Quotient of sum of exps

Percentage Accurate: 98.9% → 99.0%

Time: 9.2s

Alternatives: 21

Speedup: 1.0×

• could not determine a ground truth

  1. a = -2.00000000000000007e154
  2. b = -2.00000000000000007e154

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: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ e^{a - \log \left(e^{a} + e^{b}\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (exp (- a (log (+ (exp a) (exp b))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

   5. prod-exp N/A

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

   6. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

      \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

   11. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

   12. exp-lowering-exp.f64 99.6%

       \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

   \[\leadsto e^{a - \log \left(e^{a} + e^{b}\right)} \]
6. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999999995:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.99999999995)
   (/ (exp a) (+ (exp a) 1.0))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.99999999995) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.99999999995d0) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.99999999995) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.99999999995:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.99999999995)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.99999999995)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.99999999995], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $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) < 0.99999999995

   1. Initial program 98.8%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 98.8%

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

   if 0.99999999995 < (exp.f64 a)

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 99.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999999995:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. unpow1 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\left({\left(e^{a}\right)}^{1}\right), \mathsf{exp.f64}\left(\color{blue}{b}\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\left({\left(e^{a}\right)}^{\left(-1 \cdot -1\right)}\right), \mathsf{exp.f64}\left(b\right)\right)\right) \]

   3. pow-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\left({\left({\left(e^{a}\right)}^{-1}\right)}^{-1}\right), \mathsf{exp.f64}\left(\color{blue}{b}\right)\right)\right) \]

   4. inv-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\left({\left(\frac{1}{e^{a}}\right)}^{-1}\right), \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. unpow-1 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\left(\frac{1}{\frac{1}{e^{a}}}\right), \mathsf{exp.f64}\left(\color{blue}{b}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{e^{a}}\right)\right), \mathsf{exp.f64}\left(\color{blue}{b}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(e^{a}\right)\right)\right), \mathsf{exp.f64}\left(b\right)\right)\right) \]

   8. exp-lowering-exp.f64 99.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right), \mathsf{exp.f64}\left(b\right)\right)\right) \]

4. Applied egg-rr 99.2%

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

5. Final simplification 99.2%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

Alternative 5: 98.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -75000000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -75000000.0) (exp a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -75000000.0) {
		tmp = exp(a);
	} 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 <= (-75000000.0d0)) then
        tmp = exp(a)
    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 <= -75000000.0) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -75000000.0:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -75000000.0)
		tmp = exp(a);
	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 <= -75000000.0)
		tmp = exp(a);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -75000000.0], N[Exp[a], $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.5e7

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 100.0%

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

   if -7.5e7 < a

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 99.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -75000000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.1 \cdot 10^{-36}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \left(b \cdot \left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right)\right) \cdot \left(2 + b \cdot \left(0.6666666666666666 + b \cdot \left(0.2222222222222222 + b \cdot 0.07407407407407407\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.1e-36)
   (exp a)
   (/
    1.0
    (+
     2.0
     (*
      b
      (+
       1.0
       (*
        (* b (- 0.25 (* (* b b) 0.027777777777777776)))
        (+
         2.0
         (*
          b
          (+
           0.6666666666666666
           (* b (+ 0.2222222222222222 (* b 0.07407407407407407)))))))))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.1e-36) {
		tmp = exp(a);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * (0.6666666666666666 + (b * (0.2222222222222222 + (b * 0.07407407407407407))))))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5.1d-36)) then
        tmp = exp(a)
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + ((b * (0.25d0 - ((b * b) * 0.027777777777777776d0))) * (2.0d0 + (b * (0.6666666666666666d0 + (b * (0.2222222222222222d0 + (b * 0.07407407407407407d0))))))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.1e-36) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * (0.6666666666666666 + (b * (0.2222222222222222 + (b * 0.07407407407407407))))))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.1e-36:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * (0.6666666666666666 + (b * (0.2222222222222222 + (b * 0.07407407407407407))))))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.1e-36)
		tmp = exp(a);
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(Float64(b * Float64(0.25 - Float64(Float64(b * b) * 0.027777777777777776))) * Float64(2.0 + Float64(b * Float64(0.6666666666666666 + Float64(b * Float64(0.2222222222222222 + Float64(b * 0.07407407407407407)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.1e-36)
		tmp = exp(a);
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * (0.6666666666666666 + (b * (0.2222222222222222 + (b * 0.07407407407407407))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.1e-36], N[Exp[a], $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(N[(b * N[(0.25 - N[(N[(b * b), $MachinePrecision] * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(b * N[(0.6666666666666666 + N[(b * N[(0.2222222222222222 + N[(b * 0.07407407407407407), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.09999999999999973e-36

   1. Initial program 98.8%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 98.9%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 96.9%

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

   if -5.09999999999999973e-36 < a

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 99.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 99.7%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 67.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 67.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + b \cdot \frac{1}{6}\right) \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)}{\frac{1}{2} - b \cdot \frac{1}{6}} \cdot b\right)\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b}{\color{blue}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right) \cdot \color{blue}{\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right), \color{blue}{\left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right), b\right), \left(\frac{\color{blue}{1}}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right), \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      9. swap-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot b\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{2} - b \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 67.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 67.8%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \color{blue}{\left(\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b\right) \cdot \frac{1}{0.5 - b \cdot 0.16666666666666666}}\right)} \]

   11. Taylor expanded in b around 0

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \color{blue}{\left(2 + b \cdot \left(\frac{2}{3} + b \cdot \left(\frac{2}{9} + \frac{2}{27} \cdot b\right)\right)\right)}\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(\frac{2}{3} + b \cdot \left(\frac{2}{9} + \frac{2}{27} \cdot b\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{2}{3} + b \cdot \left(\frac{2}{9} + \frac{2}{27} \cdot b\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \color{blue}{\left(b \cdot \left(\frac{2}{9} + \frac{2}{27} \cdot b\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{2}{9} + \frac{2}{27} \cdot b\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{9}, \color{blue}{\left(\frac{2}{27} \cdot b\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{9}, \left(b \cdot \color{blue}{\frac{2}{27}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 73.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{9}, \mathsf{*.f64}\left(b, \color{blue}{\frac{2}{27}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 73.9%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \left(\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b\right) \cdot \color{blue}{\left(2 + b \cdot \left(0.6666666666666666 + b \cdot \left(0.2222222222222222 + b \cdot 0.07407407407407407\right)\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.1 \cdot 10^{-36}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \left(b \cdot \left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right)\right) \cdot \left(2 + b \cdot \left(0.6666666666666666 + b \cdot \left(0.2222222222222222 + b \cdot 0.07407407407407407\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right)\\ \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -4.3 \cdot 10^{-72}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-289}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + t\_0 \cdot \frac{-6 + \frac{-18 - \frac{54 + \frac{162}{b}}{b}}{b}}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + t\_0 \cdot \left(2 + b \cdot \left(0.6666666666666666 + b \cdot \left(0.2222222222222222 + b \cdot 0.07407407407407407\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (- 0.25 (* (* b b) 0.027777777777777776)))))
   (if (<= b -1.1)
     1.0
     (if (<= b -4.3e-72)
       0.5
       (if (<= b -1.6e-289)
         (/
          1.0
          (+
           2.0
           (*
            b
            (+
             1.0
             (*
              t_0
              (/ (+ -6.0 (/ (- -18.0 (/ (+ 54.0 (/ 162.0 b)) b)) b)) b))))))
         (/
          1.0
          (+
           2.0
           (*
            b
            (+
             1.0
             (*
              t_0
              (+
               2.0
               (*
                b
                (+
                 0.6666666666666666
                 (*
                  b
                  (+
                   0.2222222222222222
                   (* b 0.07407407407407407))))))))))))))))

