Quotient of sum of exps

Percentage Accurate: 99.0% → 98.6%

Time: 8.6s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. a = 3.939817306312814e+233
  2. b = 6.073602558680105e-137

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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3000:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3000.0) (* (exp a) 0.5) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3000.0) {
		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 (a <= (-3000.0d0)) 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 (a <= -3000.0) {
		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 a <= -3000.0:
		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 (a <= -3000.0)
		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 (a <= -3000.0)
		tmp = exp(a) * 0.5;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3000.0], 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 a < -3e3

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{1}} \]
   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 -3e3 < a

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3000:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

Alternative 3: 79.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 2.8 \cdot 10^{+36}:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b + -2, 4\right)}{\mathsf{fma}\left(b, \left(b \cdot b\right) \cdot t\_0, -64\right)} \cdot \mathsf{fma}\left(b, b \cdot b, -8\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= b 2.8e+36)
     (* (exp a) 0.5)
     (if (<= b 5.8e+102)
       (*
        (/ (fma b (+ b -2.0) 4.0) (fma b (* (* b b) t_0) -64.0))
        (fma b (* b b) -8.0))
       (/ 6.0 t_0)))))

C

double code(double a, double b) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= 2.8e+36) {
		tmp = exp(a) * 0.5;
	} else if (b <= 5.8e+102) {
		tmp = (fma(b, (b + -2.0), 4.0) / fma(b, ((b * b) * t_0), -64.0)) * fma(b, (b * b), -8.0);
	} else {
		tmp = 6.0 / t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (b <= 2.8e+36)
		tmp = Float64(exp(a) * 0.5);
	elseif (b <= 5.8e+102)
		tmp = Float64(Float64(fma(b, Float64(b + -2.0), 4.0) / fma(b, Float64(Float64(b * b) * t_0), -64.0)) * fma(b, Float64(b * b), -8.0));
	else
		tmp = Float64(6.0 / t_0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.8e+36], N[(N[Exp[a], $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[b, 5.8e+102], N[(N[(N[(b * N[(b + -2.0), $MachinePrecision] + 4.0), $MachinePrecision] / N[(b * N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision] + -64.0), $MachinePrecision]), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision] + -8.0), $MachinePrecision]), $MachinePrecision], N[(6.0 / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.8000000000000001e36

   1. Initial program 97.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 76.9%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 76.3%

      \[\leadsto \frac{e^{a}}{\color{blue}{2}} \]
   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 76.3

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

   10. Applied egg-rr 76.3%

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

   if 2.8000000000000001e36 < b < 5.8000000000000005e102

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 3.7

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

   8. Simplified 3.7%

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

   10. Applied egg-rr 83.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, b + -2, 4\right)}{\mathsf{fma}\left(b, \left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right), -64\right)} \cdot \mathsf{fma}\left(b, b \cdot b, -8\right)} \]

   if 5.8000000000000005e102 < 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 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. 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 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 3 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{+36}:\\ \;\;\;\;e^{a} \cdot 0.5\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b + -2, 4\right)}{\mathsf{fma}\left(b, \left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right), -64\right)} \cdot \mathsf{fma}\left(b, b \cdot b, -8\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 2.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, a \cdot 0.0625, -0.25\right)}{\mathsf{fma}\left(a \cdot \left(a \cdot a\right), 0.015625, -0.125\right)} \cdot \mathsf{fma}\left(a, a \cdot 0.0625, \mathsf{fma}\left(a, 0.125, 0.25\right)\right)\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b + -2, 4\right)}{\mathsf{fma}\left(b, \left(b \cdot b\right) \cdot t\_0, -64\right)} \cdot \mathsf{fma}\left(b, b \cdot b, -8\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= b 2.5e+36)
     (*
      (/ (fma a (* a 0.0625) -0.25) (fma (* a (* a a)) 0.015625 -0.125))
      (fma a (* a 0.0625) (fma a 0.125 0.25)))
     (if (<= b 5.8e+102)
       (*
        (/ (fma b (+ b -2.0) 4.0) (fma b (* (* b b) t_0) -64.0))
        (fma b (* b b) -8.0))
       (/ 6.0 t_0)))))

