Quotient of sum of exps

Percentage Accurate: 98.9% → 98.8%

Time: 6.5s

Alternatives: 19

Speedup: 1.0×

• could not determine a ground truth

  1. a = 9.31476454356097e+183
  2. b = -7.775711133752296e+265

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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - a, e^{b}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05)
   (/ (exp a) (+ (exp a) 1.0))
   (/ 1.0 (fma (- 1.0 a) (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = 1.0 / fma((1.0 - a), exp(b), 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(1.0 / fma(Float64(1.0 - a), exp(b), 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.05], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 - a), $MachinePrecision] * N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 98.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 98.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 98.8

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

   7. Applied rewrites 98.8%

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

Alternative 2: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - a, 0.16666666666666666, \frac{\mathsf{fma}\left(-0.5, 1 - a, \frac{a - 1}{b}\right)}{-b}\right) \cdot b\right) \cdot b, b, 2 - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + 1}{2 + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.0)
   (/
    1.0
    (fma
     (*
      (*
       (fma
        (- 1.0 a)
        0.16666666666666666
        (/ (fma -0.5 (- 1.0 a) (/ (- a 1.0) b)) (- b)))
       b)
      b)
     b
     (- 2.0 a)))
   (/ (+ a 1.0) (+ 2.0 a))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.0) {
		tmp = 1.0 / fma(((fma((1.0 - a), 0.16666666666666666, (fma(-0.5, (1.0 - a), ((a - 1.0) / b)) / -b)) * b) * b), b, (2.0 - a));
	} else {
		tmp = (a + 1.0) / (2.0 + a);
	}
	return tmp;
}

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(N[(N[(N[(N[(1.0 - a), $MachinePrecision] * 0.16666666666666666 + N[(N[(-0.5 * N[(1.0 - a), $MachinePrecision] + N[(N[(a - 1.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * b + N[(2.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + 1.0), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. 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. lower-/.f64 N/A

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

      4. lower-/.f64 100.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 65.6

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

   7. Applied rewrites 65.6%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 53.4%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 80.0%

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

   if 0.0 < (/.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

      \[\leadsto \frac{e^{a}}{\color{blue}{1 + e^{a}}} \]
   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 72.1

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

   5. Applied rewrites 72.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 71.1%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 71.3

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

   11. Applied rewrites 71.3%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(\mathsf{fma}\left(1 - a, 0.16666666666666666, \frac{\mathsf{fma}\left(-0.5, 1 - a, \frac{a - 1}{b}\right)}{-b}\right) \cdot b\right) \cdot b, b, 2 - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + 1}{2 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.4999999999999983:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(\left(\left(1 - a\right) \cdot b\right) \cdot b\right) \cdot 0.16666666666666666, b, 2 - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + 1}{2 + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.4999999999999983)
   (/ 1.0 (fma (* (* (* (- 1.0 a) b) b) 0.16666666666666666) b (- 2.0 a)))
   (/ (+ a 1.0) (+ 2.0 a))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.4999999999999983) {
		tmp = 1.0 / fma(((((1.0 - a) * b) * b) * 0.16666666666666666), b, (2.0 - a));
	} else {
		tmp = (a + 1.0) / (2.0 + a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.4999999999999983)
		tmp = Float64(1.0 / fma(Float64(Float64(Float64(Float64(1.0 - a) * b) * b) * 0.16666666666666666), b, Float64(2.0 - a)));
	else
		tmp = Float64(Float64(a + 1.0) / Float64(2.0 + a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.4999999999999983], N[(1.0 / N[(N[(N[(N[(N[(1.0 - a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * b + N[(2.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + 1.0), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.499999999999998279

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 100.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 65.9

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

   7. Applied rewrites 65.9%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 54.0%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 54.0%

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

   if 0.499999999999998279 < (/.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

      \[\leadsto \frac{e^{a}}{\color{blue}{1 + e^{a}}} \]
   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 71.2

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 71.2%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 71.3