C

double code(double a, double b) {
	double t_0 = b * (0.25 - ((b * b) * 0.027777777777777776));
	double tmp;
	if (b <= -1.1) {
		tmp = 1.0;
	} else if (b <= -4.3e-72) {
		tmp = 0.5;
	} else if (b <= -1.6e-289) {
		tmp = 1.0 / (2.0 + (b * (1.0 + (t_0 * ((-6.0 + ((-18.0 - ((54.0 + (162.0 / b)) / b)) / b)) / b)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (t_0 * (2.0 + (b * (0.6666666666666666 + (b * (0.2222222222222222 + (b * 0.07407407407407407))))))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (0.25d0 - ((b * b) * 0.027777777777777776d0))
    if (b <= (-1.1d0)) then
        tmp = 1.0d0
    else if (b <= (-4.3d-72)) then
        tmp = 0.5d0
    else if (b <= (-1.6d-289)) then
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (t_0 * (((-6.0d0) + (((-18.0d0) - ((54.0d0 + (162.0d0 / b)) / b)) / b)) / b)))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (t_0 * (2.0d0 + (b * (0.6666666666666666d0 + (b * (0.2222222222222222d0 + (b * 0.07407407407407407d0))))))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (0.25 - ((b * b) * 0.027777777777777776));
	double tmp;
	if (b <= -1.1) {
		tmp = 1.0;
	} else if (b <= -4.3e-72) {
		tmp = 0.5;
	} else if (b <= -1.6e-289) {
		tmp = 1.0 / (2.0 + (b * (1.0 + (t_0 * ((-6.0 + ((-18.0 - ((54.0 + (162.0 / b)) / b)) / b)) / b)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (t_0 * (2.0 + (b * (0.6666666666666666 + (b * (0.2222222222222222 + (b * 0.07407407407407407))))))))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (0.25 - ((b * b) * 0.027777777777777776))
	tmp = 0
	if b <= -1.1:
		tmp = 1.0
	elif b <= -4.3e-72:
		tmp = 0.5
	elif b <= -1.6e-289:
		tmp = 1.0 / (2.0 + (b * (1.0 + (t_0 * ((-6.0 + ((-18.0 - ((54.0 + (162.0 / b)) / b)) / b)) / b)))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (t_0 * (2.0 + (b * (0.6666666666666666 + (b * (0.2222222222222222 + (b * 0.07407407407407407))))))))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(0.25 - Float64(Float64(b * b) * 0.027777777777777776)))
	tmp = 0.0
	if (b <= -1.1)
		tmp = 1.0;
	elseif (b <= -4.3e-72)
		tmp = 0.5;
	elseif (b <= -1.6e-289)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(t_0 * Float64(Float64(-6.0 + Float64(Float64(-18.0 - Float64(Float64(54.0 + Float64(162.0 / b)) / b)) / b)) / b))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(t_0 * Float64(2.0 + Float64(b * Float64(0.6666666666666666 + Float64(b * Float64(0.2222222222222222 + Float64(b * 0.07407407407407407)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (0.25 - ((b * b) * 0.027777777777777776));
	tmp = 0.0;
	if (b <= -1.1)
		tmp = 1.0;
	elseif (b <= -4.3e-72)
		tmp = 0.5;
	elseif (b <= -1.6e-289)
		tmp = 1.0 / (2.0 + (b * (1.0 + (t_0 * ((-6.0 + ((-18.0 - ((54.0 + (162.0 / b)) / b)) / b)) / b)))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (t_0 * (2.0 + (b * (0.6666666666666666 + (b * (0.2222222222222222 + (b * 0.07407407407407407))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(0.25 - N[(N[(b * b), $MachinePrecision] * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.1], 1.0, If[LessEqual[b, -4.3e-72], 0.5, If[LessEqual[b, -1.6e-289], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(t$95$0 * N[(N[(-6.0 + N[(N[(-18.0 - N[(N[(54.0 + N[(162.0 / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(t$95$0 * N[(2.0 + N[(b * N[(0.6666666666666666 + N[(b * N[(0.2222222222222222 + N[(b * 0.07407407407407407), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.1000000000000001

   1. Initial program 97.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 97.9%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.6%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{1} \]

   if -1.1000000000000001 < b < -4.2999999999999999e-72

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 62.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 62.3%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 62.3%

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

   if -4.2999999999999999e-72 < b < -1.6000000000000001e-289

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 40.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 40.6%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 40.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 40.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + b \cdot \frac{1}{6}\right) \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)}{\frac{1}{2} - b \cdot \frac{1}{6}} \cdot b\right)\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b}{\color{blue}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right) \cdot \color{blue}{\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right), \color{blue}{\left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right), b\right), \left(\frac{\color{blue}{1}}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right), \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      9. swap-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot b\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{2} - b \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 40.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 40.6%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \color{blue}{\left(\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b\right) \cdot \frac{1}{0.5 - b \cdot 0.16666666666666666}}\right)} \]

   11. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \color{blue}{\left(\frac{-1 \cdot \frac{54 + 162 \cdot \frac{1}{b}}{{b}^{2}} - \left(6 + 18 \cdot \frac{1}{b}\right)}{b}\right)}\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(\left(-1 \cdot \frac{54 + 162 \cdot \frac{1}{b}}{{b}^{2}} - \left(6 + 18 \cdot \frac{1}{b}\right)\right), \color{blue}{b}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 57.7%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \left(\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b\right) \cdot \color{blue}{\frac{-6 + \frac{-18 - \frac{54 + \frac{162}{b}}{b}}{b}}{b}}\right)} \]

   if -1.6000000000000001e-289 < b

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 82.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 82.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 71.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 71.2%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + b \cdot \frac{1}{6}\right) \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)}{\frac{1}{2} - b \cdot \frac{1}{6}} \cdot b\right)\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b}{\color{blue}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right) \cdot \color{blue}{\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right), \color{blue}{\left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right), b\right), \left(\frac{\color{blue}{1}}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right), \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      9. swap-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot b\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{2} - b \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 71.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 71.2%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \color{blue}{\left(\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b\right) \cdot \frac{1}{0.5 - b \cdot 0.16666666666666666}}\right)} \]

