8%

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

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. associate-/r/ N/A

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

      4. flip-- N/A

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

      5. lift-fma.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

   10. Applied egg-rr 48.7%

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

   if 1.6000000000000001e77 < b < 2.00000000000000007e154

   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. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 6.1

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

   8. Simplified 6.1%

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

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

   10. Applied egg-rr 100.0%

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

   if 2.00000000000000007e154 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

       3. lower-*.f64 100.0

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{2}{b \cdot b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.6 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a \cdot 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(0.5, b \cdot b, b\right), -4\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(0.5, b \cdot b, b\right), -4\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(0.5, b \cdot b, b\right), -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a \cdot 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{+94}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.1e-9)
   (fma
    a
    (fma
     (* a a)
     (fma a (* a 0.0020833333333333333) -0.020833333333333332)
     0.25)
    0.5)
   (if (<= b 1.65e+94)
     (* -0.020833333333333332 (* a (* a a)))
     (/ 1.0 (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.1e-9) {
		tmp = fma(a, fma((a * a), fma(a, (a * 0.0020833333333333333), -0.020833333333333332), 0.25), 0.5);
	} else if (b <= 1.65e+94) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = 1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.1e-9)
		tmp = fma(a, fma(Float64(a * a), fma(a, Float64(a * 0.0020833333333333333), -0.020833333333333332), 0.25), 0.5);
	elseif (b <= 1.65e+94)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	else
		tmp = Float64(1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.1e-9], N[(a * N[(N[(a * a), $MachinePrecision] * N[(a * N[(a * 0.0020833333333333333), $MachinePrecision] + -0.020833333333333332), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision], If[LessEqual[b, 1.65e+94], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.0999999999999999e-9

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 75.8%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right), \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right) + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left({a}^{2}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\frac{1}{480} \cdot {a}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{{a}^{2} \cdot \frac{1}{480}} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{480} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot \left(a \cdot \frac{1}{480}\right)} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, a \cdot \left(a \cdot \frac{1}{480}\right) + \color{blue}{\frac{-1}{48}}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\mathsf{fma}\left(a, a \cdot \frac{1}{480}, \frac{-1}{48}\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      13. lower-*.f64 52.0

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, \color{blue}{a \cdot 0.0020833333333333333}, -0.020833333333333332\right), 0.25\right), 0.5\right) \]

   8. Simplified 52.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a \cdot 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)} \]

   if 1.0999999999999999e-9 < b < 1.65e94

   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. Simplified 44.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

       6. lower-*.f64 40.1

          \[\leadsto -0.020833333333333332 \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   11. Simplified 40.1%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]

   if 1.65e94 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 97.8

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

   8. Simplified 97.8%

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

Alternative 7: 61.0% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a \cdot 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, b, -4\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.1e-9)
   (fma
    a
    (fma
     (* a a)
     (fma a (* a 0.0020833333333333333) -0.020833333333333332)
     0.25)
    0.5)
   (if (<= b 5.5e+102)
     (* -0.020833333333333332 (* a (* a a)))
     (/ (fma 2.0 b -4.0) (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.1e-9) {
		tmp = fma(a, fma((a * a), fma(a, (a * 0.0020833333333333333), -0.020833333333333332), 0.25), 0.5);
	} else if (b <= 5.5e+102) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = fma(2.0, b, -4.0) / (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.1e-9)
		tmp = fma(a, fma(Float64(a * a), fma(a, Float64(a * 0.0020833333333333333), -0.020833333333333332), 0.25), 0.5);
	elseif (b <= 5.5e+102)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	else
		tmp = Float64(fma(2.0, b, -4.0) / Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.1e-9], N[(a * N[(N[(a * a), $MachinePrecision] * N[(a * N[(a * 0.0020833333333333333), $MachinePrecision] + -0.020833333333333332), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision], If[LessEqual[b, 5.5e+102], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * b + -4.0), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.0999999999999999e-9

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 75.8%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right), \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{{a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right) + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left({a}^{2}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\color{blue}{a \cdot a}, \frac{1}{480} \cdot {a}^{2} - \frac{1}{48}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\frac{1}{480} \cdot {a}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{{a}^{2} \cdot \frac{1}{480}} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\left(a \cdot a\right)} \cdot \frac{1}{480} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{a \cdot \left(a \cdot \frac{1}{480}\right)} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), \frac{1}{4}\right), \frac{1}{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, a \cdot \left(a \cdot \frac{1}{480}\right) + \color{blue}{\frac{-1}{48}}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \color{blue}{\mathsf{fma}\left(a, a \cdot \frac{1}{480}, \frac{-1}{48}\right)}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      13. lower-*.f64 52.0

