Quotient of sum of exps

Percentage Accurate: 98.9% → 98.4%

Time: 8.9s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. a = 2.1329901025108994e+302
  2. b = 1.4086477425659784e-198

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a) * 0.5;
	} 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) * 0.5d0
    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) * 0.5;
	} 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) * 0.5
	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) * 0.5);
	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) * 0.5;
	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] * 0.5), $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. Simplified 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 100.0%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      2. metadata-eval N/A

         \[\leadsto e^{a} \cdot \color{blue}{\frac{1}{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      4. exp-lowering-exp.f64 100.0

         \[\leadsto \color{blue}{e^{a}} \cdot 0.5 \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{e^{a} \cdot 0.5} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 98.9

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

   5. Simplified 98.9%

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

Alternative 2: 61.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.49999961:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot \left(0.16666666666666666 - \frac{\left(-0.5 + \frac{-1}{b}\right) + \frac{-2}{b \cdot b}}{b}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.49999961)
   (/
    1.0
    (*
     (* b b)
     (*
      b
      (- 0.16666666666666666 (/ (+ (+ -0.5 (/ -1.0 b)) (/ -2.0 (* b b))) b)))))
   (fma a 0.25 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.49999961) {
		tmp = 1.0 / ((b * b) * (b * (0.16666666666666666 - (((-0.5 + (-1.0 / b)) + (-2.0 / (b * b))) / b))));
	} else {
		tmp = fma(a, 0.25, 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.49999961)
		tmp = Float64(1.0 / Float64(Float64(b * b) * Float64(b * Float64(0.16666666666666666 - Float64(Float64(Float64(-0.5 + Float64(-1.0 / b)) + Float64(-2.0 / Float64(b * b))) / b)))));
	else
		tmp = fma(a, 0.25, 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.49999961], N[(1.0 / N[(N[(b * b), $MachinePrecision] * N[(b * N[(0.16666666666666666 - N[(N[(N[(-0.5 + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * 0.25 + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.499999609999999983

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 65.1

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

   5. Simplified 65.1%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 45.7

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

   8. Simplified 45.7%

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

   9. Taylor expanded in b around -inf

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

       1. mul-1-neg N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left({b}^{3} \cdot \left(-1 \cdot \frac{\frac{1}{2} + \left(\frac{1}{b} + \frac{2}{{b}^{2}}\right)}{b} - \frac{1}{6}\right)\right)}} \]

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. distribute-rgt-neg-in N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. *-commutative N/A

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

       11. distribute-rgt-neg-in N/A

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

   11. Simplified 52.1%

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

   if 0.499999609999999983 < (/.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. Simplified 65.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot \frac{1}{4}} + \frac{1}{2} \]

      3. accelerator-lowering-fma.f64 64.5

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

   8. Simplified 64.5%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. div-inv N/A

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

   2. flip3-+ N/A

      \[\leadsto e^{a} \cdot \frac{1}{\color{blue}{\frac{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}{e^{a} \cdot e^{a} + \left(e^{b} \cdot e^{b} - e^{a} \cdot e^{b}\right)}}} \]

   3. clear-num N/A

      \[\leadsto e^{a} \cdot \color{blue}{\frac{e^{a} \cdot e^{a} + \left(e^{b} \cdot e^{b} - e^{a} \cdot e^{b}\right)}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{e^{a} \cdot e^{a} + \left(e^{b} \cdot e^{b} - e^{a} \cdot e^{b}\right)}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}} \cdot e^{a}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{e^{a} \cdot e^{a} + \left(e^{b} \cdot e^{b} - e^{a} \cdot e^{b}\right)}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}} \cdot e^{a}} \]

4. Applied egg-rr 99.2%

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

5. Final simplification 99.2%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

Alternative 5: 78.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \mathsf{fma}\left(b, 0.5, 1\right)\\ \mathbf{if}\;b \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), -2\right)}{\mathsf{fma}\left(t\_0, t\_0, -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (fma b 0.5 1.0))))
   (if (<= b 1.6e+77)
     (* (exp a) 0.5)
     (if (<= b 2e+153)
       (/ (fma b (fma b 0.5 1.0) -2.0) (fma t_0 t_0 -4.0))
       (/ 2.0 (* b b))))))