   11. Taylor expanded in b around 0

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \color{blue}{\left(2 + b \cdot \left(\frac{2}{3} + b \cdot \left(\frac{2}{9} + \frac{2}{27} \cdot b\right)\right)\right)}\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(\frac{2}{3} + b \cdot \left(\frac{2}{9} + \frac{2}{27} \cdot b\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{2}{3} + b \cdot \left(\frac{2}{9} + \frac{2}{27} \cdot b\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \color{blue}{\left(b \cdot \left(\frac{2}{9} + \frac{2}{27} \cdot b\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{2}{9} + \frac{2}{27} \cdot b\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{9}, \color{blue}{\left(\frac{2}{27} \cdot b\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{9}, \left(b \cdot \color{blue}{\frac{2}{27}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 79.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{9}, \mathsf{*.f64}\left(b, \color{blue}{\frac{2}{27}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 79.6%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \left(\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b\right) \cdot \color{blue}{\left(2 + b \cdot \left(0.6666666666666666 + b \cdot \left(0.2222222222222222 + b \cdot 0.07407407407407407\right)\right)\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -4.3 \cdot 10^{-72}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-289}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \left(b \cdot \left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right)\right) \cdot \frac{-6 + \frac{-18 - \frac{54 + \frac{162}{b}}{b}}{b}}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \left(b \cdot \left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right)\right) \cdot \left(2 + b \cdot \left(0.6666666666666666 + b \cdot \left(0.2222222222222222 + b \cdot 0.07407407407407407\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -9.6 \cdot 10^{-204}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \left(b \cdot \left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right)\right) \cdot \left(2 + b \cdot \left(0.6666666666666666 + b \cdot \left(0.2222222222222222 + b \cdot 0.07407407407407407\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -6.5e-9)
   1.0
   (if (<= b -9.6e-204)
     (+ 0.5 (* a 0.25))
     (if (<= b -5e-289)
       (* 0.020833333333333332 (* b (* b b)))
       (/
        1.0
        (+
         2.0
         (*
          b
          (+
           1.0
           (*
            (* b (- 0.25 (* (* b b) 0.027777777777777776)))
            (+
             2.0
             (*
              b
              (+
               0.6666666666666666
               (*
                b
                (+ 0.2222222222222222 (* b 0.07407407407407407)))))))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -9.6e-204) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -5e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * (0.6666666666666666 + (b * (0.2222222222222222 + (b * 0.07407407407407407))))))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-6.5d-9)) then
        tmp = 1.0d0
    else if (b <= (-9.6d-204)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= (-5d-289)) then
        tmp = 0.020833333333333332d0 * (b * (b * b))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + ((b * (0.25d0 - ((b * b) * 0.027777777777777776d0))) * (2.0d0 + (b * (0.6666666666666666d0 + (b * (0.2222222222222222d0 + (b * 0.07407407407407407d0))))))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -9.6e-204) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -5e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * (0.6666666666666666 + (b * (0.2222222222222222 + (b * 0.07407407407407407))))))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -6.5e-9:
		tmp = 1.0
	elif b <= -9.6e-204:
		tmp = 0.5 + (a * 0.25)
	elif b <= -5e-289:
		tmp = 0.020833333333333332 * (b * (b * b))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * (0.6666666666666666 + (b * (0.2222222222222222 + (b * 0.07407407407407407))))))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -9.6e-204)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= -5e-289)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(Float64(b * Float64(0.25 - Float64(Float64(b * b) * 0.027777777777777776))) * Float64(2.0 + Float64(b * Float64(0.6666666666666666 + Float64(b * Float64(0.2222222222222222 + Float64(b * 0.07407407407407407)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -9.6e-204)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= -5e-289)
		tmp = 0.020833333333333332 * (b * (b * b));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * (0.6666666666666666 + (b * (0.2222222222222222 + (b * 0.07407407407407407))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -6.5e-9], 1.0, If[LessEqual[b, -9.6e-204], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -5e-289], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(N[(b * N[(0.25 - N[(N[(b * b), $MachinePrecision] * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(b * N[(0.6666666666666666 + N[(b * N[(0.2222222222222222 + N[(b * 0.07407407407407407), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.5000000000000003e-9

   1. Initial program 97.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 98.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 98.0%

       \[\leadsto \color{blue}{1} \]

   if -6.5000000000000003e-9 < b < -9.6e-204

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   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

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 54.8%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

   if -9.6e-204 < b < -5.00000000000000029e-289

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 36.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 36.7%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \frac{-1}{4}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{-1}{4} + \color{blue}{\frac{1}{48} \cdot {b}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left({b}^{2} \cdot \color{blue}{\frac{1}{48}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(b \cdot b\right) \cdot \frac{1}{48}\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(b \cdot \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 36.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 36.7%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 66.5%

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

   if -5.00000000000000029e-289 < b

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 82.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 82.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 71.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 71.2%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + b \cdot \frac{1}{6}\right) \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)}{\frac{1}{2} - b \cdot \frac{1}{6}} \cdot b\right)\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b}{\color{blue}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right) \cdot \color{blue}{\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right), \color{blue}{\left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right), b\right), \left(\frac{\color{blue}{1}}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right), \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      9. swap-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot b\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{2} - b \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 71.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 71.2%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \color{blue}{\left(\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b\right) \cdot \frac{1}{0.5 - b \cdot 0.16666666666666666}}\right)} \]

   11. Taylor expanded in b around 0

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \color{blue}{\left(2 + b \cdot \left(\frac{2}{3} + b \cdot \left(\frac{2}{9} + \frac{2}{27} \cdot b\right)\right)\right)}\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(\frac{2}{3} + b \cdot \left(\frac{2}{9} + \frac{2}{27} \cdot b\right)\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{2}{3} + b \cdot \left(\frac{2}{9} + \frac{2}{27} \cdot b\right)\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \color{blue}{\left(b \cdot \left(\frac{2}{9} + \frac{2}{27} \cdot b\right)\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{2}{9} + \frac{2}{27} \cdot b\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{9}, \color{blue}{\left(\frac{2}{27} \cdot b\right)}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{9}, \left(b \cdot \color{blue}{\frac{2}{27}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 79.6%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{9}, \mathsf{*.f64}\left(b, \color{blue}{\frac{2}{27}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 79.6%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \left(\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b\right) \cdot \color{blue}{\left(2 + b \cdot \left(0.6666666666666666 + b \cdot \left(0.2222222222222222 + b \cdot 0.07407407407407407\right)\right)\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -9.6 \cdot 10^{-204}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \left(b \cdot \left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right)\right) \cdot \left(2 + b \cdot \left(0.6666666666666666 + b \cdot \left(0.2222222222222222 + b \cdot 0.07407407407407407\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -9.6 \cdot 10^{-204}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -8 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \left(b \cdot \left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right)\right) \cdot \left(2 + b \cdot \left(0.6666666666666666 + b \cdot 0.2222222222222222\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -6.5e-9)
   1.0
   (if (<= b -9.6e-204)
     (+ 0.5 (* a 0.25))
     (if (<= b -8e-289)
       (* 0.020833333333333332 (* b (* b b)))
       (/
        1.0
        (+
         2.0
         (*
          b
          (+
           1.0
           (*
            (* b (- 0.25 (* (* b b) 0.027777777777777776)))
            (+
             2.0
             (* b (+ 0.6666666666666666 (* b 0.2222222222222222)))))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -9.6e-204) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -8e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * (0.6666666666666666 + (b * 0.2222222222222222))))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-6.5d-9)) then
        tmp = 1.0d0
    else if (b <= (-9.6d-204)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= (-8d-289)) then
        tmp = 0.020833333333333332d0 * (b * (b * b))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + ((b * (0.25d0 - ((b * b) * 0.027777777777777776d0))) * (2.0d0 + (b * (0.6666666666666666d0 + (b * 0.2222222222222222d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -9.6e-204) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -8e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * (0.6666666666666666 + (b * 0.2222222222222222))))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -6.5e-9:
		tmp = 1.0
	elif b <= -9.6e-204:
		tmp = 0.5 + (a * 0.25)
	elif b <= -8e-289:
		tmp = 0.020833333333333332 * (b * (b * b))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * (0.6666666666666666 + (b * 0.2222222222222222))))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -9.6e-204)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= -8e-289)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(Float64(b * Float64(0.25 - Float64(Float64(b * b) * 0.027777777777777776))) * Float64(2.0 + Float64(b * Float64(0.6666666666666666 + Float64(b * 0.2222222222222222)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -9.6e-204)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= -8e-289)
		tmp = 0.020833333333333332 * (b * (b * b));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * (0.6666666666666666 + (b * 0.2222222222222222))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -6.5e-9], 1.0, If[LessEqual[b, -9.6e-204], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -8e-289], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(N[(b * N[(0.25 - N[(N[(b * b), $MachinePrecision] * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(b * N[(0.6666666666666666 + N[(b * 0.2222222222222222), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.5000000000000003e-9