          \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, \color{blue}{a \cdot 0.0020833333333333333}, -0.020833333333333332\right), 0.25\right), 0.5\right) \]

   8. Simplified 52.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(a, a \cdot 0.0020833333333333333, -0.020833333333333332\right), 0.25\right), 0.5\right)} \]

   if 1.0999999999999999e-9 < b < 5.49999999999999981e102

   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. Simplified 42.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

       6. lower-*.f64 38.9

          \[\leadsto -0.020833333333333332 \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   11. Simplified 38.9%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]

   if 5.49999999999999981e102 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 83.7

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

   8. Simplified 83.7%

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

   9. Taylor expanded in b around inf

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

       1. unpow2 N/A

          \[\leadsto \frac{2 - 4 \cdot \frac{1}{b}}{\color{blue}{b \cdot b}} \]

       2. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{2 - 4 \cdot \frac{1}{b}}{b}}{b}} \]

       3. *-lft-identity N/A

          \[\leadsto \frac{\frac{2 - 4 \cdot \frac{1}{b}}{\color{blue}{1 \cdot b}}}{b} \]

       4. lft-mult-inverse N/A

          \[\leadsto \frac{\frac{2 - 4 \cdot \frac{1}{b}}{\color{blue}{\left(\frac{1}{b} \cdot b\right)} \cdot b}}{b} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\frac{2 - 4 \cdot \frac{1}{b}}{\color{blue}{\frac{1}{b} \cdot \left(b \cdot b\right)}}}{b} \]

       6. unpow2 N/A

          \[\leadsto \frac{\frac{2 - 4 \cdot \frac{1}{b}}{\frac{1}{b} \cdot \color{blue}{{b}^{2}}}}{b} \]

       7. associate-/r* N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{2 - 4 \cdot \frac{1}{b}}{\frac{1}{b}}}{{b}^{2}}}}{b} \]

       8. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{2 - 4 \cdot \frac{1}{b}}{\frac{1}{b}}}{b \cdot {b}^{2}}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{2 - 4 \cdot \frac{1}{b}}{\frac{1}{b}}}{b \cdot \color{blue}{\left(b \cdot b\right)}} \]

       10. cube-mult N/A

           \[\leadsto \frac{\frac{2 - 4 \cdot \frac{1}{b}}{\frac{1}{b}}}{\color{blue}{{b}^{3}}} \]

       11. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\frac{2 - 4 \cdot \frac{1}{b}}{\frac{1}{b}}}{{b}^{3}}} \]

       12. div-sub N/A

           \[\leadsto \frac{\color{blue}{\frac{2}{\frac{1}{b}} - \frac{4 \cdot \frac{1}{b}}{\frac{1}{b}}}}{{b}^{3}} \]

       13. associate-/l* N/A

           \[\leadsto \frac{\frac{2}{\frac{1}{b}} - \color{blue}{4 \cdot \frac{\frac{1}{b}}{\frac{1}{b}}}}{{b}^{3}} \]

       14. *-inverses N/A

           \[\leadsto \frac{\frac{2}{\frac{1}{b}} - 4 \cdot \color{blue}{1}}{{b}^{3}} \]

       15. metadata-eval N/A

           \[\leadsto \frac{\frac{2}{\frac{1}{b}} - \color{blue}{4}}{{b}^{3}} \]

       16. sub-neg N/A

           \[\leadsto \frac{\color{blue}{\frac{2}{\frac{1}{b}} + \left(\mathsf{neg}\left(4\right)\right)}}{{b}^{3}} \]

       17. associate-/r/ N/A

           \[\leadsto \frac{\color{blue}{\frac{2}{1} \cdot b} + \left(\mathsf{neg}\left(4\right)\right)}{{b}^{3}} \]

       18. metadata-eval N/A

           \[\leadsto \frac{\color{blue}{2} \cdot b + \left(\mathsf{neg}\left(4\right)\right)}{{b}^{3}} \]

       19. lower-fma.f64 N/A

           \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, b, \mathsf{neg}\left(4\right)\right)}}{{b}^{3}} \]

       20. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(2, b, \color{blue}{-4}\right)}{{b}^{3}} \]

