Quotient of sum of exps

Percentage Accurate: 99.0% → 99.0%

Time: 7.1s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. a = -2.343644056224519e+182
  2. b = -1.6304917372439575e+29

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. inv-pow N/A

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

   5. unpow-1 N/A

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

   6. lower-/.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. rec-exp N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 98.8

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

4. Applied rewrites 98.8%

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

Alternative 2: 48.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;b \cdot \left(0.0625 \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.0)
   (* b (* 0.0625 (* a a)))
   (fma 0.25 a 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.0) {
		tmp = b * (0.0625 * (a * a));
	} else {
		tmp = fma(0.25, a, 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.0)
		tmp = Float64(b * Float64(0.0625 * Float64(a * a)));
	else
		tmp = fma(0.25, a, 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(b * N[(0.0625 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * a + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \color{blue}{a \cdot \left(\left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) + a \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot b + \left(\frac{1}{8} \cdot b + \left(\frac{1}{4} \cdot b + \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right) - \left(\frac{1}{8} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)} \]

   8. Applied rewrites 2.0%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0 - b \cdot -0.0625, 0.25\right)}, \mathsf{fma}\left(b, -0.25, 0.5\right)\right) \]

   9. Taylor expanded in a around inf

      \[\leadsto \frac{1}{16} \cdot \left({a}^{2} \cdot \color{blue}{b}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 19.2%

       \[\leadsto b \cdot \left(0.0625 \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   if 0.0 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 97.6%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \color{blue}{a \cdot \left(\left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) + a \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot b + \left(\frac{1}{8} \cdot b + \left(\frac{1}{4} \cdot b + \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right) - \left(\frac{1}{8} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)} \]

   8. Applied rewrites 63.9%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0 - b \cdot -0.0625, 0.25\right)}, \mathsf{fma}\left(b, -0.25, 0.5\right)\right) \]

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 67.3%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

Alternative 4: 98.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 97.8

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

   5. Applied rewrites 97.8%

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

Alternative 5: 82.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{b}}\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{+203}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq -8.6 \cdot 10^{+141}:\\ \;\;\;\;-0.0020833333333333333 \cdot {b}^{5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp b)))))
   (if (<= a -3.5e+203)
     t_0
     (if (<= a -8.6e+141) (* -0.0020833333333333333 (pow b 5.0)) t_0))))

C

double code(double a, double b) {
	double t_0 = 1.0 / (1.0 + exp(b));
	double tmp;
	if (a <= -3.5e+203) {
		tmp = t_0;
	} else if (a <= -8.6e+141) {
		tmp = -0.0020833333333333333 * pow(b, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (1.0d0 + exp(b))
    if (a <= (-3.5d+203)) then
        tmp = t_0
    else if (a <= (-8.6d+141)) then
        tmp = (-0.0020833333333333333d0) * (b ** 5.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 1.0 / (1.0 + Math.exp(b));
	double tmp;
	if (a <= -3.5e+203) {
		tmp = t_0;
	} else if (a <= -8.6e+141) {
		tmp = -0.0020833333333333333 * Math.pow(b, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 1.0 / (1.0 + math.exp(b))
	tmp = 0
	if a <= -3.5e+203:
		tmp = t_0
	elif a <= -8.6e+141:
		tmp = -0.0020833333333333333 * math.pow(b, 5.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(1.0 / Float64(1.0 + exp(b)))
	tmp = 0.0
	if (a <= -3.5e+203)
		tmp = t_0;
	elseif (a <= -8.6e+141)
		tmp = Float64(-0.0020833333333333333 * (b ^ 5.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 1.0 / (1.0 + exp(b));
	tmp = 0.0;
	if (a <= -3.5e+203)
		tmp = t_0;
	elseif (a <= -8.6e+141)
		tmp = -0.0020833333333333333 * (b ^ 5.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.5e+203], t$95$0, If[LessEqual[a, -8.6e+141], N[(-0.0020833333333333333 * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.50000000000000031e203 or -8.5999999999999997e141 < a

   1. Initial program 98.7%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 87.3

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

   5. Applied rewrites 87.3%

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

   if -3.50000000000000031e203 < a < -8.5999999999999997e141

   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 28.6

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

   5. Applied rewrites 28.6%

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

   7. Applied rewrites 28.6%

      \[\leadsto {\left({\left(1 + e^{b}\right)}^{-0.5}\right)}^{\color{blue}{2}} \]

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 2.8%

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

   11. Taylor expanded in b around inf

       \[\leadsto \frac{-1}{480} \cdot {b}^{\color{blue}{5}} \]
   12. Step-by-step derivation

   13. Applied rewrites 74.1%

       \[\leadsto -0.0020833333333333333 \cdot {b}^{\color{blue}{5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 59.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{+141}:\\ \;\;\;\;-0.0020833333333333333 \cdot {b}^{5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.6e+141)
   (* -0.0020833333333333333 (pow b 5.0))
   (/ 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 (a <= -8.6e+141) {
		tmp = -0.0020833333333333333 * pow(b, 5.0);
	} 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 (a <= -8.6e+141)
		tmp = Float64(-0.0020833333333333333 * (b ^ 5.0));
	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[a, -8.6e+141], N[(-0.0020833333333333333 * N[Power[b, 5.0], $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 2 regimes

2. if a < -8.5999999999999997e141

   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 46.0

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

   5. Applied rewrites 46.0%

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

   7. Applied rewrites 46.0%

      \[\leadsto {\left({\left(1 + e^{b}\right)}^{-0.5}\right)}^{\color{blue}{2}} \]