C

double code(double a, double b) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= 2.5e+36) {
		tmp = (fma(a, (a * 0.0625), -0.25) / fma((a * (a * a)), 0.015625, -0.125)) * fma(a, (a * 0.0625), fma(a, 0.125, 0.25));
	} else if (b <= 5.8e+102) {
		tmp = (fma(b, (b + -2.0), 4.0) / fma(b, ((b * b) * t_0), -64.0)) * fma(b, (b * b), -8.0);
	} else {
		tmp = 6.0 / t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (b <= 2.5e+36)
		tmp = Float64(Float64(fma(a, Float64(a * 0.0625), -0.25) / fma(Float64(a * Float64(a * a)), 0.015625, -0.125)) * fma(a, Float64(a * 0.0625), fma(a, 0.125, 0.25)));
	elseif (b <= 5.8e+102)
		tmp = Float64(Float64(fma(b, Float64(b + -2.0), 4.0) / fma(b, Float64(Float64(b * b) * t_0), -64.0)) * fma(b, Float64(b * b), -8.0));
	else
		tmp = Float64(6.0 / t_0);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.5e+36], N[(N[(N[(a * N[(a * 0.0625), $MachinePrecision] + -0.25), $MachinePrecision] / N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * 0.015625 + -0.125), $MachinePrecision]), $MachinePrecision] * N[(a * N[(a * 0.0625), $MachinePrecision] + N[(a * 0.125 + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e+102], N[(N[(N[(b * N[(b + -2.0), $MachinePrecision] + 4.0), $MachinePrecision] / N[(b * N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision] + -64.0), $MachinePrecision]), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision] + -8.0), $MachinePrecision]), $MachinePrecision], N[(6.0 / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.49999999999999988e36

   1. Initial program 97.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 76.9%

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

   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. lower-fma.f64 49.5

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

   8. Simplified 49.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, 0.25, 0.5\right)} \]
   9. Step-by-step derivation

      1. flip-+ N/A

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

      2. flip3-- N/A

         \[\leadsto \frac{\left(a \cdot \frac{1}{4}\right) \cdot \left(a \cdot \frac{1}{4}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\color{blue}{\frac{{\left(a \cdot \frac{1}{4}\right)}^{3} - {\frac{1}{2}}^{3}}{\left(a \cdot \frac{1}{4}\right) \cdot \left(a \cdot \frac{1}{4}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} + \left(a \cdot \frac{1}{4}\right) \cdot \frac{1}{2}\right)}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(a \cdot \frac{1}{4}\right) \cdot \left(a \cdot \frac{1}{4}\right) - \frac{1}{2} \cdot \frac{1}{2}}{{\left(a \cdot \frac{1}{4}\right)}^{3} - {\frac{1}{2}}^{3}} \cdot \left(\left(a \cdot \frac{1}{4}\right) \cdot \left(a \cdot \frac{1}{4}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} + \left(a \cdot \frac{1}{4}\right) \cdot \frac{1}{2}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(a \cdot \frac{1}{4}\right) \cdot \left(a \cdot \frac{1}{4}\right) - \frac{1}{2} \cdot \frac{1}{2}}{{\left(a \cdot \frac{1}{4}\right)}^{3} - {\frac{1}{2}}^{3}} \cdot \left(\left(a \cdot \frac{1}{4}\right) \cdot \left(a \cdot \frac{1}{4}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} + \left(a \cdot \frac{1}{4}\right) \cdot \frac{1}{2}\right)\right)} \]

