Quotient of sum of exps

Percentage Accurate: 98.9% → 98.5%

Time: 6.2s

Alternatives: 19

Speedup: 1.0×

• could not determine a ground truth

  1. a = 2.0494017481785893e+102
  2. b = 6.408028576950376e+209

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: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999998:\\ \;\;\;\;\frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, e^{a}\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999999998)
   (/ (exp a) (+ (fma (fma 0.5 b 1.0) b (exp a)) 1.0))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999999998) {
		tmp = exp(a) / (fma(fma(0.5, b, 1.0), b, exp(a)) + 1.0);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.999999998)
		tmp = Float64(exp(a) / Float64(fma(fma(0.5, b, 1.0), b, exp(a)) + 1.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.999999998], N[(N[Exp[a], $MachinePrecision] / N[(N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + N[Exp[a], $MachinePrecision]), $MachinePrecision] + 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) < 0.999999997999999946

   1. Initial program 97.1%

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

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1 + e^{a}\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

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

   if 0.999999997999999946 < (exp.f64 a)

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999998:\\ \;\;\;\;\frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, e^{a}\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-neg.f64 98.8

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

4. Applied rewrites 98.8%

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

5. Final simplification 98.8%

   \[\leadsto \frac{--1}{e^{a} + e^{b}} \cdot e^{a} \]
6. Add Preprocessing

Alternative 3: 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 98.8%

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

Alternative 4: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999998:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 + b, e^{-a}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999999998)
   (/ 1.0 (fma (+ 1.0 b) (exp (- a)) 1.0))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999999998) {
		tmp = 1.0 / fma((1.0 + b), exp(-a), 1.0);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.999999998)
		tmp = Float64(1.0 / fma(Float64(1.0 + b), exp(Float64(-a)), 1.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.999999998], N[(1.0 / N[(N[(1.0 + b), $MachinePrecision] * N[Exp[(-a)], $MachinePrecision] + 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) < 0.999999997999999946

   1. Initial program 97.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.1

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

   5. Applied rewrites 98.1%

      \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right)}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 98.2

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. rec-exp N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 97.8

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

   10. Applied rewrites 97.8%

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

   if 0.999999997999999946 < (exp.f64 a)

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999998:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 + b, e^{-a}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 98.7

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

5. Applied rewrites 98.7%

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

6. Final simplification 98.7%

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

Alternative 6: 98.5% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ (exp a) (+ (+ 1.0 b) 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 + b) + 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 + b) + 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 + b) + 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 + b) + 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(Float64(1.0 + b) + 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 + b) + 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[(N[(1.0 + b), $MachinePrecision] + 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 96.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 98.7%

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

Alternative 7: 93.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)\\ \mathbf{if}\;a \leq -1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t\_0 + \left(1 + b\right)}\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{+96}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 a 1.0) a 1.0)))
   (if (<= a -1.9e+154)
     (/ 1.0 (+ t_0 (+ 1.0 b)))
     (if (<= a -6.2e+96)
       (/
        t_0
        (fma
         (fma 0.5 b 1.0)
         b
         (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 2.0)))
       (/ 1.0 (+ 1.0 (exp b)))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(0.5, a, 1.0), a, 1.0);
	double tmp;
	if (a <= -1.9e+154) {
		tmp = 1.0 / (t_0 + (1.0 + b));
	} else if (a <= -6.2e+96) {
		tmp = t_0 / fma(fma(0.5, b, 1.0), b, fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 2.0));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(0.5, a, 1.0), a, 1.0)
	tmp = 0.0
	if (a <= -1.9e+154)
		tmp = Float64(1.0 / Float64(t_0 + Float64(1.0 + b)));
	elseif (a <= -6.2e+96)
		tmp = Float64(t_0 / fma(fma(0.5, b, 1.0), b, fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 2.0)));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision]}, If[LessEqual[a, -1.9e+154], N[(1.0 / N[(t$95$0 + N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -6.2e+96], N[(t$95$0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.8999999999999999e154

   1. Initial program 96.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 100.0%

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

   if -1.8999999999999999e154 < a < -6.1999999999999996e96

   1. Initial program 92.9%

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

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 17.0

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

   11. Applied rewrites 17.0%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 79.4%

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

   if -6.1999999999999996e96 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 92.1