   1. Initial program 97.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 98.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 98.0%

       \[\leadsto \color{blue}{1} \]

   if -6.5000000000000003e-9 < b < -9.6e-204

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   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

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 54.8%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

   if -9.6e-204 < b < -8.0000000000000001e-289

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 36.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 36.7%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \frac{-1}{4}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{-1}{4} + \color{blue}{\frac{1}{48} \cdot {b}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left({b}^{2} \cdot \color{blue}{\frac{1}{48}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(b \cdot b\right) \cdot \frac{1}{48}\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(b \cdot \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 36.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 36.7%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 66.5%

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

   if -8.0000000000000001e-289 < b

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 82.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 82.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 71.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 71.2%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + b \cdot \frac{1}{6}\right) \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)}{\frac{1}{2} - b \cdot \frac{1}{6}} \cdot b\right)\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b}{\color{blue}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right) \cdot \color{blue}{\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right), \color{blue}{\left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right), b\right), \left(\frac{\color{blue}{1}}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right), \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      9. swap-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot b\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{2} - b \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 71.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 71.2%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \color{blue}{\left(\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b\right) \cdot \frac{1}{0.5 - b \cdot 0.16666666666666666}}\right)} \]

   11. Taylor expanded in b around 0

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \color{blue}{\left(2 + b \cdot \left(\frac{2}{3} + \frac{2}{9} \cdot b\right)\right)}\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(\frac{2}{3} + \frac{2}{9} \cdot b\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{2}{3} + \frac{2}{9} \cdot b\right)}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \color{blue}{\left(\frac{2}{9} \cdot b\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \left(b \cdot \color{blue}{\frac{2}{9}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 78.3%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(b, \color{blue}{\frac{2}{9}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 78.3%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \left(\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b\right) \cdot \color{blue}{\left(2 + b \cdot \left(0.6666666666666666 + b \cdot 0.2222222222222222\right)\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -9.6 \cdot 10^{-204}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -8 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \left(b \cdot \left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right)\right) \cdot \left(2 + b \cdot \left(0.6666666666666666 + b \cdot 0.2222222222222222\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-203}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -8.1 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \left(b \cdot \left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right)\right) \cdot \left(2 + b \cdot 0.6666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -6.5e-9)
   1.0
   (if (<= b -1.2e-203)
     (+ 0.5 (* a 0.25))
     (if (<= b -8.1e-289)
       (* 0.020833333333333332 (* b (* b b)))
       (/
        1.0
        (+
         2.0
         (*
          b
          (+
           1.0
           (*
            (* b (- 0.25 (* (* b b) 0.027777777777777776)))
            (+ 2.0 (* b 0.6666666666666666)))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -1.2e-203) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -8.1e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * 0.6666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-6.5d-9)) then
        tmp = 1.0d0
    else if (b <= (-1.2d-203)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= (-8.1d-289)) then
        tmp = 0.020833333333333332d0 * (b * (b * b))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + ((b * (0.25d0 - ((b * b) * 0.027777777777777776d0))) * (2.0d0 + (b * 0.6666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -1.2e-203) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -8.1e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * 0.6666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -6.5e-9:
		tmp = 1.0
	elif b <= -1.2e-203:
		tmp = 0.5 + (a * 0.25)
	elif b <= -8.1e-289:
		tmp = 0.020833333333333332 * (b * (b * b))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * 0.6666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -1.2e-203)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= -8.1e-289)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(Float64(b * Float64(0.25 - Float64(Float64(b * b) * 0.027777777777777776))) * Float64(2.0 + Float64(b * 0.6666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -1.2e-203)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= -8.1e-289)
		tmp = 0.020833333333333332 * (b * (b * b));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + ((b * (0.25 - ((b * b) * 0.027777777777777776))) * (2.0 + (b * 0.6666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -6.5e-9], 1.0, If[LessEqual[b, -1.2e-203], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -8.1e-289], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(N[(b * N[(0.25 - N[(N[(b * b), $MachinePrecision] * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(b * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.5000000000000003e-9

   1. Initial program 97.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 98.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 98.0%

       \[\leadsto \color{blue}{1} \]

   if -6.5000000000000003e-9 < b < -1.1999999999999999e-203

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   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

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 54.8%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

   if -1.1999999999999999e-203 < b < -8.1000000000000003e-289

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 36.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 36.7%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \frac{-1}{4}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{-1}{4} + \color{blue}{\frac{1}{48} \cdot {b}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left({b}^{2} \cdot \color{blue}{\frac{1}{48}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(b \cdot b\right) \cdot \frac{1}{48}\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(b \cdot \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 36.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 36.7%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 66.5%

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

   if -8.1000000000000003e-289 < b

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 82.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 82.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 71.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 71.2%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + b \cdot \frac{1}{6}\right) \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)}{\frac{1}{2} - b \cdot \frac{1}{6}} \cdot b\right)\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b}{\color{blue}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right) \cdot \color{blue}{\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right), \color{blue}{\left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right), b\right), \left(\frac{\color{blue}{1}}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right), \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      9. swap-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot b\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{2} - b \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 71.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 71.2%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \color{blue}{\left(\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b\right) \cdot \frac{1}{0.5 - b \cdot 0.16666666666666666}}\right)} \]

   11. Taylor expanded in b around 0

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \color{blue}{\left(2 + \frac{2}{3} \cdot b\right)}\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{2}{3} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \left(b \cdot \color{blue}{\frac{2}{3}}\right)\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 78.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\frac{2}{3}}\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 78.2%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \left(\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b\right) \cdot \color{blue}{\left(2 + b \cdot 0.6666666666666666\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-203}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -8.1 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \left(b \cdot \left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right)\right) \cdot \left(2 + b \cdot 0.6666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -2.4 \cdot 10^{-202}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -4.9 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + 2 \cdot \left(b \cdot \left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -6.5e-9)
   1.0
   (if (<= b -2.4e-202)
     (+ 0.5 (* a 0.25))
     (if (<= b -4.9e-289)
       (* 0.020833333333333332 (* b (* b b)))
       (/
        1.0
        (+
         2.0
         (*
          b
          (+
           1.0
           (* 2.0 (* b (- 0.25 (* (* b b) 0.027777777777777776))))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -2.4e-202) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -4.9e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (2.0 * (b * (0.25 - ((b * b) * 0.027777777777777776)))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-6.5d-9)) then
        tmp = 1.0d0
    else if (b <= (-2.4d-202)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= (-4.9d-289)) then
        tmp = 0.020833333333333332d0 * (b * (b * b))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (2.0d0 * (b * (0.25d0 - ((b * b) * 0.027777777777777776d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -2.4e-202) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -4.9e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (2.0 * (b * (0.25 - ((b * b) * 0.027777777777777776)))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -6.5e-9:
		tmp = 1.0
	elif b <= -2.4e-202:
		tmp = 0.5 + (a * 0.25)
	elif b <= -4.9e-289:
		tmp = 0.020833333333333332 * (b * (b * b))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (2.0 * (b * (0.25 - ((b * b) * 0.027777777777777776)))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -2.4e-202)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= -4.9e-289)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(2.0 * Float64(b * Float64(0.25 - Float64(Float64(b * b) * 0.027777777777777776))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -2.4e-202)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= -4.9e-289)
		tmp = 0.020833333333333332 * (b * (b * b));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (2.0 * (b * (0.25 - ((b * b) * 0.027777777777777776)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -6.5e-9], 1.0, If[LessEqual[b, -2.4e-202], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -4.9e-289], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(2.0 * N[(b * N[(0.25 - N[(N[(b * b), $MachinePrecision] * 0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.5000000000000003e-9