C

double code(double a, double b) {
	double t_0 = b * fma(b, 0.5, 1.0);
	double tmp;
	if (b <= 1.6e+77) {
		tmp = exp(a) * 0.5;
	} else if (b <= 2e+153) {
		tmp = fma(b, fma(b, 0.5, 1.0), -2.0) / fma(t_0, t_0, -4.0);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(b * fma(b, 0.5, 1.0))
	tmp = 0.0
	if (b <= 1.6e+77)
		tmp = Float64(exp(a) * 0.5);
	elseif (b <= 2e+153)
		tmp = Float64(fma(b, fma(b, 0.5, 1.0), -2.0) / fma(t_0, t_0, -4.0));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.6e+77], N[(N[Exp[a], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[b, 2e+153], N[(N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + -2.0), $MachinePrecision] / N[(t$95$0 * t$95$0 + -4.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.6000000000000001e77

   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. Simplified 72.4%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 71.4%

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

      1. div-inv N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      2. metadata-eval N/A

         \[\leadsto e^{a} \cdot \color{blue}{\frac{1}{2}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{a} \cdot \frac{1}{2}} \]

      4. exp-lowering-exp.f64 71.4

         \[\leadsto \color{blue}{e^{a}} \cdot 0.5 \]

   10. Applied egg-rr 71.4%

       \[\leadsto \color{blue}{e^{a} \cdot 0.5} \]

   if 1.6000000000000001e77 < b < 2e153

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 6.5

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

   8. Simplified 6.5%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}} \]
   9. Step-by-step derivation

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. sub-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2}, 1\right), -2\right)}{\mathsf{fma}\left(b \cdot \mathsf{fma}\left(b, \frac{1}{2}, 1\right), b \cdot \mathsf{fma}\left(b, \frac{1}{2}, 1\right), \mathsf{neg}\left(\color{blue}{4}\right)\right)} \]

      15. metadata-eval 100.0

          \[\leadsto \frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), -2\right)}{\mathsf{fma}\left(b \cdot \mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, 0.5, 1\right), \color{blue}{-4}\right)} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), -2\right)}{\mathsf{fma}\left(b \cdot \mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, 0.5, 1\right), -4\right)}} \]

   if 2e153 < 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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

       3. *-lowering-*.f64 100.0

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

   11. Simplified 100.0%

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

Alternative 6: 63.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right)\\ \mathbf{if}\;b \leq 26000000000000:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, t\_0, -2\right)}{\mathsf{fma}\left(t\_0, \left(b \cdot b\right) \cdot t\_0, -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma b (fma b 0.16666666666666666 0.5) 1.0)))
   (if (<= b 26000000000000.0)
     (fma a 0.25 0.5)
     (if (<= b 1.65e+38)
       (* -0.020833333333333332 (* a (* a a)))
       (if (<= b 5e+102)
         (/ (fma b t_0 -2.0) (fma t_0 (* (* b b) t_0) -4.0))
         (/ 6.0 (* b (* b b))))))))

C

double code(double a, double b) {
	double t_0 = fma(b, fma(b, 0.16666666666666666, 0.5), 1.0);
	double tmp;
	if (b <= 26000000000000.0) {
		tmp = fma(a, 0.25, 0.5);
	} else if (b <= 1.65e+38) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else if (b <= 5e+102) {
		tmp = fma(b, t_0, -2.0) / fma(t_0, ((b * b) * t_0), -4.0);
	} else {
		tmp = 6.0 / (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(b, fma(b, 0.16666666666666666, 0.5), 1.0)
	tmp = 0.0
	if (b <= 26000000000000.0)
		tmp = fma(a, 0.25, 0.5);
	elseif (b <= 1.65e+38)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	elseif (b <= 5e+102)
		tmp = Float64(fma(b, t_0, -2.0) / fma(t_0, Float64(Float64(b * b) * t_0), -4.0));
	else
		tmp = Float64(6.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[b, 26000000000000.0], N[(a * 0.25 + 0.5), $MachinePrecision], If[LessEqual[b, 1.65e+38], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+102], N[(N[(b * t$95$0 + -2.0), $MachinePrecision] / N[(t$95$0 * N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision] + -4.0), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 2.6e13