       21. cube-mult N/A

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

   11. Simplified 100.0%

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

Alternative 8: 60.9% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, b, -4\right)}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.1e-9)
   (fma a (fma -0.020833333333333332 (* a a) 0.25) 0.5)
   (if (<= b 5.5e+102)
     (* -0.020833333333333332 (* a (* a a)))
     (/ (fma 2.0 b -4.0) (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.1e-9) {
		tmp = fma(a, fma(-0.020833333333333332, (a * a), 0.25), 0.5);
	} else if (b <= 5.5e+102) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = fma(2.0, b, -4.0) / (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.1e-9)
		tmp = fma(a, fma(-0.020833333333333332, Float64(a * a), 0.25), 0.5);
	elseif (b <= 5.5e+102)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	else
		tmp = Float64(fma(2.0, b, -4.0) / Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.1e-9], N[(a * N[(-0.020833333333333332 * N[(a * a), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision], If[LessEqual[b, 5.5e+102], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * b + -4.0), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.0999999999999999e-9

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 75.8%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 51.9

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

   8. Simplified 51.9%

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

   if 1.0999999999999999e-9 < b < 5.49999999999999981e102

   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. Simplified 42.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

       6. lower-*.f64 38.9

          \[\leadsto -0.020833333333333332 \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   11. Simplified 38.9%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]

   if 5.49999999999999981e102 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 83.7

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

   8. Simplified 83.7%

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

   9. Taylor expanded in b around inf

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

       1. unpow2 N/A

          \[\leadsto \frac{2 - 4 \cdot \frac{1}{b}}{\color{blue}{b \cdot b}} \]

       2. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{2 - 4 \cdot \frac{1}{b}}{b}}{b}} \]

       3. *-lft-identity N/A

          \[\leadsto \frac{\frac{2 - 4 \cdot \frac{1}{b}}{\color{blue}{1 \cdot b}}}{b} \]

       4. lft-mult-inverse N/A

          \[\leadsto \frac{\frac{2 - 4 \cdot \frac{1}{b}}{\color{blue}{\left(\frac{1}{b} \cdot b\right)} \cdot b}}{b} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\frac{2 - 4 \cdot \frac{1}{b}}{\color{blue}{\frac{1}{b} \cdot \left(b \cdot b\right)}}}{b} \]

       6. unpow2 N/A

          \[\leadsto \frac{\frac{2 - 4 \cdot \frac{1}{b}}{\frac{1}{b} \cdot \color{blue}{{b}^{2}}}}{b} \]

       7. associate-/r* N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{2 - 4 \cdot \frac{1}{b}}{\frac{1}{b}}}{{b}^{2}}}}{b} \]

       8. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{2 - 4 \cdot \frac{1}{b}}{\frac{1}{b}}}{b \cdot {b}^{2}}} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{2 - 4 \cdot \frac{1}{b}}{\frac{1}{b}}}{b \cdot \color{blue}{\left(b \cdot b\right)}} \]

       10. cube-mult N/A

           \[\leadsto \frac{\frac{2 - 4 \cdot \frac{1}{b}}{\frac{1}{b}}}{\color{blue}{{b}^{3}}} \]

       11. lower-/.f64 N/A

           \[\leadsto \color{blue}{\frac{\frac{2 - 4 \cdot \frac{1}{b}}{\frac{1}{b}}}{{b}^{3}}} \]

       12. div-sub N/A

           \[\leadsto \frac{\color{blue}{\frac{2}{\frac{1}{b}} - \frac{4 \cdot \frac{1}{b}}{\frac{1}{b}}}}{{b}^{3}} \]

       13. associate-/l* N/A

           \[\leadsto \frac{\frac{2}{\frac{1}{b}} - \color{blue}{4 \cdot \frac{\frac{1}{b}}{\frac{1}{b}}}}{{b}^{3}} \]

       14. *-inverses N/A

           \[\leadsto \frac{\frac{2}{\frac{1}{b}} - 4 \cdot \color{blue}{1}}{{b}^{3}} \]

       15. metadata-eval N/A

           \[\leadsto \frac{\frac{2}{\frac{1}{b}} - \color{blue}{4}}{{b}^{3}} \]

       16. sub-neg N/A

           \[\leadsto \frac{\color{blue}{\frac{2}{\frac{1}{b}} + \left(\mathsf{neg}\left(4\right)\right)}}{{b}^{3}} \]

       17. associate-/r/ N/A

           \[\leadsto \frac{\color{blue}{\frac{2}{1} \cdot b} + \left(\mathsf{neg}\left(4\right)\right)}{{b}^{3}} \]

