Quotient of sum of exps

Percentage Accurate: 98.6% → 99.4%

Time: 6.2s

Alternatives: 17

Speedup: 1.3×

• could not determine a ground truth

  1. a = 1.0437838731202357e+132
  2. b = 5.887458696797108e-89

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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right)\\ \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, a, 1\right)}{\mathsf{fma}\left(t\_0, a, 1 + e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma 0.16666666666666666 a 0.5) a 1.0)))
   (if (<= (exp a) 0.0)
     (/ (exp a) (+ 1.0 1.0))
     (/ (fma t_0 a 1.0) (fma t_0 a (+ 1.0 (exp b)))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(0.16666666666666666, a, 0.5), a, 1.0);
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a) / (1.0 + 1.0);
	} else {
		tmp = fma(t_0, a, 1.0) / fma(t_0, a, (1.0 + exp(b)));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(0.16666666666666666, a, 0.5), a, 1.0)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(exp(a) / Float64(1.0 + 1.0));
	else
		tmp = Float64(fma(t_0, a, 1.0) / fma(t_0, a, Float64(1.0 + exp(b))));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision]}, If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * a + 1.0), $MachinePrecision] / N[(t$95$0 * a + N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. lft-mult-inverse N/A

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

      9. associate-*l* N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-rgt-in N/A

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

      13. associate-*l* N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 99.4

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

   8. Applied rewrites 99.4%

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

4. Final simplification 99.6%

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

Alternative 2: 57.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.4:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(b \cdot b\right) \cdot 0.16666666666666666, b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.4)
   (/ 1.0 (fma (* (* b b) 0.16666666666666666) b 2.0))
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.4) {
		tmp = 1.0 / fma(((b * b) * 0.16666666666666666), b, 2.0);
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.4)
		tmp = Float64(1.0 / fma(Float64(Float64(b * b) * 0.16666666666666666), b, 2.0));
	else
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	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], 0.4], N[(1.0 / N[(N[(N[(b * b), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.40000000000000002

   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 59.4

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

   5. Applied rewrites 59.4%

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

   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 41.3%

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

   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 41.2%

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

   if 0.40000000000000002 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 67.8

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

   8. Applied rewrites 67.8%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 68.8

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

   11. Applied rewrites 68.8%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.4:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(b \cdot b\right) \cdot 0.16666666666666666, b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.4:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.4)
   (/ 1.0 (fma (fma 0.5 b 1.0) b 2.0))
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.4) {
		tmp = 1.0 / fma(fma(0.5, b, 1.0), b, 2.0);
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.4)
		tmp = Float64(1.0 / fma(fma(0.5, b, 1.0), b, 2.0));
	else
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	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], 0.4], N[(1.0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.40000000000000002

   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 59.4

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

   5. Applied rewrites 59.4%

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

   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 29.9%

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

   if 0.40000000000000002 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 67.8

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

   8. Applied rewrites 67.8%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 68.8

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

   11. Applied rewrites 68.8%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.4:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% 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 99.2%

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

3. Final simplification 99.2%

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

Alternative 5: 99.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-exp.f64 96.9

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.2

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

   8. Applied rewrites 99.2%

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

4. Final simplification 99.5%

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

Alternative 6: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

4. Final simplification 98.2%

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

Alternative 7: 93.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right) \cdot a\right) \cdot a}\\ \mathbf{elif}\;a \leq -105000000000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.05e+113)
   (/ 1.0 (* (* (fma 0.16666666666666666 a 0.5) a) a))
   (if (<= a -105000000000.0)
     (/ 1.0 (* (fma (+ (/ 2.0 (* b b)) 0.5) b 1.0) b))
     (/ 1.0 (+ 1.0 (exp b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.05e+113) {
		tmp = 1.0 / ((fma(0.16666666666666666, a, 0.5) * a) * a);
	} else if (a <= -105000000000.0) {
		tmp = 1.0 / (fma(((2.0 / (b * b)) + 0.5), b, 1.0) * b);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.05e+113)
		tmp = Float64(1.0 / Float64(Float64(fma(0.16666666666666666, a, 0.5) * a) * a));
	elseif (a <= -105000000000.0)
		tmp = Float64(1.0 / Float64(fma(Float64(Float64(2.0 / Float64(b * b)) + 0.5), b, 1.0) * b));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.05e+113], N[(1.0 / N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -105000000000.0], 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[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.04999999999999996e113

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. lft-mult-inverse N/A

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

      9. associate-*l* N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-rgt-in N/A

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

      13. associate-*l* N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 0.0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 0.0%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 100.0%