   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. Simplified 73.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot \frac{1}{4}} + \frac{1}{2} \]

      3. accelerator-lowering-fma.f64 47.9

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

   8. Simplified 47.9%

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

   if 2.6e13 < b < 1.65e38

   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. Simplified 32.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

       6. *-lowering-*.f64 70.5

          \[\leadsto -0.020833333333333332 \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   11. Simplified 70.5%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]

   if 1.65e38 < b < 5e102

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 7.5

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

   8. Simplified 7.5%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}} \]
   9. Step-by-step derivation

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. sub-neg N/A

         \[\leadsto \frac{\color{blue}{b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right) + \left(\mathsf{neg}\left(2\right)\right)}}{\left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right)\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\right)\right) - 2 \cdot 2} \]

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

   10. Applied egg-rr 85.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right) \cdot \left(b \cdot b\right), -4\right)}} \]

   if 5e102 < 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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{6}{{b}^{3}}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{6}{{b}^{3}}} \]

       2. cube-mult N/A

          \[\leadsto \frac{6}{\color{blue}{b \cdot \left(b \cdot b\right)}} \]

       3. unpow2 N/A

          \[\leadsto \frac{6}{b \cdot \color{blue}{{b}^{2}}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{6}{\color{blue}{b \cdot {b}^{2}}} \]

       5. unpow2 N/A

          \[\leadsto \frac{6}{b \cdot \color{blue}{\left(b \cdot b\right)}} \]

       6. *-lowering-*.f64 100.0

          \[\leadsto \frac{6}{b \cdot \color{blue}{\left(b \cdot b\right)}} \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{6}{b \cdot \left(b \cdot b\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 26000000000000:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), -2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)\\ \mathbf{if}\;b \leq 26000000000000:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \frac{\mathsf{fma}\left(t\_0, t\_0, -1\right)}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), -1\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (fma b 0.16666666666666666 0.5))))
   (if (<= b 26000000000000.0)
     (fma a 0.25 0.5)
     (if (<= b 1.65e+38)
       (* -0.020833333333333332 (* a (* a a)))
       (if (<= b 2e+153)
         (/
          1.0
          (fma
           b
           (/ (fma t_0 t_0 -1.0) (fma b (fma b 0.16666666666666666 0.5) -1.0))
           2.0))
         (/ 2.0 (* b b)))))))

C

double code(double a, double b) {
	double t_0 = b * fma(b, 0.16666666666666666, 0.5);
	double tmp;
	if (b <= 26000000000000.0) {
		tmp = fma(a, 0.25, 0.5);
	} else if (b <= 1.65e+38) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else if (b <= 2e+153) {
		tmp = 1.0 / fma(b, (fma(t_0, t_0, -1.0) / fma(b, fma(b, 0.16666666666666666, 0.5), -1.0)), 2.0);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(b * fma(b, 0.16666666666666666, 0.5))
	tmp = 0.0
	if (b <= 26000000000000.0)
		tmp = fma(a, 0.25, 0.5);
	elseif (b <= 1.65e+38)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	elseif (b <= 2e+153)
		tmp = Float64(1.0 / fma(b, Float64(fma(t_0, t_0, -1.0) / fma(b, fma(b, 0.16666666666666666, 0.5), -1.0)), 2.0));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 26000000000000.0], N[(a * 0.25 + 0.5), $MachinePrecision], If[LessEqual[b, 1.65e+38], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+153], N[(1.0 / N[(b * N[(N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision] / N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 2.6e13

   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. Simplified 73.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot \frac{1}{4}} + \frac{1}{2} \]

      3. accelerator-lowering-fma.f64 47.9

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

   8. Simplified 47.9%

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

   if 2.6e13 < b < 1.65e38

   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. Simplified 32.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

       6. *-lowering-*.f64 70.5

          \[\leadsto -0.020833333333333332 \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   11. Simplified 70.5%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]

   if 1.65e38 < b < 2e153

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 62.4

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

   8. Simplified 62.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}} \]
   9. Step-by-step derivation

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. accelerator-lowering-fma.f64 85.2

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

   10. Applied egg-rr 85.2%

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

   if 2e153 < 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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