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

   11. Applied rewrites 71.3%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.4999999999999983:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(\left(\left(1 - a\right) \cdot b\right) \cdot b\right) \cdot 0.16666666666666666, b, 2 - a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + 1}{2 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.05:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + 1}{2 + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.05)
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))
   (/ (+ a 1.0) (+ 2.0 a))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.05) {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	} else {
		tmp = (a + 1.0) / (2.0 + a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.05)
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	else
		tmp = Float64(Float64(a + 1.0) / Float64(2.0 + a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.05], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a + 1.0), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.050000000000000003

   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 64.5

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

   5. Applied rewrites 64.5%

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

   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 49.9%

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

   if 0.050000000000000003 < (/.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

      \[\leadsto \frac{e^{a}}{\color{blue}{1 + e^{a}}} \]
   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 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 71.6%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 71.7

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

   11. Applied rewrites 71.7%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.05:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + 1}{2 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.05:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + 1}{2 + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.05)
   (/ 1.0 (fma (fma 0.5 b 1.0) b 2.0))
   (/ (+ a 1.0) (+ 2.0 a))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.05) {
		tmp = 1.0 / fma(fma(0.5, b, 1.0), b, 2.0);
	} else {
		tmp = (a + 1.0) / (2.0 + a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.05)
		tmp = Float64(1.0 / fma(fma(0.5, b, 1.0), b, 2.0));
	else
		tmp = Float64(Float64(a + 1.0) / Float64(2.0 + a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.05], N[(1.0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a + 1.0), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.050000000000000003

   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 64.5

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

   5. Applied rewrites 64.5%

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

   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 37.3%

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

   if 0.050000000000000003 < (/.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

      \[\leadsto \frac{e^{a}}{\color{blue}{1 + e^{a}}} \]
   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 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 71.6%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 71.7

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

   11. Applied rewrites 71.7%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.05:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + 1}{2 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 1.0× speedup

Math

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

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(1.0 / Float64(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[(1.0 / N[(N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision] / N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

   1. 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. lower-/.f64 N/A

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

   4. lower-/.f64 98.8

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 98.8

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

4. Applied rewrites 98.8%

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

5. Final simplification 98.8%

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

Alternative 7: 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 8: 98.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b + 1, e^{-a}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - a, e^{b}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05)
   (/ 1.0 (fma (+ b 1.0) (exp (- a)) 1.0))
   (/ 1.0 (fma (- 1.0 a) (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = 1.0 / fma((b + 1.0), exp(-a), 1.0);
	} else {
		tmp = 1.0 / fma((1.0 - a), exp(b), 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(1.0 / fma(Float64(b + 1.0), exp(Float64(-a)), 1.0));
	else
		tmp = Float64(1.0 / fma(Float64(1.0 - a), exp(b), 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.05], N[(1.0 / N[(N[(b + 1.0), $MachinePrecision] * N[Exp[(-a)], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 - a), $MachinePrecision] * N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 98.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 98.5

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{1}{\color{blue}{1 + \left(\frac{1}{e^{a}} + \frac{b}{e^{a}}\right)}} \]
   6. 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. lower-+.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 98.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 98.8

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

   7. Applied rewrites 98.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b + 1, e^{-a}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - a, e^{b}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - a, e^{b}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05) (/ (exp a) 2.0) (/ 1.0 (fma (- 1.0 a) (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 1.0 / fma((1.0 - a), exp(b), 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(1.0 / fma(Float64(1.0 - a), exp(b), 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.05], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(N[(1.0 - a), $MachinePrecision] * N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 98.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 98.8%

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

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 98.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 98.8

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

   7. Applied rewrites 98.8%

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

Alternative 10: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05) (/ (exp a) 2.0) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.05d0) then
        tmp = exp(a) / 2.0d0
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.05) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.05:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.05)
		tmp = exp(a) / 2.0;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.05], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 98.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 98.8%

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

   if 0.050000000000000003 < (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 98.7