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

   5. Applied rewrites 92.1%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right) + \left(1 + b\right)}\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{+96}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.8% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, \mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 2\right)\right)\\ \mathbf{if}\;b \leq -680000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 720:\\ \;\;\;\;\frac{1 + a}{t\_0}\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{+101}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot 0.5}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 b 1.0) b (fma (fma 0.5 a 1.0) a 2.0))))
   (if (<= b -680000000.0)
     0.5
     (if (<= b 720.0)
       (/ (+ 1.0 a) t_0)
       (if (<= b 2.65e+101)
         (/ (* (* a a) 0.5) t_0)
         (/ 1.0 (* (fma (* 0.16666666666666666 b) b 1.0) b)))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(0.5, b, 1.0), b, fma(fma(0.5, a, 1.0), a, 2.0));
	double tmp;
	if (b <= -680000000.0) {
		tmp = 0.5;
	} else if (b <= 720.0) {
		tmp = (1.0 + a) / t_0;
	} else if (b <= 2.65e+101) {
		tmp = ((a * a) * 0.5) / t_0;
	} else {
		tmp = 1.0 / (fma((0.16666666666666666 * b), b, 1.0) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(0.5, b, 1.0), b, fma(fma(0.5, a, 1.0), a, 2.0))
	tmp = 0.0
	if (b <= -680000000.0)
		tmp = 0.5;
	elseif (b <= 720.0)
		tmp = Float64(Float64(1.0 + a) / t_0);
	elseif (b <= 2.65e+101)
		tmp = Float64(Float64(Float64(a * a) * 0.5) / t_0);
	else
		tmp = Float64(1.0 / Float64(fma(Float64(0.16666666666666666 * b), b, 1.0) * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -680000000.0], 0.5, If[LessEqual[b, 720.0], N[(N[(1.0 + a), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[b, 2.65e+101], N[(N[(N[(a * a), $MachinePrecision] * 0.5), $MachinePrecision] / t$95$0), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b), $MachinePrecision] * b + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.8e8

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -6.8e8 < b < 720

   1. Initial program 97.8%

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

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 82.3

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

   11. Applied rewrites 82.3%

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

   if 720 < b < 2.65000000000000003e101

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 32.1

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

   5. Applied rewrites 32.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 32.1%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 3.6

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

   11. Applied rewrites 3.6%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 50.9%

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

   if 2.65000000000000003e101 < 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 98.2%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 98.2%

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

   12. Taylor expanded in b around inf

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1}{6} \cdot b, b, 1\right) \cdot b} \]
   13. Step-by-step derivation

   14. Applied rewrites 98.2%

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

Alternative 9: 68.1% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -680000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + a}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, \mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -680000000.0)
   0.5
   (if (<= b 1.05e+34)
     (/ (+ 1.0 a) (fma (fma 0.5 b 1.0) b (fma (fma 0.5 a 1.0) a 2.0)))
     (/ 1.0 (* (* (fma 0.16666666666666666 b 0.5) b) b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -680000000.0) {
		tmp = 0.5;
	} else if (b <= 1.05e+34) {
		tmp = (1.0 + a) / fma(fma(0.5, b, 1.0), b, fma(fma(0.5, a, 1.0), a, 2.0));
	} else {
		tmp = 1.0 / ((fma(0.16666666666666666, b, 0.5) * b) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -680000000.0)
		tmp = 0.5;
	elseif (b <= 1.05e+34)
		tmp = Float64(Float64(1.0 + a) / fma(fma(0.5, b, 1.0), b, fma(fma(0.5, a, 1.0), a, 2.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -680000000.0], 0.5, If[LessEqual[b, 1.05e+34], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.8e8

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -6.8e8 < b < 1.05000000000000009e34

   1. Initial program 97.9%

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

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-exp.f64 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 95.1%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 79.4

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

   11. Applied rewrites 79.4%

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

   if 1.05000000000000009e34 < 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 75.4%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 75.4%

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

Alternative 10: 67.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -680000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + a}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right) + \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -680000000.0)
   0.5
   (if (<= b 1.05e+34)
     (/ (+ 1.0 a) (+ (fma (fma 0.5 a 1.0) a 1.0) (+ 1.0 b)))
     (/ 1.0 (* (* (fma 0.16666666666666666 b 0.5) b) b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -680000000.0) {
		tmp = 0.5;
	} else if (b <= 1.05e+34) {
		tmp = (1.0 + a) / (fma(fma(0.5, a, 1.0), a, 1.0) + (1.0 + b));
	} else {
		tmp = 1.0 / ((fma(0.16666666666666666, b, 0.5) * b) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -680000000.0)
		tmp = 0.5;
	elseif (b <= 1.05e+34)
		tmp = Float64(Float64(1.0 + a) / Float64(fma(fma(0.5, a, 1.0), a, 1.0) + Float64(1.0 + b)));
	else
		tmp = Float64(1.0 / Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -680000000.0], 0.5, If[LessEqual[b, 1.05e+34], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.8e8

