Quotient of sum of exps

Percentage Accurate: 98.9% → 99.0%

Time: 5.4s

Alternatives: 12

Speedup: 1.3×

• could not determine a ground truth

  1. a = 9.905509480779674e+272
  2. b = 5.098497673433094e-282

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: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99998:\\ \;\;\;\;\frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, e^{b} + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.99998)
   (/ (exp a) (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 2.0))
   (/ (fma (fma 0.5 a 1.0) a 1.0) (fma (fma 0.5 a 1.0) a (+ (exp b) 1.0)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.99998) {
		tmp = exp(a) / fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 2.0);
	} else {
		tmp = fma(fma(0.5, a, 1.0), a, 1.0) / fma(fma(0.5, a, 1.0), a, (exp(b) + 1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.99998)
		tmp = Float64(exp(a) / fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 2.0));
	else
		tmp = Float64(fma(fma(0.5, a, 1.0), a, 1.0) / fma(fma(0.5, a, 1.0), a, Float64(exp(b) + 1.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.99998], N[(N[Exp[a], $MachinePrecision] / N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] / N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.99997999999999998

   1. Initial program 97.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if 0.99997999999999998 < (exp.f64 a)

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.2

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

   8. Applied rewrites 99.2%

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

Alternative 2: 53.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.500000000005:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot b, b, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.500000000005)
   (pow (fma (* 0.5 b) b 2.0) -1.0)
   0.5))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.500000000005) {
		tmp = pow(fma((0.5 * b), b, 2.0), -1.0);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.500000000005)
		tmp = fma(Float64(0.5 * b), b, 2.0) ^ -1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.500000000005], N[Power[N[(N[(0.5 * b), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.500000000005

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 60.4%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 60.4%

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

   if 0.500000000005 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 94.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 13.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 22.0%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.500000000005:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot b, b, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999998:\\ \;\;\;\;\frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999998)
   (/ (exp a) (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 2.0))
   (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999998) {
		tmp = exp(a) / fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 2.0);
	} else {
		tmp = pow((exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.999998)
		tmp = Float64(exp(a) / fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 2.0));
	else
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.999998], N[(N[Exp[a], $MachinePrecision] / N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.999998000000000054

   1. Initial program 97.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if 0.999998000000000054 < (exp.f64 a)

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999998:\\ \;\;\;\;\frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999998:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999998) (/ (exp a) 2.0) (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999998) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = pow((exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.999998d0) then
        tmp = exp(a) / 2.0d0
    else
        tmp = (exp(b) + 1.0d0) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.999998) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.999998:
		tmp = math.exp(a) / 2.0
	else:
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.999998)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.999998)
		tmp = exp(a) / 2.0;
	else
		tmp = (exp(b) + 1.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.999998], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.999998000000000054

   1. Initial program 97.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 97.9%

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

   if 0.999998000000000054 < (exp.f64 a)

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999998:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, e^{b} + 1\right)} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (/ (exp a) (fma (fma 0.5 a 1.0) a (+ (exp b) 1.0))))

C

double code(double a, double b) {
	return exp(a) / fma(fma(0.5, a, 1.0), a, (exp(b) + 1.0));
}

Julia

function code(a, b)
	return Float64(exp(a) / fma(fma(0.5, a, 1.0), a, Float64(exp(b) + 1.0)))
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower-exp.f64 99.2

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

5. Applied rewrites 99.2%

   \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, e^{b} + 1\right)}} \]
6. Add Preprocessing

Alternative 6: 77.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{+88}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.22e+88)
   (/ (exp a) 2.0)
   (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.22e+88) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.22e+88)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.22e+88], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.22e88

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 80.0

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

   5. Applied rewrites 80.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 78.2%

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

   if 1.22e88 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 96.8%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{+88}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{+88}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.22e+88)
   (/ 1.0 (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 2.0))
   (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.22e+88) {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 2.0);
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.22e+88)
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 2.0));
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.22e+88], N[(1.0 / N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.22e88

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 80.0

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

   5. Applied rewrites 80.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.2%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 68.4%

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

   if 1.22e88 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 96.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{+88}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -260000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -260000000.0)
   0.5
   (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -260000000.0) {
		tmp = 0.5;
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -260000000.0)
		tmp = 0.5;
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -260000000.0], 0.5, N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.6e8

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 5.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 18.8%

       \[\leadsto 0.5 \]

   if -2.6e8 < b

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 67.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -260000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -260000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -260000000.0) 0.5 (pow (fma (fma 0.5 b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -260000000.0) {
		tmp = 0.5;
	} else {
		tmp = pow(fma(fma(0.5, b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -260000000.0)
		tmp = 0.5;
	else
		tmp = fma(fma(0.5, b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -260000000.0], 0.5, N[Power[N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.6e8

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 5.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 18.8%

       \[\leadsto 0.5 \]

   if -2.6e8 < b

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 59.9%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -260000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, b, 1\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.25) 0.5 (pow (* (fma 0.5 b 1.0) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.25) {
		tmp = 0.5;
	} else {
		tmp = pow((fma(0.5, b, 1.0) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.25)
		tmp = 0.5;
	else
		tmp = Float64(fma(0.5, b, 1.0) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.25], 0.5, N[Power[N[(N[(0.5 * b + 1.0), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.25

   1. Initial program 98.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 70.9

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 54.2%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 55.9%

       \[\leadsto 0.5 \]

   if 1.25 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 51.3%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 51.3%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, b, 1\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.2% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.0) 0.5 (pow (* (* 0.5 b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = pow(((0.5 * b) * 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 (b <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = ((0.5d0 * b) * b) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = Math.pow(((0.5 * b) * b), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.0:
		tmp = 0.5
	else:
		tmp = math.pow(((0.5 * b) * b), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(Float64(0.5 * b) * b) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.0)
		tmp = 0.5;
	else
		tmp = ((0.5 * b) * b) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.0], 0.5, N[Power[N[(N[(0.5 * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 98.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 70.9

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 54.2%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 55.9%

       \[\leadsto 0.5 \]

   if 2 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 51.3%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 51.3%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.5 \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.5% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 79.7

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

5. Applied rewrites 79.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 38.6%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 40.0%

    \[\leadsto 0.5 \]
12. 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 2024296 
(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))))