   10. Applied egg-rr 53.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, a \cdot 0.0625, -0.25\right)}{\mathsf{fma}\left(a \cdot \left(a \cdot a\right), 0.015625, -0.125\right)} \cdot \mathsf{fma}\left(a, a \cdot 0.0625, \mathsf{fma}\left(a, 0.125, 0.25\right)\right)} \]

   if 2.49999999999999988e36 < b < 5.8000000000000005e102

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 3.7

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

   8. Simplified 3.7%

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

   10. Applied egg-rr 83.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, b + -2, 4\right)}{\mathsf{fma}\left(b, \left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right), -64\right)} \cdot \mathsf{fma}\left(b, b \cdot b, -8\right)} \]

   if 5.8000000000000005e102 < 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 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. 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 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 3 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, a \cdot 0.0625, -0.25\right)}{\mathsf{fma}\left(a \cdot \left(a \cdot a\right), 0.015625, -0.125\right)} \cdot \mathsf{fma}\left(a, a \cdot 0.0625, \mathsf{fma}\left(a, 0.125, 0.25\right)\right)\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b + -2, 4\right)}{\mathsf{fma}\left(b, \left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right), -64\right)} \cdot \mathsf{fma}\left(b, b \cdot b, -8\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, a \cdot 0.0625, -0.25\right)}{\mathsf{fma}\left(a \cdot \left(a \cdot a\right), 0.015625, -0.125\right)} \cdot \mathsf{fma}\left(a, a \cdot 0.0625, \mathsf{fma}\left(a, 0.125, 0.25\right)\right)\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{+68}:\\ \;\;\;\;a \cdot \left(a \cdot 0.25\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{b + -2}{\mathsf{fma}\left(b \cdot b, b \cdot b, -16\right)} \cdot \mathsf{fma}\left(b, b, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b + -2}{\mathsf{fma}\left(b, b, -4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2e+36)
   (*
    (/ (fma a (* a 0.0625) -0.25) (fma (* a (* a a)) 0.015625 -0.125))
    (fma a (* a 0.0625) (fma a 0.125 0.25)))
   (if (<= b 3.9e+68)
     (* a (* a 0.25))
     (if (<= b 1.35e+154)
       (* (/ (+ b -2.0) (fma (* b b) (* b b) -16.0)) (fma b b 4.0))
       (/ (+ b -2.0) (fma b b -4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2e+36) {
		tmp = (fma(a, (a * 0.0625), -0.25) / fma((a * (a * a)), 0.015625, -0.125)) * fma(a, (a * 0.0625), fma(a, 0.125, 0.25));
	} else if (b <= 3.9e+68) {
		tmp = a * (a * 0.25);
	} else if (b <= 1.35e+154) {
		tmp = ((b + -2.0) / fma((b * b), (b * b), -16.0)) * fma(b, b, 4.0);
	} else {
		tmp = (b + -2.0) / fma(b, b, -4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2e+36)
		tmp = Float64(Float64(fma(a, Float64(a * 0.0625), -0.25) / fma(Float64(a * Float64(a * a)), 0.015625, -0.125)) * fma(a, Float64(a * 0.0625), fma(a, 0.125, 0.25)));
	elseif (b <= 3.9e+68)
		tmp = Float64(a * Float64(a * 0.25));
	elseif (b <= 1.35e+154)
		tmp = Float64(Float64(Float64(b + -2.0) / fma(Float64(b * b), Float64(b * b), -16.0)) * fma(b, b, 4.0));
	else
		tmp = Float64(Float64(b + -2.0) / fma(b, b, -4.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2e+36], N[(N[(N[(a * N[(a * 0.0625), $MachinePrecision] + -0.25), $MachinePrecision] / N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * 0.015625 + -0.125), $MachinePrecision]), $MachinePrecision] * N[(a * N[(a * 0.0625), $MachinePrecision] + N[(a * 0.125 + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.9e+68], N[(a * N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(N[(N[(b + -2.0), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -16.0), $MachinePrecision]), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(b + -2.0), $MachinePrecision] / N[(b * b + -4.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < 2.00000000000000008e36

   1. Initial program 97.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 76.9%

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

   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. lower-fma.f64 49.5

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

   8. Simplified 49.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, 0.25, 0.5\right)} \]
   9. Step-by-step derivation

      1. flip-+ N/A

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

      2. flip3-- N/A

         \[\leadsto \frac{\left(a \cdot \frac{1}{4}\right) \cdot \left(a \cdot \frac{1}{4}\right) - \frac{1}{2} \cdot \frac{1}{2}}{\color{blue}{\frac{{\left(a \cdot \frac{1}{4}\right)}^{3} - {\frac{1}{2}}^{3}}{\left(a \cdot \frac{1}{4}\right) \cdot \left(a \cdot \frac{1}{4}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} + \left(a \cdot \frac{1}{4}\right) \cdot \frac{1}{2}\right)}}} \]