       3. *-lowering-*.f64 100.0

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

   11. Simplified 100.0%

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

Alternative 8: 61.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \mathsf{fma}\left(b, 0.5, 1\right)\\ \mathbf{if}\;b \leq 26000000000000:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), -2\right)}{\mathsf{fma}\left(t\_0, t\_0, -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (fma b 0.5 1.0))))
   (if (<= b 26000000000000.0)
     (fma a 0.25 0.5)
     (if (<= b 1.65e+38)
       (* -0.020833333333333332 (* a (* a a)))
       (if (<= b 2e+153)
         (/ (fma b (fma b 0.5 1.0) -2.0) (fma t_0 t_0 -4.0))
         (/ 2.0 (* b b)))))))

C

double code(double a, double b) {
	double t_0 = b * fma(b, 0.5, 1.0);
	double tmp;
	if (b <= 26000000000000.0) {
		tmp = fma(a, 0.25, 0.5);
	} else if (b <= 1.65e+38) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else if (b <= 2e+153) {
		tmp = fma(b, fma(b, 0.5, 1.0), -2.0) / fma(t_0, t_0, -4.0);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(b * fma(b, 0.5, 1.0))
	tmp = 0.0
	if (b <= 26000000000000.0)
		tmp = fma(a, 0.25, 0.5);
	elseif (b <= 1.65e+38)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	elseif (b <= 2e+153)
		tmp = Float64(fma(b, fma(b, 0.5, 1.0), -2.0) / fma(t_0, t_0, -4.0));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 26000000000000.0], N[(a * 0.25 + 0.5), $MachinePrecision], If[LessEqual[b, 1.65e+38], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+153], N[(N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + -2.0), $MachinePrecision] / N[(t$95$0 * t$95$0 + -4.0), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 2.6e13

   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. Simplified 73.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot \frac{1}{4}} + \frac{1}{2} \]

      3. accelerator-lowering-fma.f64 47.9

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

   8. Simplified 47.9%

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

   if 2.6e13 < b < 1.65e38

   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. Simplified 32.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

       6. *-lowering-*.f64 70.5

          \[\leadsto -0.020833333333333332 \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   11. Simplified 70.5%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]

   if 1.65e38 < b < 2e153

   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 6.1

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

   8. Simplified 6.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 2\right)}} \]
   9. Step-by-step derivation

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. sub-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \frac{1}{2}, 1\right), -2\right)}{\mathsf{fma}\left(b \cdot \mathsf{fma}\left(b, \frac{1}{2}, 1\right), b \cdot \mathsf{fma}\left(b, \frac{1}{2}, 1\right), \mathsf{neg}\left(\color{blue}{4}\right)\right)} \]

      15. metadata-eval 85.0

          \[\leadsto \frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), -2\right)}{\mathsf{fma}\left(b \cdot \mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, 0.5, 1\right), \color{blue}{-4}\right)} \]

   10. Applied egg-rr 85.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), -2\right)}{\mathsf{fma}\left(b \cdot \mathsf{fma}\left(b, 0.5, 1\right), b \cdot \mathsf{fma}\left(b, 0.5, 1\right), -4\right)}} \]

   if 2e153 < 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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

       3. *-lowering-*.f64 100.0

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

   11. Simplified 100.0%

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

Alternative 9: 61.2% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 26000000000000:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{+101}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 26000000000000.0)
   (fma a 0.25 0.5)
   (if (<= b 2.55e+101)
     (* -0.020833333333333332 (* a (* a a)))
     (/ 6.0 (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 26000000000000.0) {
		tmp = fma(a, 0.25, 0.5);
	} else if (b <= 2.55e+101) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = 6.0 / (b * (b * b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 26000000000000.0)
		tmp = fma(a, 0.25, 0.5);
	elseif (b <= 2.55e+101)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	else
		tmp = Float64(6.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 26000000000000.0], N[(a * 0.25 + 0.5), $MachinePrecision], If[LessEqual[b, 2.55e+101], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.6e13

   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. Simplified 73.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot \frac{1}{4}} + \frac{1}{2} \]

      3. accelerator-lowering-fma.f64 47.9

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

   8. Simplified 47.9%

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

   if 2.6e13 < b < 2.54999999999999997e101

   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. Simplified 49.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 2.5