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -6.8e8 < b < 1.05000000000000009e34

   1. Initial program 97.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 95.0

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

   8. Applied rewrites 95.0%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 79.3

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

   11. Applied rewrites 79.3%

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

   if 1.05000000000000009e34 < 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 75.4%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 75.4%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -680000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + a}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right) + \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -680000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right) + \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -680000000.0)
   0.5
   (if (<= b 7.5e+51)
     (/ 1.0 (+ (fma (fma 0.5 a 1.0) a 1.0) (+ 1.0 b)))
     (/ 1.0 (* (fma (* 0.16666666666666666 b) b 1.0) b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -680000000.0) {
		tmp = 0.5;
	} else if (b <= 7.5e+51) {
		tmp = 1.0 / (fma(fma(0.5, a, 1.0), a, 1.0) + (1.0 + b));
	} else {
		tmp = 1.0 / (fma((0.16666666666666666 * b), b, 1.0) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -680000000.0)
		tmp = 0.5;
	elseif (b <= 7.5e+51)
		tmp = Float64(1.0 / Float64(fma(fma(0.5, a, 1.0), a, 1.0) + Float64(1.0 + b)));
	else
		tmp = Float64(1.0 / Float64(fma(Float64(0.16666666666666666 * b), b, 1.0) * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -680000000.0], 0.5, If[LessEqual[b, 7.5e+51], N[(1.0 / N[(N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + N[(1.0 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b), $MachinePrecision] * b + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.8e8

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -6.8e8 < b < 7.4999999999999999e51

   1. Initial program 98.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 93.9

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

   8. Applied rewrites 93.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 77.0%

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

   if 7.4999999999999999e51 < 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 80.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 80.0%

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

   12. Taylor expanded in b around inf

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1}{6} \cdot b, b, 1\right) \cdot b} \]
   13. Step-by-step derivation

   14. Applied rewrites 80.0%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -680000000:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right) + \left(1 + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -100000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -100000000.0)
   0.5
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -100000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -100000000.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -100000000.0], 0.5, N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1e8

   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

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -1e8 < b

   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. +-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 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 68.2%

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

Alternative 13: 58.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -100000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -100000000.0)
   0.5
   (/ 1.0 (fma (fma (* 0.16666666666666666 b) b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -100000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / fma(fma((0.16666666666666666 * b), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -100000000.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / fma(fma(Float64(0.16666666666666666 * b), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -100000000.0], 0.5, N[(1.0 / N[(N[(N[(0.16666666666666666 * b), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1e8

   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

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -1e8 < b

   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. +-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 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 68.2%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 68.2%

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

Alternative 14: 57.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -100000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b, b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -100000000.0)
   0.5
   (/ 1.0 (fma (* (fma 0.16666666666666666 b 0.5) b) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -100000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / fma((fma(0.16666666666666666, b, 0.5) * b), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -100000000.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / fma(Float64(fma(0.16666666666666666, b, 0.5) * b), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -100000000.0], 0.5, N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1e8

   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

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -1e8 < b

   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. +-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 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 68.2%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 67.7%

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

Alternative 15: 57.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -100000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(b \cdot b\right) \cdot 0.16666666666666666, b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -100000000.0)
   0.5
   (/ 1.0 (fma (* (* b b) 0.16666666666666666) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -100000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / fma(((b * b) * 0.16666666666666666), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -100000000.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / fma(Float64(Float64(b * b) * 0.16666666666666666), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -100000000.0], 0.5, N[(1.0 / N[(N[(N[(b * b), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1e8

   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

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -1e8 < b

   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. +-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 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 68.2%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 67.7%

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

Alternative 16: 57.7% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.6) 0.5 (/ 1.0 (* (* (fma 0.16666666666666666 b 0.5) b) b))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001

   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. +-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 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 56.2%

      \[\leadsto 0.5 \]

   if 1.6000000000000001 < 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 67.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 67.5%

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

Alternative 17: 53.8% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -100000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -100000000.0) 0.5 (/ 1.0 (fma (fma 0.5 b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -100000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / fma(fma(0.5, b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -100000000.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / fma(fma(0.5, b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -100000000.0], 0.5, N[(1.0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1e8

   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

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -1e8 < b

   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. +-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 79.3

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 62.9%

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

Alternative 18: 53.3% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 1.25

   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. +-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 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 56.2%

      \[\leadsto 0.5 \]

   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

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

   8. Applied rewrites 67.5%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 67.5%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 52.1%

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

Alternative 19: 39.7% 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. +-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 82.8

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

5. Applied rewrites 82.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 41.1%

   \[\leadsto 0.5 \]
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 2024276 
(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))))