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\left(a \cdot \frac{1}{4}\right) \cdot \left(a \cdot \frac{1}{4}\right) - \frac{1}{2} \cdot \frac{1}{2}}{{\left(a \cdot \frac{1}{4}\right)}^{3} - {\frac{1}{2}}^{3}} \cdot \left(\left(a \cdot \frac{1}{4}\right) \cdot \left(a \cdot \frac{1}{4}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} + \left(a \cdot \frac{1}{4}\right) \cdot \frac{1}{2}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(a \cdot \frac{1}{4}\right) \cdot \left(a \cdot \frac{1}{4}\right) - \frac{1}{2} \cdot \frac{1}{2}}{{\left(a \cdot \frac{1}{4}\right)}^{3} - {\frac{1}{2}}^{3}} \cdot \left(\left(a \cdot \frac{1}{4}\right) \cdot \left(a \cdot \frac{1}{4}\right) + \left(\frac{1}{2} \cdot \frac{1}{2} + \left(a \cdot \frac{1}{4}\right) \cdot \frac{1}{2}\right)\right)} \]

   10. Applied egg-rr 53.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, a \cdot 0.0625, -0.25\right)}{\mathsf{fma}\left(a \cdot \left(a \cdot a\right), 0.015625, -0.125\right)} \cdot \mathsf{fma}\left(a, a \cdot 0.0625, \mathsf{fma}\left(a, 0.125, 0.25\right)\right)} \]

   if 2.00000000000000008e36 < b < 3.90000000000000019e68

   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 22.5%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 22.5%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 2.8

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

   11. Simplified 2.8%

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

   12. Taylor expanded in a around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 52.5

          \[\leadsto a \cdot \color{blue}{\left(a \cdot 0.25\right)} \]

   14. Simplified 52.5%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot 0.25\right)} \]

   if 3.90000000000000019e68 < b < 1.35000000000000003e154

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 4.1

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

   8. Simplified 4.1%

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

      1. flip-+ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b - 2 \cdot 2}{b - 2}}} \]

      2. clear-num N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. lower-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{b + -2}{\mathsf{fma}\left(b \cdot b, b \cdot b, \mathsf{neg}\left(\color{blue}{16}\right)\right)} \cdot \left(b \cdot b + 2 \cdot 2\right) \]

      17. metadata-eval N/A

          \[\leadsto \frac{b + -2}{\mathsf{fma}\left(b \cdot b, b \cdot b, \color{blue}{-16}\right)} \cdot \left(b \cdot b + 2 \cdot 2\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{b + -2}{\mathsf{fma}\left(b \cdot b, b \cdot b, -16\right)} \cdot \color{blue}{\mathsf{fma}\left(b, b, 2 \cdot 2\right)} \]

   10. Applied egg-rr 88.0%

       \[\leadsto \color{blue}{\frac{b + -2}{\mathsf{fma}\left(b \cdot b, b \cdot b, -16\right)} \cdot \mathsf{fma}\left(b, b, 4\right)} \]

   if 1.35000000000000003e154 < 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

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

      1. +-commutative N/A

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

      2. lower-+.f64 6.6

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

   8. Simplified 6.6%

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

      1. flip-+ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b - 2 \cdot 2}{b - 2}}} \]