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

   8. Simplified 2.5%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

       6. *-lowering-*.f64 48.7

          \[\leadsto -0.020833333333333332 \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   11. Simplified 48.7%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]

   if 2.54999999999999997e101 < 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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 96.8

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

   8. Simplified 96.8%

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

   9. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{6}{{b}^{3}}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{6}{{b}^{3}}} \]

       2. cube-mult N/A

          \[\leadsto \frac{6}{\color{blue}{b \cdot \left(b \cdot b\right)}} \]

       3. unpow2 N/A

          \[\leadsto \frac{6}{b \cdot \color{blue}{{b}^{2}}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{6}{\color{blue}{b \cdot {b}^{2}}} \]

       5. unpow2 N/A

          \[\leadsto \frac{6}{b \cdot \color{blue}{\left(b \cdot b\right)}} \]

       6. *-lowering-*.f64 96.8

          \[\leadsto \frac{6}{b \cdot \color{blue}{\left(b \cdot b\right)}} \]

   11. Simplified 96.8%

       \[\leadsto \color{blue}{\frac{6}{b \cdot \left(b \cdot b\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 58.8% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 26000000000000:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 26000000000000.0)
   (fma a 0.25 0.5)
   (if (<= b 1.4e+154)
     (* -0.020833333333333332 (* a (* a a)))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 26000000000000.0) {
		tmp = fma(a, 0.25, 0.5);
	} else if (b <= 1.4e+154) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 26000000000000.0)
		tmp = fma(a, 0.25, 0.5);
	elseif (b <= 1.4e+154)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 26000000000000.0], N[(a * 0.25 + 0.5), $MachinePrecision], If[LessEqual[b, 1.4e+154], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.6e13

   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. Simplified 73.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot \frac{1}{4}} + \frac{1}{2} \]

      3. accelerator-lowering-fma.f64 47.9

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

   8. Simplified 47.9%

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

   if 2.6e13 < b < 1.4e154

   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. Simplified 35.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

       6. *-lowering-*.f64 53.6

          \[\leadsto -0.020833333333333332 \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   11. Simplified 53.6%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]

   if 1.4e154 < 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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. exp-lowering-exp.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{2}{{b}^{2}}} \]

       2. unpow2 N/A

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

       3. *-lowering-*.f64 100.0

          \[\leadsto \frac{2}{\color{blue}{b \cdot b}} \]

   11. Simplified 100.0%

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

Alternative 11: 50.9% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 26000000000000:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 26000000000000.0)
   (fma a 0.25 0.5)
   (* -0.020833333333333332 (* a (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 26000000000000.0) {
		tmp = fma(a, 0.25, 0.5);
	} else {
		tmp = -0.020833333333333332 * (a * (a * a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 26000000000000.0)
		tmp = fma(a, 0.25, 0.5);
	else
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 26000000000000.0], N[(a * 0.25 + 0.5), $MachinePrecision], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6e13

   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. Simplified 73.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{4} \cdot a + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot \frac{1}{4}} + \frac{1}{2} \]

      3. accelerator-lowering-fma.f64 47.9

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

   8. Simplified 47.9%

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

   if 2.6e13 < b

   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. Simplified 37.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right) + \frac{1}{2}} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}, \frac{1}{2}\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{-1}{48} \cdot {a}^{2} + \frac{1}{4}}, \frac{1}{2}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(\frac{-1}{48}, {a}^{2}, \frac{1}{4}\right)}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{-1}{48}, \color{blue}{a \cdot a}, \frac{1}{4}\right), \frac{1}{2}\right) \]

      6. *-lowering-*.f64 2.6

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

   8. Simplified 2.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{48} \cdot {a}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{-1}{48} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{48} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

       6. *-lowering-*.f64 47.5

          \[\leadsto -0.020833333333333332 \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   11. Simplified 47.5%

       \[\leadsto \color{blue}{-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 40.0% 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. /-lowering-/.f64 N/A

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

   2. +-lowering-+.f64 N/A

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

   3. exp-lowering-exp.f64 82.2

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

5. Simplified 82.2%

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

6. Taylor expanded in b around 0

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

8. Simplified 34.5%

   \[\leadsto \color{blue}{0.5} \]
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 2024196 
(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))))