   1. Initial program 97.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 98.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 98.0%

       \[\leadsto \color{blue}{1} \]

   if -6.5000000000000003e-9 < b < -2.4000000000000001e-202

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   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

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 54.8%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

   if -2.4000000000000001e-202 < b < -4.90000000000000008e-289

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 36.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 36.7%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \frac{-1}{4}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{-1}{4} + \color{blue}{\frac{1}{48} \cdot {b}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left({b}^{2} \cdot \color{blue}{\frac{1}{48}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(b \cdot b\right) \cdot \frac{1}{48}\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(b \cdot \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 36.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 36.7%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 66.5%

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

   if -4.90000000000000008e-289 < b

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 82.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 82.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 71.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 71.2%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} + b \cdot \frac{1}{6}\right) \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      2. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)}{\frac{1}{2} - b \cdot \frac{1}{6}} \cdot b\right)\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b}{\color{blue}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right) \cdot \color{blue}{\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right) \cdot b\right), \color{blue}{\left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right), b\right), \left(\frac{\color{blue}{1}}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right), \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot \frac{1}{6}\right) \cdot \left(b \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      9. swap-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \left(\left(b \cdot b\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{6} \cdot \frac{1}{6}\right)\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \left(\frac{1}{\frac{1}{2} - b \cdot \frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{2} - b \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \frac{1}{6}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 71.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 71.2%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \color{blue}{\left(\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b\right) \cdot \frac{1}{0.5 - b \cdot 0.16666666666666666}}\right)} \]

   11. Taylor expanded in b around 0

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \frac{1}{36}\right)\right), b\right), \color{blue}{2}\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 76.2%

       \[\leadsto \frac{1}{2 + b \cdot \left(1 + \left(\left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right) \cdot b\right) \cdot \color{blue}{2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -2.4 \cdot 10^{-202}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -4.9 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + 2 \cdot \left(b \cdot \left(0.25 - \left(b \cdot b\right) \cdot 0.027777777777777776\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.0% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -2.5 \cdot 10^{-203}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;b \leq 2.4:\\ \;\;\;\;0.5 + b \cdot \left(-0.25 + b \cdot \left(b \cdot 0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot \left(0.5 + b \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -6.5e-9)
   1.0
   (if (<= b -2.5e-203)
     (+ 0.5 (* a 0.25))
     (if (<= b -5e-289)
       (* 0.020833333333333332 (* b (* b b)))
       (if (<= b 2.4)
         (+ 0.5 (* b (+ -0.25 (* b (* b 0.020833333333333332)))))
         (/ 1.0 (* (* b b) (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -2.5e-203) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -5e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else if (b <= 2.4) {
		tmp = 0.5 + (b * (-0.25 + (b * (b * 0.020833333333333332))));
	} else {
		tmp = 1.0 / ((b * b) * (0.5 + (b * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-6.5d-9)) then
        tmp = 1.0d0
    else if (b <= (-2.5d-203)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= (-5d-289)) then
        tmp = 0.020833333333333332d0 * (b * (b * b))
    else if (b <= 2.4d0) then
        tmp = 0.5d0 + (b * ((-0.25d0) + (b * (b * 0.020833333333333332d0))))
    else
        tmp = 1.0d0 / ((b * b) * (0.5d0 + (b * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -2.5e-203) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -5e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else if (b <= 2.4) {
		tmp = 0.5 + (b * (-0.25 + (b * (b * 0.020833333333333332))));
	} else {
		tmp = 1.0 / ((b * b) * (0.5 + (b * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -6.5e-9:
		tmp = 1.0
	elif b <= -2.5e-203:
		tmp = 0.5 + (a * 0.25)
	elif b <= -5e-289:
		tmp = 0.020833333333333332 * (b * (b * b))
	elif b <= 2.4:
		tmp = 0.5 + (b * (-0.25 + (b * (b * 0.020833333333333332))))
	else:
		tmp = 1.0 / ((b * b) * (0.5 + (b * 0.16666666666666666)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -2.5e-203)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= -5e-289)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	elseif (b <= 2.4)
		tmp = Float64(0.5 + Float64(b * Float64(-0.25 + Float64(b * Float64(b * 0.020833333333333332)))));
	else
		tmp = Float64(1.0 / Float64(Float64(b * b) * Float64(0.5 + Float64(b * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -2.5e-203)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= -5e-289)
		tmp = 0.020833333333333332 * (b * (b * b));
	elseif (b <= 2.4)
		tmp = 0.5 + (b * (-0.25 + (b * (b * 0.020833333333333332))));
	else
		tmp = 1.0 / ((b * b) * (0.5 + (b * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -6.5e-9], 1.0, If[LessEqual[b, -2.5e-203], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -5e-289], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4], N[(0.5 + N[(b * N[(-0.25 + N[(b * N[(b * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -6.5000000000000003e-9

   1. Initial program 97.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 98.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 98.0%

       \[\leadsto \color{blue}{1} \]

   if -6.5000000000000003e-9 < b < -2.5000000000000001e-203

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   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

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 54.8%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

   if -2.5000000000000001e-203 < b < -5.00000000000000029e-289

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 36.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 36.7%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \frac{-1}{4}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{-1}{4} + \color{blue}{\frac{1}{48} \cdot {b}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left({b}^{2} \cdot \color{blue}{\frac{1}{48}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(b \cdot b\right) \cdot \frac{1}{48}\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(b \cdot \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 36.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 36.7%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 66.5%

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

   if -5.00000000000000029e-289 < b < 2.39999999999999991

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 67.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 67.7%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \frac{-1}{4}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{-1}{4} + \color{blue}{\frac{1}{48} \cdot {b}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left({b}^{2} \cdot \color{blue}{\frac{1}{48}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(b \cdot b\right) \cdot \frac{1}{48}\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(b \cdot \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 67.8%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 67.8%

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

   if 2.39999999999999991 < b

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 75.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 75.1%

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

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left({b}^{3} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. unpow3 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left({b}^{2} \cdot b\right) \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left({b}^{2} \cdot \color{blue}{\left(b \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left(b \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{b} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{b} \cdot \left(\frac{1}{6} + \frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(b \cdot \left(\frac{1}{2} \cdot \frac{1}{b} + \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(\frac{1}{2} \cdot \frac{1}{b}\right) \cdot b + \color{blue}{\frac{1}{6} \cdot b}\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot \left(\frac{1}{b} \cdot b\right) + \color{blue}{\frac{1}{6}} \cdot b\right)\right)\right) \]