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

   5. Applied rewrites 98.7%

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

Alternative 11: 63.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, b, \frac{b}{a}\right) \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), a, 1\right) - a, b, 2 - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ 1.0 (* (fma (+ (/ 2.0 (* b b)) 0.5) b 1.0) b))
   (/
    1.0
    (fma
     (- (fma (* (fma -1.0 b (/ b a)) (fma 0.16666666666666666 b 0.5)) a 1.0) a)
     b
     (- 2.0 a)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = 1.0 / (fma(((2.0 / (b * b)) + 0.5), b, 1.0) * b);
	} else {
		tmp = 1.0 / fma((fma((fma(-1.0, b, (b / a)) * fma(0.16666666666666666, b, 0.5)), a, 1.0) - a), b, (2.0 - a));
	}
	return tmp;
}

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], 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[(N[(N[(-1.0 * b + N[(b / a), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 * b + 0.5), $MachinePrecision]), $MachinePrecision] * a + 1.0), $MachinePrecision] - a), $MachinePrecision] * b + N[(2.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 33.5

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

   5. Applied rewrites 33.5%

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

   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 18.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 51.2%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 98.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 69.7%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 72.7%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, b, \frac{b}{a}\right) \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), a, 1\right) - a, b, 2 - a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (exp(b) <= 2.0) {
		tmp = (a + 1.0) / (2.0 + a);
	} 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 = (a + 1.0d0) / (2.0d0 + a)
    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 = (a + 1.0) / (2.0 + a);
	} else {
		tmp = 1.0 / ((b * b) * 0.5);
	}
	return tmp;
}

Python

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

Julia

function code(a, b)
	tmp = 0.0
	if (exp(b) <= 2.0)
		tmp = Float64(Float64(a + 1.0) / Float64(2.0 + a));
	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 = (a + 1.0) / (2.0 + a);
	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], N[(N[(a + 1.0), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision], 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 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 78.4%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 54.0

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

   11. Applied rewrites 54.0%

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

   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 57.0%

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

   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 57.0%

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

4. Final simplification 54.9%

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

Alternative 13: 77.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{+17}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, b, \frac{b}{a}\right) \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), a, 1\right) - a, b, 2 - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.95e+17)
   (/ (exp a) 2.0)
   (/
    1.0
    (fma
     (- (fma (* (fma -1.0 b (/ b a)) (fma 0.16666666666666666 b 0.5)) a 1.0) a)
     b
     (- 2.0 a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.95e+17) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 1.0 / fma((fma((fma(-1.0, b, (b / a)) * fma(0.16666666666666666, b, 0.5)), a, 1.0) - a), b, (2.0 - a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.95e+17)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(1.0 / fma(Float64(fma(Float64(fma(-1.0, b, Float64(b / a)) * fma(0.16666666666666666, b, 0.5)), a, 1.0) - a), b, Float64(2.0 - a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.95e+17], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(N[(-1.0 * b + N[(b / a), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 * b + 0.5), $MachinePrecision]), $MachinePrecision] * a + 1.0), $MachinePrecision] - a), $MachinePrecision] * b + N[(2.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.95e17

   1. Initial program 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 79.0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 78.1%

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

   if 1.95e17 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 100.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 82.5%

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

   11. Taylor expanded in a around inf

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

   13. Applied rewrites 90.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.95 \cdot 10^{+17}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1, b, \frac{b}{a}\right) \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right), a, 1\right) - a, b, 2 - a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -310:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot b\right) \cdot \left(1 - a\right), b, 1 - a\right), b, 2 - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -310.0)
   (/ 1.0 (* (fma (+ (/ 2.0 (* b b)) 0.5) b 1.0) b))
   (/
    1.0
    (fma
     (fma (* (* 0.16666666666666666 b) (- 1.0 a)) b (- 1.0 a))
     b
     (- 2.0 a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -310.0) {
		tmp = 1.0 / (fma(((2.0 / (b * b)) + 0.5), b, 1.0) * b);
	} else {
		tmp = 1.0 / fma(fma(((0.16666666666666666 * b) * (1.0 - a)), b, (1.0 - a)), b, (2.0 - a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -310.0)
		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(Float64(Float64(0.16666666666666666 * b) * Float64(1.0 - a)), b, Float64(1.0 - a)), b, Float64(2.0 - a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -310.0], 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[(N[(0.16666666666666666 * b), $MachinePrecision] * N[(1.0 - a), $MachinePrecision]), $MachinePrecision] * b + N[(1.0 - a), $MachinePrecision]), $MachinePrecision] * b + N[(2.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -310