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval 100.0

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

   10. Applied egg-rr 100.0%

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

Alternative 6: 60.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+19}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{+68}:\\ \;\;\;\;a \cdot \left(a \cdot 0.25\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{b + -2}{\mathsf{fma}\left(b \cdot b, b \cdot b, -16\right)} \cdot \mathsf{fma}\left(b, b, 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b + -2}{\mathsf{fma}\left(b, b, -4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5e+19)
   0.5
   (if (<= b 3.9e+68)
     (* a (* a 0.25))
     (if (<= b 1.35e+154)
       (* (/ (+ b -2.0) (fma (* b b) (* b b) -16.0)) (fma b b 4.0))
       (/ (+ b -2.0) (fma b b -4.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5e+19) {
		tmp = 0.5;
	} else if (b <= 3.9e+68) {
		tmp = a * (a * 0.25);
	} else if (b <= 1.35e+154) {
		tmp = ((b + -2.0) / fma((b * b), (b * b), -16.0)) * fma(b, b, 4.0);
	} else {
		tmp = (b + -2.0) / fma(b, b, -4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5e+19)
		tmp = 0.5;
	elseif (b <= 3.9e+68)
		tmp = Float64(a * Float64(a * 0.25));
	elseif (b <= 1.35e+154)
		tmp = Float64(Float64(Float64(b + -2.0) / fma(Float64(b * b), Float64(b * b), -16.0)) * fma(b, b, 4.0));
	else
		tmp = Float64(Float64(b + -2.0) / fma(b, b, -4.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5e+19], 0.5, If[LessEqual[b, 3.9e+68], N[(a * N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(N[(N[(b + -2.0), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision] + -16.0), $MachinePrecision]), $MachinePrecision] * N[(b * b + 4.0), $MachinePrecision]), $MachinePrecision], N[(N[(b + -2.0), $MachinePrecision] / N[(b * b + -4.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < 5e19

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 73.2

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

   5. Simplified 73.2%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 50.5%

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

   if 5e19 < b < 3.90000000000000019e68

   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 25.5%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 25.5%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 2.8

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

   11. Simplified 2.8%

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

   12. Taylor expanded in a around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 41.2

          \[\leadsto a \cdot \color{blue}{\left(a \cdot 0.25\right)} \]

   14. Simplified 41.2%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot 0.25\right)} \]

   if 3.90000000000000019e68 < b < 1.35000000000000003e154

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 4.1

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

   8. Simplified 4.1%

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

      1. flip-+ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b - 2 \cdot 2}{b - 2}}} \]

      2. clear-num N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. lower-+.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{b + -2}{\mathsf{fma}\left(b \cdot b, b \cdot b, \mathsf{neg}\left(\color{blue}{16}\right)\right)} \cdot \left(b \cdot b + 2 \cdot 2\right) \]

      17. metadata-eval N/A

          \[\leadsto \frac{b + -2}{\mathsf{fma}\left(b \cdot b, b \cdot b, \color{blue}{-16}\right)} \cdot \left(b \cdot b + 2 \cdot 2\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{b + -2}{\mathsf{fma}\left(b \cdot b, b \cdot b, -16\right)} \cdot \color{blue}{\mathsf{fma}\left(b, b, 2 \cdot 2\right)} \]

   10. Applied egg-rr 88.0%

       \[\leadsto \color{blue}{\frac{b + -2}{\mathsf{fma}\left(b \cdot b, b \cdot b, -16\right)} \cdot \mathsf{fma}\left(b, b, 4\right)} \]

   if 1.35000000000000003e154 < 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

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

      1. +-commutative N/A

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

      2. lower-+.f64 6.6

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

   8. Simplified 6.6%

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

      1. flip-+ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b - 2 \cdot 2}{b - 2}}} \]

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval 100.0

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

   10. Applied egg-rr 100.0%

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

Alternative 7: 58.3% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+19}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{+68}:\\ \;\;\;\;a \cdot \left(a \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{b \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5e+19)
   0.5
   (if (<= b 3.9e+68) (* a (* a 0.25)) (/ 6.0 (* b (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5e+19) {
		tmp = 0.5;
	} else if (b <= 3.9e+68) {
		tmp = a * (a * 0.25);
	} 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 (b <= 5d+19) then
        tmp = 0.5d0
    else if (b <= 3.9d+68) then
        tmp = a * (a * 0.25d0)
    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 (b <= 5e+19) {
		tmp = 0.5;
	} else if (b <= 3.9e+68) {
		tmp = a * (a * 0.25);
	} else {
		tmp = 6.0 / (b * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5e+19:
		tmp = 0.5
	elif b <= 3.9e+68:
		tmp = a * (a * 0.25)
	else:
		tmp = 6.0 / (b * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5e+19)
		tmp = 0.5;
	elseif (b <= 3.9e+68)
		tmp = Float64(a * Float64(a * 0.25));
	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 (b <= 5e+19)
		tmp = 0.5;
	elseif (b <= 3.9e+68)
		tmp = a * (a * 0.25);
	else
		tmp = 6.0 / (b * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5e+19], 0.5, If[LessEqual[b, 3.9e+68], N[(a * N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 5e19