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

   if -2.04999999999999996e113 < a < -1.05e11

   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 28.2

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

   5. Applied rewrites 28.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 21.2%

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

   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 57.0%

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

   if -1.05e11 < a

   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 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right) \cdot a\right) \cdot a}\\ \mathbf{elif}\;a \leq -105000000000:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right)\\ \mathbf{if}\;a \leq -2.05 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right) \cdot a\right) \cdot a}\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b}\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, a, 1\right)}{\mathsf{fma}\left(t\_0, a, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma 0.16666666666666666 a 0.5) a 1.0)))
   (if (<= a -2.05e+113)
     (/ 1.0 (* (* (fma 0.16666666666666666 a 0.5) a) a))
     (if (<= a -1.45e+19)
       (/ 1.0 (* (fma (+ (/ 2.0 (* b b)) 0.5) b 1.0) b))
       (if (<= a 5.8e-58)
         (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))
         (/ (fma t_0 a 1.0) (fma t_0 a 2.0)))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(0.16666666666666666, a, 0.5), a, 1.0);
	double tmp;
	if (a <= -2.05e+113) {
		tmp = 1.0 / ((fma(0.16666666666666666, a, 0.5) * a) * a);
	} else if (a <= -1.45e+19) {
		tmp = 1.0 / (fma(((2.0 / (b * b)) + 0.5), b, 1.0) * b);
	} else if (a <= 5.8e-58) {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	} else {
		tmp = fma(t_0, a, 1.0) / fma(t_0, a, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(0.16666666666666666, a, 0.5), a, 1.0)
	tmp = 0.0
	if (a <= -2.05e+113)
		tmp = Float64(1.0 / Float64(Float64(fma(0.16666666666666666, a, 0.5) * a) * a));
	elseif (a <= -1.45e+19)
		tmp = Float64(1.0 / Float64(fma(Float64(Float64(2.0 / Float64(b * b)) + 0.5), b, 1.0) * b));
	elseif (a <= 5.8e-58)
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	else
		tmp = Float64(fma(t_0, a, 1.0) / fma(t_0, a, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision]}, If[LessEqual[a, -2.05e+113], N[(1.0 / N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -1.45e+19], N[(1.0 / N[(N[(N[(N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.8e-58], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * a + 1.0), $MachinePrecision] / N[(t$95$0 * a + 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.04999999999999996e113

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. lft-mult-inverse N/A

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

      9. associate-*l* N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-rgt-in N/A

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

      13. associate-*l* N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 0.0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 0.0%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 100.0%

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

   if -2.04999999999999996e113 < a < -1.45e19

   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 31.4

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

   5. Applied rewrites 31.4%

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

   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 23.4%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 63.8%

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

   if -1.45e19 < a < 5.7999999999999998e-58

   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 96.3

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

   5. Applied rewrites 96.3%

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

   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 64.2%

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

   if 5.7999999999999998e-58 < a

   1. Initial program 88.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. lft-mult-inverse N/A

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

      9. associate-*l* N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-rgt-in N/A

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

      13. associate-*l* N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 72.9%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right) \cdot a\right) \cdot a}\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b}\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right) \cdot a\right) \cdot a}\\ \mathbf{elif}\;a \leq -8.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.05e+113)
   (/ 1.0 (* (* (fma 0.16666666666666666 a 0.5) a) a))
   (if (<= a -8.8e+17)
     (/ 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 <= -2.05e+113) {
		tmp = 1.0 / ((fma(0.16666666666666666, a, 0.5) * a) * a);
	} else if (a <= -8.8e+17) {
		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 <= -2.05e+113)
		tmp = Float64(1.0 / Float64(Float64(fma(0.16666666666666666, a, 0.5) * a) * a));
	elseif (a <= -8.8e+17)
		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, -2.05e+113], N[(1.0 / N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -8.8e+17], 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 3 regimes

2. if a < -2.04999999999999996e113

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. lft-mult-inverse N/A

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

      9. associate-*l* N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-rgt-in N/A

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

      13. associate-*l* N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 0.0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 0.0%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 100.0%

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

   if -2.04999999999999996e113 < a < -8.8e17

   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 30.3

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

   5. Applied rewrites 30.3%

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

   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 22.6%

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

   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 61.4%

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

   if -8.8e17 < 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 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 63.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right) \cdot a\right) \cdot a}\\ \mathbf{elif}\;a \leq -8.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{2}{b \cdot b} + 0.5, b, 1\right) \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right) \cdot a\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.8e+102)
   (/ 1.0 (* (* (fma 0.16666666666666666 a 0.5) a) a))
   (/ 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 <= -3.8e+102) {
		tmp = 1.0 / ((fma(0.16666666666666666, a, 0.5) * a) * a);
	} 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 <= -3.8e+102)
		tmp = Float64(1.0 / Float64(Float64(fma(0.16666666666666666, a, 0.5) * a) * a));
	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, -3.8e+102], N[(1.0 / N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a), $MachinePrecision] * a), $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 < -3.79999999999999979e102

