Quotient of sum of exps

Percentage Accurate: 98.9% → 98.4%

Time: 9.1s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. a = 8.871682802632695e+87
  2. b = -1.4922260749245836e-174

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

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = exp(a) * 0.5d0
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = Math.exp(a) * 0.5;
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	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 / (math.exp(b) + 1.0)
	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(exp(b) + 1.0));
	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 / (exp(b) + 1.0);
	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[(N[Exp[b], $MachinePrecision] + 1.0), $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. lift-exp.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.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 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 99.5

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

   5. Simplified 99.5%

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

4. Final simplification 99.6%

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

Alternative 2: 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 99.6%

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

Alternative 3: 57.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 2.0) 0.5 (/ 6.0 (* b (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(b) <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 6.0 / (b * (b * 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(b) <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = 6.0d0 / (b * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(b) <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 6.0 / (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(b) <= 2.0:
		tmp = 0.5
	else:
		tmp = 6.0 / (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(b) <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(6.0 / Float64(b * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(b) <= 2.0)
		tmp = 0.5;
	else
		tmp = 6.0 / (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[b], $MachinePrecision], 2.0], 0.5, N[(6.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 2

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 75.0

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

   5. Simplified 75.0%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 51.6%

      \[\leadsto \color{blue}{0.5} \]

   if 2 < (exp.f64 b)

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-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. lower-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. lower-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. lower-fma.f64 66.8

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

   8. Simplified 66.8%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.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. lower-*.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. lower-*.f64 66.8

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

   11. Simplified 66.8%

       \[\leadsto \color{blue}{\frac{6}{b \cdot \left(b \cdot b\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 79.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.4 \cdot 10^{+52}:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b \cdot \left(b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)\right), -1\right)}{-1}, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5.4e+52)
   (* (exp a) 0.5)
   (/
    1.0
    (fma
     b
     (/
      (fma
       (fma b 0.16666666666666666 0.5)
       (* b (* b (fma b 0.16666666666666666 0.5)))
       -1.0)
      -1.0)
     2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5.4e+52) {
		tmp = exp(a) * 0.5;
	} else {
		tmp = 1.0 / fma(b, (fma(fma(b, 0.16666666666666666, 0.5), (b * (b * fma(b, 0.16666666666666666, 0.5))), -1.0) / -1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5.4e+52)
		tmp = Float64(exp(a) * 0.5);
	else
		tmp = Float64(1.0 / fma(b, Float64(fma(fma(b, 0.16666666666666666, 0.5), Float64(b * Float64(b * fma(b, 0.16666666666666666, 0.5))), -1.0) / -1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5.4e+52], N[(N[Exp[a], $MachinePrecision] * 0.5), $MachinePrecision], N[(1.0 / N[(b * N[(N[(N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.4e52

   1. Initial program 99.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 74.2%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 73.6%

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

      1. lift-exp.f64 N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 73.6

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

   10. Applied egg-rr 73.6%

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

   if 5.4e52 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-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. lower-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. lower-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. lower-fma.f64 76.6

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

   8. Simplified 76.6%

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

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 27.4

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

   10. Applied egg-rr 27.4%

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

   11. Taylor expanded in b around 0

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

   13. Simplified 97.2%

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

Alternative 5: 62.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{+34}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \frac{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), b \cdot \left(b \cdot \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right)\right), -1\right)}{-1}, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.15e+34)
   (* 0.020833333333333332 (* b (* b b)))
   (/
    1.0
    (fma
     b
     (/
      (fma
       (fma b 0.16666666666666666 0.5)
       (* b (* b (fma b 0.16666666666666666 0.5)))
       -1.0)
      -1.0)
     2.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.15e+34) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / fma(b, (fma(fma(b, 0.16666666666666666, 0.5), (b * (b * fma(b, 0.16666666666666666, 0.5))), -1.0) / -1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.15e+34)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(1.0 / fma(b, Float64(fma(fma(b, 0.16666666666666666, 0.5), Float64(b * Float64(b * fma(b, 0.16666666666666666, 0.5))), -1.0) / -1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.15e+34], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / -1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1499999999999999e34