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 73.2

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

   5. Simplified 73.2%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 50.5%

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

   if 5e19 < b < 3.90000000000000019e68

   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 25.5%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 25.5%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 2.8

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

   11. Simplified 2.8%

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

   12. Taylor expanded in a around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 41.2

          \[\leadsto a \cdot \color{blue}{\left(a \cdot 0.25\right)} \]

   14. Simplified 41.2%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot 0.25\right)} \]

   if 3.90000000000000019e68 < 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 87.2

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

   8. Simplified 87.2%

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

   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 87.2

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

   11. Simplified 87.2%

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

Alternative 8: 55.9% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+19}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+139}:\\ \;\;\;\;a \cdot \left(a \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b + -2}{\mathsf{fma}\left(b, b, -4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 5e+19)
   0.5
   (if (<= b 1.35e+139) (* a (* a 0.25)) (/ (+ b -2.0) (fma b b -4.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5e+19) {
		tmp = 0.5;
	} else if (b <= 1.35e+139) {
		tmp = a * (a * 0.25);
	} else {
		tmp = (b + -2.0) / fma(b, b, -4.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5e+19)
		tmp = 0.5;
	elseif (b <= 1.35e+139)
		tmp = Float64(a * Float64(a * 0.25));
	else
		tmp = Float64(Float64(b + -2.0) / fma(b, b, -4.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5e+19], 0.5, If[LessEqual[b, 1.35e+139], N[(a * N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(N[(b + -2.0), $MachinePrecision] / N[(b * b + -4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 5e19

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 73.2

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

   5. Simplified 73.2%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 50.5%

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

   if 5e19 < b < 1.3499999999999999e139

   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 32.9%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 32.9%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 2.7

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

   11. Simplified 2.7%

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

   12. Taylor expanded in a around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 29.9

          \[\leadsto a \cdot \color{blue}{\left(a \cdot 0.25\right)} \]

   14. Simplified 29.9%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot 0.25\right)} \]

   if 1.3499999999999999e139 < 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

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

      1. +-commutative N/A

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

      2. lower-+.f64 6.4

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

   8. Simplified 6.4%

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

      1. flip-+ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot b - 2 \cdot 2}{b - 2}}} \]

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval 92.5

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

   10. Applied egg-rr 92.5%

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

Alternative 9: 47.4% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 5e+19) 0.5 (* a (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 5e+19) {
		tmp = 0.5;
	} else {
		tmp = a * (a * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 5d+19) then
        tmp = 0.5d0
    else
        tmp = a * (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 5e+19) {
		tmp = 0.5;
	} else {
		tmp = a * (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 5e+19:
		tmp = 0.5
	else:
		tmp = a * (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 5e+19)
		tmp = 0.5;
	else
		tmp = Float64(a * Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 5e+19)
		tmp = 0.5;
	else
		tmp = a * (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5e+19], 0.5, N[(a * N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5e19

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 73.2

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

   5. Simplified 73.2%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 50.5%

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

   if 5e19 < 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 33.4%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 33.4%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 2.8

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

   11. Simplified 2.8%

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

   12. Taylor expanded in a around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 38.3

          \[\leadsto a \cdot \color{blue}{\left(a \cdot 0.25\right)} \]

   14. Simplified 38.3%

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

Alternative 10: 38.7% 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 98.4%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-exp.f64 79.9

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

5. Simplified 79.9%

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

6. Taylor expanded in b around 0

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

8. Simplified 38.6%

   \[\leadsto \color{blue}{0.5} \]
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 2024208 
(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))))