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 76.5

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. lft-mult-inverse N/A

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

      9. associate-*l* N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-rgt-in N/A

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

      13. associate-*l* N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 0.0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 0.0%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 100.0%

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

   if -3.79999999999999979e102 < a

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 88.3

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

   5. Applied rewrites 88.3%

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

   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 58.3%

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

Alternative 11: 68.3% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right) \cdot a\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(b \cdot b\right) \cdot 0.16666666666666666, b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.8e+102)
   (/ 1.0 (* (* (fma 0.16666666666666666 a 0.5) a) a))
   (/ 1.0 (fma (* (* b b) 0.16666666666666666) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.8e+102) {
		tmp = 1.0 / ((fma(0.16666666666666666, a, 0.5) * a) * a);
	} else {
		tmp = 1.0 / fma(((b * b) * 0.16666666666666666), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.8e+102)
		tmp = Float64(1.0 / Float64(Float64(fma(0.16666666666666666, a, 0.5) * a) * a));
	else
		tmp = Float64(1.0 / fma(Float64(Float64(b * b) * 0.16666666666666666), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.8e+102], N[(1.0 / N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(b * b), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.79999999999999979e102

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 76.5

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. lft-mult-inverse N/A

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

      9. associate-*l* N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-rgt-in N/A

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

      13. associate-*l* N/A

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

      14. lft-mult-inverse N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 0.0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 0.0%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 100.0%

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

   if -3.79999999999999979e102 < a

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 88.3

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

   5. Applied rewrites 88.3%

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

   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 58.3%

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

   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 58.1%

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

Alternative 12: 57.8% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.3:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.3)
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))
   (/ 1.0 (* (* (fma 0.16666666666666666 b 0.5) b) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.3) {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	} else {
		tmp = 1.0 / ((fma(0.16666666666666666, b, 0.5) * b) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.3)
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	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, 7.3], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $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 < 7.29999999999999982

   1. Initial program 98.9%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 76.6

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

   8. Applied rewrites 76.6%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 50.1

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

   11. Applied rewrites 50.1%

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

   if 7.29999999999999982 < 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 69.4%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 69.4%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.3:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.1% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.3:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, 0.5, 1\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.3) (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0)) (/ 1.0 (* (fma b 0.5 1.0) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.3) {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	} else {
		tmp = 1.0 / (fma(b, 0.5, 1.0) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.3)
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	else
		tmp = Float64(1.0 / Float64(fma(b, 0.5, 1.0) * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.3], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b * 0.5 + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.29999999999999982

   1. Initial program 98.9%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 76.6

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

   8. Applied rewrites 76.6%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 50.1

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

   11. Applied rewrites 50.1%

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

   if 7.29999999999999982 < 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 49.7%

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

   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 49.7%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.3:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, 0.5, 1\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.3:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.3) (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0)) (/ 1.0 (* (* b b) 0.5))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.3) {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	} 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 (b <= 7.3d0) then
        tmp = (1.0d0 + a) / ((1.0d0 + a) + 1.0d0)
    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 (b <= 7.3) {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	} else {
		tmp = 1.0 / ((b * b) * 0.5);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7.3:
		tmp = (1.0 + a) / ((1.0 + a) + 1.0)
	else:
		tmp = 1.0 / ((b * b) * 0.5)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.3)
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	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 (b <= 7.3)
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	else
		tmp = 1.0 / ((b * b) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.3], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.29999999999999982

   1. Initial program 98.9%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 76.6

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

   8. Applied rewrites 76.6%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 50.1

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

   11. Applied rewrites 50.1%

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

   if 7.29999999999999982 < 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 49.7%

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

   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 49.7%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.3:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 40.1% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 + a}{\left(1 + a\right) + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return (1.0 + a) / ((1.0 + a) + 1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 66.8%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 65.7

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

8. Applied rewrites 65.7%

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

9. Taylor expanded in a around 0

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

    1. lower-+.f64 37.1

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

11. Applied rewrites 37.1%

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

12. Final simplification 37.1%

    \[\leadsto \frac{1 + a}{\left(1 + a\right) + 1} \]
13. Add Preprocessing

Alternative 16: 39.9% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + a\right) + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 1.0 / ((1.0 + a) + 1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 66.8%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 65.7

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

8. Applied rewrites 65.7%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 36.5%

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

12. Final simplification 36.5%

    \[\leadsto \frac{1}{\left(1 + a\right) + 1} \]
13. Add Preprocessing

Alternative 17: 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 99.2%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 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

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

8. Applied rewrites 36.1%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024248 
(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))))