       18. metadata-eval N/A

           \[\leadsto \frac{\color{blue}{2} \cdot b + \left(\mathsf{neg}\left(4\right)\right)}{{b}^{3}} \]

       19. lower-fma.f64 N/A

           \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(2, b, \mathsf{neg}\left(4\right)\right)}}{{b}^{3}} \]

       20. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(2, b, \color{blue}{-4}\right)}{{b}^{3}} \]

       21. cube-mult N/A

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

   11. Simplified 100.0%

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

Alternative 9: 57.1% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+34}:\\ \;\;\;\;b \cdot \left(\left(b \cdot b\right) \cdot 0.020833333333333332\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.5e+34)
   (* b (* (* b b) 0.020833333333333332))
   (/ 1.0 (fma b (fma b 0.5 1.0) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.5e+34) {
		tmp = b * ((b * b) * 0.020833333333333332);
	} 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 (a <= -8.5e+34)
		tmp = Float64(b * Float64(Float64(b * b) * 0.020833333333333332));
	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[a, -8.5e+34], N[(b * N[(N[(b * b), $MachinePrecision] * 0.020833333333333332), $MachinePrecision]), $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 a < -8.5000000000000003e34

   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 33.5

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

   5. Simplified 33.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{1}{48} \cdot {b}^{2} - \frac{1}{4}, \frac{1}{2}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{1}{48} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{{b}^{2} \cdot \frac{1}{48}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(b \cdot b\right)} \cdot \frac{1}{48} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{b \cdot \left(b \cdot \frac{1}{48}\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(b, b \cdot \left(b \cdot \frac{1}{48}\right) + \color{blue}{\frac{-1}{4}}, \frac{1}{2}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, b \cdot \frac{1}{48}, \frac{-1}{4}\right)}, \frac{1}{2}\right) \]

      9. lower-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in b around inf

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

       1. unpow3 N/A

          \[\leadsto \frac{1}{48} \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot b\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{48} \cdot \left(\color{blue}{{b}^{2}} \cdot b\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right) \cdot b} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2}\right)} \]

       6. *-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left({b}^{2} \cdot \frac{1}{48}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto b \cdot \color{blue}{\left({b}^{2} \cdot \frac{1}{48}\right)} \]

       8. unpow2 N/A

          \[\leadsto b \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \frac{1}{48}\right) \]

       9. lower-*.f64 45.0

          \[\leadsto b \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot 0.020833333333333332\right) \]

   11. Simplified 45.0%

       \[\leadsto \color{blue}{b \cdot \left(\left(b \cdot b\right) \cdot 0.020833333333333332\right)} \]

   if -8.5000000000000003e34 < a

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 97.1

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

   5. Simplified 97.1%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 58.6

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

   8. Simplified 58.6%

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

Alternative 10: 57.2% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), 0.5\right)\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+113}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.1e-9)
   (fma a (fma -0.020833333333333332 (* a a) 0.25) 0.5)
   (if (<= b 1.15e+113)
     (* -0.020833333333333332 (* a (* a a)))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.1e-9) {
		tmp = fma(a, fma(-0.020833333333333332, (a * a), 0.25), 0.5);
	} else if (b <= 1.15e+113) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.1e-9)
		tmp = fma(a, fma(-0.020833333333333332, Float64(a * a), 0.25), 0.5);
	elseif (b <= 1.15e+113)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.1e-9], N[(a * N[(-0.020833333333333332 * N[(a * a), $MachinePrecision] + 0.25), $MachinePrecision] + 0.5), $MachinePrecision], If[LessEqual[b, 1.15e+113], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.0999999999999999e-9

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 75.8%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 51.9

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

   8. Simplified 51.9%

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

   if 1.0999999999999999e-9 < b < 1.14999999999999998e113

   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. Simplified 42.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

       6. lower-*.f64 38.9

          \[\leadsto -0.020833333333333332 \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   11. Simplified 38.9%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]

   if 1.14999999999999998e113 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 83.7

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

   8. Simplified 83.7%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

       3. lower-*.f64 83.7

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

   11. Simplified 83.7%

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

Alternative 11: 57.4% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 340:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+113}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 340.0)
   (fma a 0.25 0.5)
   (if (<= b 1.15e+113)
     (* -0.020833333333333332 (* a (* a a)))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 340.0) {
		tmp = fma(a, 0.25, 0.5);
	} else if (b <= 1.15e+113) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 340.0)
		tmp = fma(a, 0.25, 0.5);
	elseif (b <= 1.15e+113)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 340.0], N[(a * 0.25 + 0.5), $MachinePrecision], If[LessEqual[b, 1.15e+113], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 340