       10. lft-mult-inverse N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} \cdot 1 + \frac{1}{6} \cdot b\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\frac{1}{2} + \color{blue}{\frac{1}{6}} \cdot b\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right) \]

       13. *-lowering-*.f64 75.1%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{b}\right)\right)\right)\right) \]

   11. Simplified 75.1%

       \[\leadsto \frac{1}{\color{blue}{\left(b \cdot b\right) \cdot \left(0.5 + 0.16666666666666666 \cdot b\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -2.5 \cdot 10^{-203}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;b \leq 2.4:\\ \;\;\;\;0.5 + b \cdot \left(-0.25 + b \cdot \left(b \cdot 0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot \left(0.5 + b \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.0% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -9.6 \cdot 10^{-204}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -8.1 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot t\_0\\ \mathbf{elif}\;b \leq 2.85:\\ \;\;\;\;0.5 + b \cdot \left(-0.25 + b \cdot \left(b \cdot 0.020833333333333332\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= b -6.5e-9)
     1.0
     (if (<= b -9.6e-204)
       (+ 0.5 (* a 0.25))
       (if (<= b -8.1e-289)
         (* 0.020833333333333332 t_0)
         (if (<= b 2.85)
           (+ 0.5 (* b (+ -0.25 (* b (* b 0.020833333333333332)))))
           (/ 6.0 t_0)))))))

C

double code(double a, double b) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -9.6e-204) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -8.1e-289) {
		tmp = 0.020833333333333332 * t_0;
	} else if (b <= 2.85) {
		tmp = 0.5 + (b * (-0.25 + (b * (b * 0.020833333333333332))));
	} else {
		tmp = 6.0 / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (b * b)
    if (b <= (-6.5d-9)) then
        tmp = 1.0d0
    else if (b <= (-9.6d-204)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= (-8.1d-289)) then
        tmp = 0.020833333333333332d0 * t_0
    else if (b <= 2.85d0) then
        tmp = 0.5d0 + (b * ((-0.25d0) + (b * (b * 0.020833333333333332d0))))
    else
        tmp = 6.0d0 / t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -9.6e-204) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -8.1e-289) {
		tmp = 0.020833333333333332 * t_0;
	} else if (b <= 2.85) {
		tmp = 0.5 + (b * (-0.25 + (b * (b * 0.020833333333333332))));
	} else {
		tmp = 6.0 / t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (b * b)
	tmp = 0
	if b <= -6.5e-9:
		tmp = 1.0
	elif b <= -9.6e-204:
		tmp = 0.5 + (a * 0.25)
	elif b <= -8.1e-289:
		tmp = 0.020833333333333332 * t_0
	elif b <= 2.85:
		tmp = 0.5 + (b * (-0.25 + (b * (b * 0.020833333333333332))))
	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 <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -9.6e-204)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= -8.1e-289)
		tmp = Float64(0.020833333333333332 * t_0);
	elseif (b <= 2.85)
		tmp = Float64(0.5 + Float64(b * Float64(-0.25 + Float64(b * Float64(b * 0.020833333333333332)))));
	else
		tmp = Float64(6.0 / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (b * b);
	tmp = 0.0;
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -9.6e-204)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= -8.1e-289)
		tmp = 0.020833333333333332 * t_0;
	elseif (b <= 2.85)
		tmp = 0.5 + (b * (-0.25 + (b * (b * 0.020833333333333332))));
	else
		tmp = 6.0 / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6.5e-9], 1.0, If[LessEqual[b, -9.6e-204], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -8.1e-289], N[(0.020833333333333332 * t$95$0), $MachinePrecision], If[LessEqual[b, 2.85], N[(0.5 + N[(b * N[(-0.25 + N[(b * N[(b * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -6.5000000000000003e-9

   1. Initial program 97.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 98.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 98.0%

       \[\leadsto \color{blue}{1} \]

   if -6.5000000000000003e-9 < b < -9.6e-204

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   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

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 54.8%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

   if -9.6e-204 < b < -8.1000000000000003e-289

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 36.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 36.7%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \frac{-1}{4}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{-1}{4} + \color{blue}{\frac{1}{48} \cdot {b}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left({b}^{2} \cdot \color{blue}{\frac{1}{48}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(b \cdot b\right) \cdot \frac{1}{48}\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(b \cdot \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 36.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 36.7%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 66.5%

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

   if -8.1000000000000003e-289 < b < 2.85000000000000009

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 67.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 67.7%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \frac{-1}{4}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{-1}{4} + \color{blue}{\frac{1}{48} \cdot {b}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left({b}^{2} \cdot \color{blue}{\frac{1}{48}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(b \cdot b\right) \cdot \frac{1}{48}\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(b \cdot \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 67.8%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 67.8%

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

   if 2.85000000000000009 < b

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 75.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 75.1%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 75.1%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 75.1%

       \[\leadsto \color{blue}{\frac{6}{b \cdot \left(b \cdot b\right)}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 14: 69.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -9.6 \cdot 10^{-204}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -6.5e-9)
   1.0
   (if (<= b -9.6e-204)
     (+ 0.5 (* a 0.25))
     (if (<= b -7e-289)
       (* 0.020833333333333332 (* b (* b b)))
       (/
        1.0
        (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -9.6e-204) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -7e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-6.5d-9)) then
        tmp = 1.0d0
    else if (b <= (-9.6d-204)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= (-7d-289)) then
        tmp = 0.020833333333333332d0 * (b * (b * b))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -9.6e-204) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -7e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -6.5e-9:
		tmp = 1.0
	elif b <= -9.6e-204:
		tmp = 0.5 + (a * 0.25)
	elif b <= -7e-289:
		tmp = 0.020833333333333332 * (b * (b * b))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -9.6e-204)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= -7e-289)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -9.6e-204)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= -7e-289)
		tmp = 0.020833333333333332 * (b * (b * b));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -6.5e-9], 1.0, If[LessEqual[b, -9.6e-204], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -7e-289], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.5000000000000003e-9

   1. Initial program 97.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 98.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 98.0%

       \[\leadsto \color{blue}{1} \]

   if -6.5000000000000003e-9 < b < -9.6e-204

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   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

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 54.8%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

   if -9.6e-204 < b < -6.9999999999999999e-289

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 36.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 36.7%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \frac{-1}{4}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{-1}{4} + \color{blue}{\frac{1}{48} \cdot {b}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left({b}^{2} \cdot \color{blue}{\frac{1}{48}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(b \cdot b\right) \cdot \frac{1}{48}\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(b \cdot \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 36.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 36.7%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 66.5%

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

   if -6.9999999999999999e-289 < b

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 82.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 82.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 71.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 71.2%

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

Alternative 15: 69.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -2.5 \cdot 10^{-203}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -8.1 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot t\_0\\ \mathbf{elif}\;b \leq 1.92:\\ \;\;\;\;\frac{1}{\frac{1}{0.5 + b \cdot -0.25}}\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= b -6.5e-9)
     1.0
     (if (<= b -2.5e-203)
       (+ 0.5 (* a 0.25))
       (if (<= b -8.1e-289)
         (* 0.020833333333333332 t_0)
         (if (<= b 1.92) (/ 1.0 (/ 1.0 (+ 0.5 (* b -0.25)))) (/ 6.0 t_0)))))))