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 33.5

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

   5. Applied rewrites 33.5%

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

   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 18.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 51.2%

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

   if -310 < a

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 98.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 69.7%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 69.7%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -310:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot b\right) \cdot \left(1 - a\right), b, 1 - a\right), b, 2 - a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -310:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(\left(\left(1 - a\right) \cdot b\right) \cdot b\right) \cdot 0.16666666666666666, b, 2 - a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -310.0)
   (/ 1.0 (* (fma (+ (/ 2.0 (* b b)) 0.5) b 1.0) b))
   (/ 1.0 (fma (* (* (* (- 1.0 a) b) b) 0.16666666666666666) b (- 2.0 a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -310.0) {
		tmp = 1.0 / (fma(((2.0 / (b * b)) + 0.5), b, 1.0) * b);
	} else {
		tmp = 1.0 / fma(((((1.0 - a) * b) * b) * 0.16666666666666666), b, (2.0 - a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -310.0)
		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(Float64(Float64(Float64(Float64(1.0 - a) * b) * b) * 0.16666666666666666), b, Float64(2.0 - a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -310.0], 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[(N[(N[(1.0 - a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * b + N[(2.0 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -310

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-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 33.5

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

   5. Applied rewrites 33.5%

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

   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 18.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 51.2%

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

   if -310 < a

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 98.9

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-rgt1-in N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 98.4

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

   7. Applied rewrites 98.4%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 69.7%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 69.7%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -310:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(\left(\left(1 - a\right) \cdot b\right) \cdot b\right) \cdot 0.16666666666666666, b, 2 - a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.6% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1800:\\ \;\;\;\;\frac{a + 1}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1800.0)
   (/ (+ a 1.0) (+ 2.0 a))
   (/ 1.0 (* (* (fma 0.16666666666666666 b 0.5) b) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1800.0) {
		tmp = (a + 1.0) / (2.0 + a);
	} else {
		tmp = 1.0 / ((fma(0.16666666666666666, b, 0.5) * b) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1800.0)
		tmp = Float64(Float64(a + 1.0) / Float64(2.0 + a));
	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, 1800.0], N[(N[(a + 1.0), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $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 < 1800

   1. Initial program 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 78.4%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 54.0

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

   11. Applied rewrites 54.0%

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

   if 1800 < 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 77.0%

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

   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 77.0%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1800:\\ \;\;\;\;\frac{a + 1}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 53.3% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1800:\\ \;\;\;\;\frac{a + 1}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, b, 1\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1800.0) (/ (+ a 1.0) (+ 2.0 a)) (/ 1.0 (* (fma 0.5 b 1.0) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1800.0) {
		tmp = (a + 1.0) / (2.0 + a);
	} else {
		tmp = 1.0 / (fma(0.5, b, 1.0) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1800.0)
		tmp = Float64(Float64(a + 1.0) / Float64(2.0 + a));
	else
		tmp = Float64(1.0 / Float64(fma(0.5, b, 1.0) * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1800.0], N[(N[(a + 1.0), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1800

   1. Initial program 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 78.4%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 54.0

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

   11. Applied rewrites 54.0%

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

   if 1800 < 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 57.0%

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

   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 57.0%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1800:\\ \;\;\;\;\frac{a + 1}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, b, 1\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 40.3% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (- 2.0 a)))

C

double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 - a)
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Python

def code(a, b):
	return 1.0 / (2.0 - a)

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 - a))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 - a);
end

Wolfram

code[a_, b_] := N[(1.0 / N[(2.0 - 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. lower-/.f64 N/A

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

   4. lower-/.f64 98.8

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. distribute-lft-neg-in N/A

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

   4. distribute-rgt1-in N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. unsub-neg N/A

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

   8. lower--.f64 N/A

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

   9. lower-exp.f64 81.8

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

7. Applied rewrites 81.8%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 38.4%

    \[\leadsto \frac{1}{2 - \color{blue}{a}} \]
11. Add Preprocessing

Alternative 19: 39.4% 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 81.3

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

5. Applied rewrites 81.3%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 37.8%

   \[\leadsto 0.5 \]
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 2024294 
(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))))