Quotient of sum of exps

Percentage Accurate: 98.9% → 98.6%

Time: 4.9s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

  1. a = 7.773185709655008e+39
  2. b = -3.194700886320209e-292

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: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ (exp a) 2.0) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a) / 2.0;
	} 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) / 2.0d0
    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) / 2.0;
	} 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) / 2.0
	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) / 2.0);
	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) / 2.0;
	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] / 2.0), $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 98.7%

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

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.5740082154945411:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.5740082154945411)
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))
   0.5))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.5740082154945411) {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.5740082154945411)
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	else
		tmp = 0.5;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5740082154945411], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.574008215494541063

   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 73.8

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

   5. Applied rewrites 73.8%

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

   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 62.8%

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

   if 0.574008215494541063 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 94.1%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 98.1

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

   5. Applied rewrites 98.1%

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

   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 5.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 18.4%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.5740082154945411:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.4998:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot b\right) \cdot b, b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.4998)
   (/ 1.0 (fma (* (* 0.16666666666666666 b) b) b 2.0))
   (fma 0.25 a 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.4998) {
		tmp = 1.0 / fma(((0.16666666666666666 * b) * b), b, 2.0);
	} else {
		tmp = fma(0.25, a, 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.4998)
		tmp = Float64(1.0 / fma(Float64(Float64(0.16666666666666666 * b) * b), b, 2.0));
	else
		tmp = fma(0.25, a, 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.4998], N[(1.0 / N[(N[(N[(0.16666666666666666 * b), $MachinePrecision] * b), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.25 * a + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.49980000000000002

   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 56.3

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

   5. Applied rewrites 56.3%

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

   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 38.3%

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

   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 37.9%

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

   if 0.49980000000000002 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 97.7%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. neg-mul-1 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 68.1%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.4998:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot b\right) \cdot b, b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.4998:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5 \cdot b, b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.4998)
   (/ 1.0 (fma (* 0.5 b) b 2.0))
   (fma 0.25 a 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.4998) {
		tmp = 1.0 / fma((0.5 * b), b, 2.0);
	} else {
		tmp = fma(0.25, a, 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.4998)
		tmp = Float64(1.0 / fma(Float64(0.5 * b), b, 2.0));
	else
		tmp = fma(0.25, a, 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.4998], N[(1.0 / N[(N[(0.5 * b), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.25 * a + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.49980000000000002

   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 56.3

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

   5. Applied rewrites 56.3%

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

   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 26.8%

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

   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 26.5%

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

   if 0.49980000000000002 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 97.7%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. neg-mul-1 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 68.1%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.4998:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5 \cdot b, b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Final simplification 98.8%

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

Alternative 6: 57.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (exp(b) <= 2.0) {
		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 (exp(b) <= 2.0)
		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[N[Exp[b], $MachinePrecision], 2.0], 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 (exp.f64 b) < 2

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 71.2

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

   5. Applied rewrites 71.2%

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

   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 45.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 48.9%

       \[\leadsto 0.5 \]

   if 2 < (exp.f64 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 66.9%

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

   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 66.9%

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

Alternative 7: 52.6% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 2.0) 0.5 (/ 1.0 (* (* b b) 0.5))))

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(b) <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / ((b * b) * 0.5d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 2

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 71.2

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

   5. Applied rewrites 71.2%

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

   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 45.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 48.9%

       \[\leadsto 0.5 \]

   if 2 < (exp.f64 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 45.8%

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

   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 45.8%

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

Alternative 8: 76.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+87}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6e+87)
   (/ (exp a) 2.0)
   (/ 1.0 (* (* (fma 0.16666666666666666 b 0.5) b) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6e+87) {
		tmp = exp(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 <= 6e+87)
		tmp = Float64(exp(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, 6e+87], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], 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 < 5.9999999999999998e87

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 73.7

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.4%

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

   if 5.9999999999999998e87 < 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 94.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 94.0%

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

Alternative 9: 60.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{+34}:\\ \;\;\;\;{b}^{5} \cdot -0.0020833333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e+34)
   (* (pow b 5.0) -0.0020833333333333333)
   (/ 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 (a <= -5.4e+34) {
		tmp = pow(b, 5.0) * -0.0020833333333333333;
	} 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 (a <= -5.4e+34)
		tmp = Float64((b ^ 5.0) * -0.0020833333333333333);
	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[a, -5.4e+34], N[(N[Power[b, 5.0], $MachinePrecision] * -0.0020833333333333333), $MachinePrecision], 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 a < -5.4000000000000001e34

   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 25.0

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

   5. Applied rewrites 25.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 2.8%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 54.2%

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

   if -5.4000000000000001e34 < a

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 99.2

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

   5. Applied rewrites 99.2%

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

   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 63.9%

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

Alternative 10: 60.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2e+37)
   (/ 1.0 (* (fma (+ (/ 2.0 (* b b)) 0.5) b 1.0) b))
   (/ 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 (a <= -2e+37) {
		tmp = 1.0 / (fma(((2.0 / (b * b)) + 0.5), b, 1.0) * b);
	} 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 (a <= -2e+37)
		tmp = Float64(1.0 / Float64(fma(Float64(Float64(2.0 / Float64(b * b)) + 0.5), b, 1.0) * b));
	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[a, -2e+37], N[(1.0 / N[(N[(N[(N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], 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 a < -1.99999999999999991e37

   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 25.3

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

   5. Applied rewrites 25.3%

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

   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 10.4%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 9.8%

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

   12. Taylor expanded in b around inf

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

   14. Applied rewrites 42.2%

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

   if -1.99999999999999991e37 < a

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 63.6%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.3e-303)
   (fma 0.25 a 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 <= 2.3e-303) {
		tmp = fma(0.25, a, 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 <= 2.3e-303)
		tmp = fma(0.25, a, 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, 2.3e-303], N[(0.25 * a + 0.5), $MachinePrecision], 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 < 2.29999999999999995e-303

   1. Initial program 97.4%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. neg-mul-1 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 34.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 40.1%

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

   if 2.29999999999999995e-303 < 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 80.7

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 65.2%

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

   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 64.9%

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

Alternative 12: 52.9% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.3e-303) (fma 0.25 a 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 <= 2.3e-303) {
		tmp = fma(0.25, a, 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 <= 2.3e-303)
		tmp = fma(0.25, a, 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, 2.3e-303], N[(0.25 * a + 0.5), $MachinePrecision], 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 < 2.29999999999999995e-303

   1. Initial program 97.4%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. associate-*r/ N/A

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

      9. neg-mul-1 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 34.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 40.1%

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

   if 2.29999999999999995e-303 < 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 80.7

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

   5. Applied rewrites 80.7%

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

   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 55.2%

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

Alternative 13: 52.6% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.22:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, b, 1\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.22) 0.5 (/ 1.0 (* (fma 0.5 b 1.0) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.22) {
		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 <= 1.22)
		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, 1.22], 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 < 1.21999999999999997

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 71.2

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

   5. Applied rewrites 71.2%

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

   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 45.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 48.9%

       \[\leadsto 0.5 \]

   if 1.21999999999999997 < 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 45.8%

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

   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 45.8%

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

Alternative 14: 39.1% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. +-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 78.6

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

5. Applied rewrites 78.6%

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

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 34.6%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 37.1%

    \[\leadsto 0.5 \]
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 2024304 
(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))))