C

double code(double a, double b) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -2.5e-203) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -8.1e-289) {
		tmp = 0.020833333333333332 * t_0;
	} else if (b <= 1.92) {
		tmp = 1.0 / (1.0 / (0.5 + (b * -0.25)));
	} else {
		tmp = 6.0 / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (b * b)
    if (b <= (-6.5d-9)) then
        tmp = 1.0d0
    else if (b <= (-2.5d-203)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= (-8.1d-289)) then
        tmp = 0.020833333333333332d0 * t_0
    else if (b <= 1.92d0) then
        tmp = 1.0d0 / (1.0d0 / (0.5d0 + (b * (-0.25d0))))
    else
        tmp = 6.0d0 / t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -2.5e-203) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -8.1e-289) {
		tmp = 0.020833333333333332 * t_0;
	} else if (b <= 1.92) {
		tmp = 1.0 / (1.0 / (0.5 + (b * -0.25)));
	} else {
		tmp = 6.0 / t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (b * b)
	tmp = 0
	if b <= -6.5e-9:
		tmp = 1.0
	elif b <= -2.5e-203:
		tmp = 0.5 + (a * 0.25)
	elif b <= -8.1e-289:
		tmp = 0.020833333333333332 * t_0
	elif b <= 1.92:
		tmp = 1.0 / (1.0 / (0.5 + (b * -0.25)))
	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 <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -2.5e-203)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= -8.1e-289)
		tmp = Float64(0.020833333333333332 * t_0);
	elseif (b <= 1.92)
		tmp = Float64(1.0 / Float64(1.0 / Float64(0.5 + Float64(b * -0.25))));
	else
		tmp = Float64(6.0 / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (b * b);
	tmp = 0.0;
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -2.5e-203)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= -8.1e-289)
		tmp = 0.020833333333333332 * t_0;
	elseif (b <= 1.92)
		tmp = 1.0 / (1.0 / (0.5 + (b * -0.25)));
	else
		tmp = 6.0 / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6.5e-9], 1.0, If[LessEqual[b, -2.5e-203], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -8.1e-289], N[(0.020833333333333332 * t$95$0), $MachinePrecision], If[LessEqual[b, 1.92], N[(1.0 / N[(1.0 / N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(6.0 / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -6.5000000000000003e-9

   1. Initial program 97.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 98.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 98.0%

       \[\leadsto \color{blue}{1} \]

   if -6.5000000000000003e-9 < b < -2.5000000000000001e-203

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   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

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 54.8%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

   if -2.5000000000000001e-203 < b < -8.1000000000000003e-289

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 36.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 36.7%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \frac{-1}{4}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{-1}{4} + \color{blue}{\frac{1}{48} \cdot {b}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left({b}^{2} \cdot \color{blue}{\frac{1}{48}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(b \cdot b\right) \cdot \frac{1}{48}\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(b \cdot \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 36.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 36.7%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 66.5%

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

   if -8.1000000000000003e-289 < b < 1.9199999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 67.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 67.7%

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

   6. Taylor expanded in b around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{4} \cdot b\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{-1}{4}}\right)\right) \]

      3. *-lowering-*.f64 67.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{-1}{4}}\right)\right) \]

   8. Simplified 67.6%

      \[\leadsto \color{blue}{0.5 + b \cdot -0.25} \]
   9. Step-by-step derivation

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{2} - b \cdot \frac{-1}{4}}{\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{-1}{4}\right) \cdot \left(b \cdot \frac{-1}{4}\right)}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(b \cdot \frac{-1}{4}\right) \cdot \left(b \cdot \frac{-1}{4}\right)}{\frac{1}{2} - b \cdot \frac{-1}{4}}}}\right)\right) \]

      5. flip-+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\frac{1}{2} + \color{blue}{b \cdot \frac{-1}{4}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{2} + b \cdot \frac{-1}{4}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \frac{-1}{4}\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 67.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{-1}{4}}\right)\right)\right)\right) \]

   10. Applied egg-rr 67.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{0.5 + b \cdot -0.25}}} \]

   if 1.9199999999999999 < b

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 75.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 75.1%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 75.1%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 75.1%

       \[\leadsto \color{blue}{\frac{6}{b \cdot \left(b \cdot b\right)}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 16: 69.0% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -9.6 \cdot 10^{-204}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -8.1 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot t\_0\\ \mathbf{elif}\;b \leq 1.92:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= b -6.5e-9)
     1.0
     (if (<= b -9.6e-204)
       (+ 0.5 (* a 0.25))
       (if (<= b -8.1e-289)
         (* 0.020833333333333332 t_0)
         (if (<= b 1.92) (+ 0.5 (* b -0.25)) (/ 6.0 t_0)))))))

C

double code(double a, double b) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -9.6e-204) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -8.1e-289) {
		tmp = 0.020833333333333332 * t_0;
	} else if (b <= 1.92) {
		tmp = 0.5 + (b * -0.25);
	} else {
		tmp = 6.0 / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (b * b)
    if (b <= (-6.5d-9)) then
        tmp = 1.0d0
    else if (b <= (-9.6d-204)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= (-8.1d-289)) then
        tmp = 0.020833333333333332d0 * t_0
    else if (b <= 1.92d0) then
        tmp = 0.5d0 + (b * (-0.25d0))
    else
        tmp = 6.0d0 / t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -9.6e-204) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -8.1e-289) {
		tmp = 0.020833333333333332 * t_0;
	} else if (b <= 1.92) {
		tmp = 0.5 + (b * -0.25);
	} else {
		tmp = 6.0 / t_0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (b * b)
	tmp = 0
	if b <= -6.5e-9:
		tmp = 1.0
	elif b <= -9.6e-204:
		tmp = 0.5 + (a * 0.25)
	elif b <= -8.1e-289:
		tmp = 0.020833333333333332 * t_0
	elif b <= 1.92:
		tmp = 0.5 + (b * -0.25)
	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 <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -9.6e-204)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= -8.1e-289)
		tmp = Float64(0.020833333333333332 * t_0);
	elseif (b <= 1.92)
		tmp = Float64(0.5 + Float64(b * -0.25));
	else
		tmp = Float64(6.0 / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (b * b);
	tmp = 0.0;
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -9.6e-204)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= -8.1e-289)
		tmp = 0.020833333333333332 * t_0;
	elseif (b <= 1.92)
		tmp = 0.5 + (b * -0.25);
	else
		tmp = 6.0 / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -6.5e-9], 1.0, If[LessEqual[b, -9.6e-204], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -8.1e-289], N[(0.020833333333333332 * t$95$0), $MachinePrecision], If[LessEqual[b, 1.92], N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision], N[(6.0 / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -6.5000000000000003e-9

   1. Initial program 97.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 98.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 98.0%

       \[\leadsto \color{blue}{1} \]

   if -6.5000000000000003e-9 < b < -9.6e-204

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   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

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 54.8%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

   if -9.6e-204 < b < -8.1000000000000003e-289

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 36.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 36.7%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \frac{-1}{4}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{-1}{4} + \color{blue}{\frac{1}{48} \cdot {b}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left({b}^{2} \cdot \color{blue}{\frac{1}{48}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(b \cdot b\right) \cdot \frac{1}{48}\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(b \cdot \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 36.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 36.7%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 66.5%