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 2.5%

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

   11. Taylor expanded in b around inf

       \[\leadsto \frac{-1}{480} \cdot {b}^{\color{blue}{5}} \]
   12. Step-by-step derivation

   13. Applied rewrites 50.5%

       \[\leadsto -0.0020833333333333333 \cdot {b}^{\color{blue}{5}} \]

   if -8.5999999999999997e141 < a

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 59.8%

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

Alternative 7: 60.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 60000000000:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{elif}\;b \leq 3.75 \cdot 10^{+102}:\\ \;\;\;\;b \cdot \left(0.0625 \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 60000000000.0)
   (fma 0.25 a 0.5)
   (if (<= b 3.75e+102)
     (* b (* 0.0625 (* 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 <= 60000000000.0) {
		tmp = fma(0.25, a, 0.5);
	} else if (b <= 3.75e+102) {
		tmp = b * (0.0625 * (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 <= 60000000000.0)
		tmp = fma(0.25, a, 0.5);
	elseif (b <= 3.75e+102)
		tmp = Float64(b * Float64(0.0625 * 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, 60000000000.0], N[(0.25 * a + 0.5), $MachinePrecision], If[LessEqual[b, 3.75e+102], N[(b * N[(0.0625 * 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 < 6e10

   1. Initial program 98.2%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \color{blue}{a \cdot \left(\left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) + a \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot b + \left(\frac{1}{8} \cdot b + \left(\frac{1}{4} \cdot b + \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right) - \left(\frac{1}{8} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)} \]

   8. Applied rewrites 48.2%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0 - b \cdot -0.0625, 0.25\right)}, \mathsf{fma}\left(b, -0.25, 0.5\right)\right) \]

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 50.7%

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

   if 6e10 < b < 3.75e102

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 27.9%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \color{blue}{a \cdot \left(\left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) + a \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot b + \left(\frac{1}{8} \cdot b + \left(\frac{1}{4} \cdot b + \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right) - \left(\frac{1}{8} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)} \]

   8. Applied rewrites 2.5%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0 - b \cdot -0.0625, 0.25\right)}, \mathsf{fma}\left(b, -0.25, 0.5\right)\right) \]

   9. Taylor expanded in a around inf

      \[\leadsto \frac{1}{16} \cdot \left({a}^{2} \cdot \color{blue}{b}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 31.7%

       \[\leadsto b \cdot \left(0.0625 \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   if 3.75e102 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 98.5%

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

Alternative 8: 57.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 60000000000:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;b \cdot \left(0.0625 \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 60000000000.0)
   (fma 0.25 a 0.5)
   (if (<= b 1.05e+153)
     (* b (* 0.0625 (* a a)))
     (/ 1.0 (fma b (fma b 0.5 1.0) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 60000000000.0) {
		tmp = fma(0.25, a, 0.5);
	} else if (b <= 1.05e+153) {
		tmp = b * (0.0625 * (a * a));
	} else {
		tmp = 1.0 / fma(b, fma(b, 0.5, 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 60000000000.0)
		tmp = fma(0.25, a, 0.5);
	elseif (b <= 1.05e+153)
		tmp = Float64(b * Float64(0.0625 * Float64(a * a)));
	else
		tmp = Float64(1.0 / fma(b, fma(b, 0.5, 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 60000000000.0], N[(0.25 * a + 0.5), $MachinePrecision], If[LessEqual[b, 1.05e+153], N[(b * N[(0.0625 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 6e10

   1. Initial program 98.2%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \color{blue}{a \cdot \left(\left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) + a \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot b + \left(\frac{1}{8} \cdot b + \left(\frac{1}{4} \cdot b + \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right) - \left(\frac{1}{8} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)} \]

   8. Applied rewrites 48.2%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0 - b \cdot -0.0625, 0.25\right)}, \mathsf{fma}\left(b, -0.25, 0.5\right)\right) \]

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 50.7%

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

   if 6e10 < b < 1.05000000000000008e153

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 37.8%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \color{blue}{a \cdot \left(\left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) + a \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot b + \left(\frac{1}{8} \cdot b + \left(\frac{1}{4} \cdot b + \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right) - \left(\frac{1}{8} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)} \]

   8. Applied rewrites 2.3%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0 - b \cdot -0.0625, 0.25\right)}, \mathsf{fma}\left(b, -0.25, 0.5\right)\right) \]

   9. Taylor expanded in a around inf

      \[\leadsto \frac{1}{16} \cdot \left({a}^{2} \cdot \color{blue}{b}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 27.6%

       \[\leadsto b \cdot \left(0.0625 \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   if 1.05000000000000008e153 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 97.6%

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

Alternative 9: 40.2% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. times-frac N/A

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

   5. distribute-lft1-in N/A

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

   6. lower-*.f64 N/A

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

5. Applied rewrites 63.8%

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

6. Taylor expanded in a around 0

   \[\leadsto \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \color{blue}{a \cdot \left(\left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) + a \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot b + \left(\frac{1}{8} \cdot b + \left(\frac{1}{4} \cdot b + \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right) - \left(\frac{1}{8} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)} \]

8. Applied rewrites 33.2%

   \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0 - b \cdot -0.0625, 0.25\right)}, \mathsf{fma}\left(b, -0.25, 0.5\right)\right) \]

9. Taylor expanded in b around 0

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

11. Applied rewrites 35.2%

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

Alternative 10: 40.1% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-exp.f64 83.0

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

5. Applied rewrites 83.0%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 35.1%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024231 
(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))))