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 76.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot \frac{1}{4}} + \frac{1}{2} \]

      3. lower-fma.f64 51.0

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

   8. Simplified 51.0%

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

   if 340 < b < 1.14999999999999998e113

   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. Simplified 35.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

       6. lower-*.f64 43.5

          \[\leadsto -0.020833333333333332 \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   11. Simplified 43.5%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]

   if 1.14999999999999998e113 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 83.7

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

   8. Simplified 83.7%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

       3. lower-*.f64 83.7

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

   11. Simplified 83.7%

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

Alternative 12: 50.9% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2:\\ \;\;\;\;b \cdot \left(\left(b \cdot b\right) \cdot 0.020833333333333332\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.0) (* b (* (* b b) 0.020833333333333332)) (fma a 0.25 0.5)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.0) {
		tmp = b * ((b * b) * 0.020833333333333332);
	} else {
		tmp = fma(a, 0.25, 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.0)
		tmp = Float64(b * Float64(Float64(b * b) * 0.020833333333333332));
	else
		tmp = fma(a, 0.25, 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.0], N[(b * N[(N[(b * b), $MachinePrecision] * 0.020833333333333332), $MachinePrecision]), $MachinePrecision], N[(a * 0.25 + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 35.9

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

   5. Simplified 35.9%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{1}{48} \cdot {b}^{2} - \frac{1}{4}, \frac{1}{2}\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{1}{48} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{{b}^{2} \cdot \frac{1}{48}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(b \cdot b\right)} \cdot \frac{1}{48} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{b \cdot \left(b \cdot \frac{1}{48}\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(b, b \cdot \left(b \cdot \frac{1}{48}\right) + \color{blue}{\frac{-1}{4}}, \frac{1}{2}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, b \cdot \frac{1}{48}, \frac{-1}{4}\right)}, \frac{1}{2}\right) \]

      9. lower-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in b around inf

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

       1. unpow3 N/A

          \[\leadsto \frac{1}{48} \cdot \color{blue}{\left(\left(b \cdot b\right) \cdot b\right)} \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{48} \cdot \left(\color{blue}{{b}^{2}} \cdot b\right) \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right) \cdot b} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2}\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2}\right)} \]

       6. *-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left({b}^{2} \cdot \frac{1}{48}\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto b \cdot \color{blue}{\left({b}^{2} \cdot \frac{1}{48}\right)} \]

       8. unpow2 N/A

          \[\leadsto b \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \frac{1}{48}\right) \]

       9. lower-*.f64 41.0

          \[\leadsto b \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot 0.020833333333333332\right) \]

   11. Simplified 41.0%

       \[\leadsto \color{blue}{b \cdot \left(\left(b \cdot b\right) \cdot 0.020833333333333332\right)} \]

   if -2 < a

   1. Initial program 99.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 54.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot \frac{1}{4}} + \frac{1}{2} \]

      3. lower-fma.f64 53.9

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

   8. Simplified 53.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, 0.25, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 51.1% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 340:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 340.0) (fma a 0.25 0.5) (* -0.020833333333333332 (* a (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 340.0) {
		tmp = fma(a, 0.25, 0.5);
	} else {
		tmp = -0.020833333333333332 * (a * (a * a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 340.0)
		tmp = fma(a, 0.25, 0.5);
	else
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 340.0], N[(a * 0.25 + 0.5), $MachinePrecision], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 340

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. Simplified 76.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot \frac{1}{4}} + \frac{1}{2} \]

      3. lower-fma.f64 51.0

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

   8. Simplified 51.0%

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

   if 340 < b

   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. Simplified 39.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. lower-*.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

       6. lower-*.f64 40.0

          \[\leadsto -0.020833333333333332 \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   11. Simplified 40.0%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 39.5% 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 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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. Simplified 67.0%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{a \cdot \frac{1}{4}} + \frac{1}{2} \]

   3. lower-fma.f64 39.0

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

8. Simplified 39.0%

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

Alternative 15: 39.3% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-exp.f64 80.5

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

5. Simplified 80.5%

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

6. Taylor expanded in b around 0

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

8. Simplified 38.6%

   \[\leadsto \color{blue}{0.5} \]
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 2024215 
(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))))