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 30.6

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

   5. Simplified 30.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 48.4

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

   11. Simplified 48.4%

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

   if -1.1499999999999999e34 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 98.0

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

   5. Simplified 98.0%

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

   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. lower-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. lower-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. lower-fma.f64 61.7

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

   8. Simplified 61.7%

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

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 50.3

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

   10. Applied egg-rr 50.3%

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

   11. Taylor expanded in b around 0

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

   13. Simplified 68.2%

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

Alternative 6: 60.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+30}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e+30)
   (* 0.020833333333333332 (* b (* b b)))
   (/ 1.0 (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e+30) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e+30)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(1.0 / fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e+30], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3999999999999999e30

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 30.6

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

   5. Simplified 30.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 48.4

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

   11. Simplified 48.4%

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

   if -2.3999999999999999e30 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 98.0

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

   5. Simplified 98.0%

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

   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. lower-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. lower-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. lower-fma.f64 61.7

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

   8. Simplified 61.7%

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

Alternative 7: 59.7% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+30}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, 0.16666666666666666 \cdot \left(b \cdot b\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e+30)
   (* 0.020833333333333332 (* b (* b b)))
   (/ 1.0 (fma b (* 0.16666666666666666 (* b b)) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e+30) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / fma(b, (0.16666666666666666 * (b * b)), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e+30)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(1.0 / fma(b, Float64(0.16666666666666666 * Float64(b * b)), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e+30], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(0.16666666666666666 * N[(b * b), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3999999999999999e30

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 30.6

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

   5. Simplified 30.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 48.4

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

   11. Simplified 48.4%

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

   if -2.3999999999999999e30 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 98.0

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

   5. Simplified 98.0%

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

   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. lower-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. lower-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. lower-fma.f64 61.7

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

   8. Simplified 61.7%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 61.1

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

   11. Simplified 61.1%

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

Alternative 8: 57.4% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+30}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \mathsf{fma}\left(0.5, b, 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e+30)
   (* 0.020833333333333332 (* b (* b b)))
   (/ 1.0 (fma b (fma 0.5 b 1.0) 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e+30) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / fma(b, fma(0.5, b, 1.0), 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e+30)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(1.0 / fma(b, fma(0.5, b, 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e+30], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(0.5 * b + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3999999999999999e30

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 30.6

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

   5. Simplified 30.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 48.4

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

   11. Simplified 48.4%

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

   if -2.3999999999999999e30 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 98.0

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

   5. Simplified 98.0%

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

   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. lower-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. lower-fma.f64 58.4

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

   8. Simplified 58.4%

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

Alternative 9: 53.5% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 75.0

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

   5. Simplified 75.0%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 51.6%

      \[\leadsto \color{blue}{0.5} \]

   if 2 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-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. lower-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. lower-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. lower-fma.f64 66.8

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

   8. Simplified 66.8%

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

   9. Taylor expanded in b around inf

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. distribute-rgt-in N/A

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

       9. associate-*l* N/A

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

       10. lft-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. lower-fma.f64 66.8

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

   11. Simplified 66.8%

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

   12. Taylor expanded in b around 0

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 52.6

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

   14. Simplified 52.6%

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

Alternative 10: 50.8% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -61000000000:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -61000000000.0)
   (* 0.020833333333333332 (* b (* b b)))
   (fma 0.25 a 0.5)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -61000000000.0) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = fma(0.25, a, 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -61000000000.0)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = fma(0.25, a, 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -61000000000.0], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * a + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.1e10

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 31.4

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

   5. Simplified 31.4%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 2.7

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

   8. Simplified 2.7%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 46.4

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

   11. Simplified 46.4%

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

   if -6.1e10 < a

   1. Initial program 99.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 50.2%

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

   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. lower-fma.f64 49.3

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

   8. Simplified 49.3%

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

Alternative 11: 39.7% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-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 37.6%

   \[\leadsto \color{blue}{0.5} \]
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 2024207 
(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))))