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

   if -8.1000000000000003e-289 < b < 1.9199999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 67.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 67.7%

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

   6. Taylor expanded in b around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{4} \cdot b\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{-1}{4}}\right)\right) \]

      3. *-lowering-*.f64 67.6%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{-1}{4}}\right)\right) \]

   8. Simplified 67.6%

      \[\leadsto \color{blue}{0.5 + b \cdot -0.25} \]

   if 1.9199999999999999 < b

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 75.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 75.1%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 75.1%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 75.1%

       \[\leadsto \color{blue}{\frac{6}{b \cdot \left(b \cdot b\right)}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 17: 68.8% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -1.08 \cdot 10^{-203}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -8 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(\left(b \cdot b\right) \cdot 0.16666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -6.5e-9)
   1.0
   (if (<= b -1.08e-203)
     (+ 0.5 (* a 0.25))
     (if (<= b -8e-289)
       (* 0.020833333333333332 (* b (* b b)))
       (/ 1.0 (+ 2.0 (* b (* (* b b) 0.16666666666666666))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -1.08e-203) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -8e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (2.0 + (b * ((b * b) * 0.16666666666666666)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-6.5d-9)) then
        tmp = 1.0d0
    else if (b <= (-1.08d-203)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= (-8d-289)) then
        tmp = 0.020833333333333332d0 * (b * (b * b))
    else
        tmp = 1.0d0 / (2.0d0 + (b * ((b * b) * 0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -1.08e-203) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -8e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (2.0 + (b * ((b * b) * 0.16666666666666666)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -6.5e-9:
		tmp = 1.0
	elif b <= -1.08e-203:
		tmp = 0.5 + (a * 0.25)
	elif b <= -8e-289:
		tmp = 0.020833333333333332 * (b * (b * b))
	else:
		tmp = 1.0 / (2.0 + (b * ((b * b) * 0.16666666666666666)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -1.08e-203)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= -8e-289)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(Float64(b * b) * 0.16666666666666666))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -1.08e-203)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= -8e-289)
		tmp = 0.020833333333333332 * (b * (b * b));
	else
		tmp = 1.0 / (2.0 + (b * ((b * b) * 0.16666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -6.5e-9], 1.0, If[LessEqual[b, -1.08e-203], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -8e-289], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(N[(b * b), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.5000000000000003e-9

   1. Initial program 97.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 98.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 98.0%

       \[\leadsto \color{blue}{1} \]

   if -6.5000000000000003e-9 < b < -1.07999999999999997e-203

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   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

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 54.8%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

   if -1.07999999999999997e-203 < b < -8.0000000000000001e-289

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 36.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 36.7%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \frac{-1}{4}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{-1}{4} + \color{blue}{\frac{1}{48} \cdot {b}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left({b}^{2} \cdot \color{blue}{\frac{1}{48}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(b \cdot b\right) \cdot \frac{1}{48}\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(b \cdot \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 36.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 36.7%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 66.5%

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

   if -8.0000000000000001e-289 < b

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 82.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 82.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot b\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 71.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 71.2%

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

   9. Taylor expanded in b around inf

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{6} \cdot {b}^{2}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{6}, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 70.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right) \]

   11. Simplified 70.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -1.08 \cdot 10^{-203}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -8 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(\left(b \cdot b\right) \cdot 0.16666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.3% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq -9.6 \cdot 10^{-204}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -8.1 \cdot 10^{-289}:\\ \;\;\;\;0.020833333333333332 \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -6.5e-9)
   1.0
   (if (<= b -9.6e-204)
     (+ 0.5 (* a 0.25))
     (if (<= b -8.1e-289)
       (* 0.020833333333333332 (* b (* b b)))
       (/ 1.0 (+ b 2.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -9.6e-204) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -8.1e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-6.5d-9)) then
        tmp = 1.0d0
    else if (b <= (-9.6d-204)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= (-8.1d-289)) then
        tmp = 0.020833333333333332d0 * (b * (b * b))
    else
        tmp = 1.0d0 / (b + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -6.5e-9) {
		tmp = 1.0;
	} else if (b <= -9.6e-204) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= -8.1e-289) {
		tmp = 0.020833333333333332 * (b * (b * b));
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -6.5e-9:
		tmp = 1.0
	elif b <= -9.6e-204:
		tmp = 0.5 + (a * 0.25)
	elif b <= -8.1e-289:
		tmp = 0.020833333333333332 * (b * (b * b))
	else:
		tmp = 1.0 / (b + 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -9.6e-204)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= -8.1e-289)
		tmp = Float64(0.020833333333333332 * Float64(b * Float64(b * b)));
	else
		tmp = Float64(1.0 / Float64(b + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -6.5e-9)
		tmp = 1.0;
	elseif (b <= -9.6e-204)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= -8.1e-289)
		tmp = 0.020833333333333332 * (b * (b * b));
	else
		tmp = 1.0 / (b + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -6.5e-9], 1.0, If[LessEqual[b, -9.6e-204], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -8.1e-289], N[(0.020833333333333332 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.5000000000000003e-9

   1. Initial program 97.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 98.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 98.0%

       \[\leadsto \color{blue}{1} \]

   if -6.5000000000000003e-9 < b < -9.6e-204

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{1}\right)\right) \]
   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

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

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{4} \cdot a\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(a \cdot \color{blue}{\frac{1}{4}}\right)\right) \]

      3. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\frac{1}{4}}\right)\right) \]

   8. Simplified 54.8%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

   if -9.6e-204 < b < -8.1000000000000003e-289

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 36.7%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 36.7%

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

   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. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{1}{48} \cdot {b}^{2} + \frac{-1}{4}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \left(\frac{-1}{4} + \color{blue}{\frac{1}{48} \cdot {b}^{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \color{blue}{\left(\frac{1}{48} \cdot {b}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left({b}^{2} \cdot \color{blue}{\frac{1}{48}}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(\left(b \cdot b\right) \cdot \frac{1}{48}\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \left(b \cdot \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \frac{1}{48}\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 36.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{\frac{1}{48}}\right)\right)\right)\right)\right) \]

   8. Simplified 36.7%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \color{blue}{\left({b}^{3}\right)}\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \left(b \cdot {b}^{\color{blue}{2}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right) \]

       6. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{48}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right) \]

   11. Simplified 66.5%

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

   if -8.1000000000000003e-289 < b

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 82.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 82.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b\right)}\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(b + \color{blue}{2}\right)\right) \]

      2. +-lowering-+.f64 38.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, \color{blue}{2}\right)\right) \]

   8. Simplified 38.6%

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

Alternative 19: 54.8% accurate, 30.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	tmp = 0
	if b <= -1.0:
		tmp = 1.0
	else:
		tmp = 1.0 / (b + 2.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1

   1. Initial program 97.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 97.9%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.6%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{1} \]

   if -1 < b

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 71.4%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 71.4%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + b\right)}\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(b + \color{blue}{2}\right)\right) \]

      2. +-lowering-+.f64 41.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(b, \color{blue}{2}\right)\right) \]

   8. Simplified 41.1%

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

Alternative 20: 53.8% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -1.1) 1.0 0.5))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.1d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.1:
		tmp = 1.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.1)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.1)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.1], 1.0, 0.5]

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001

   1. Initial program 97.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. prod-exp N/A

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

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1 + a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\left(\log \left(e^{a} + e^{b}\right) \cdot -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log \left(e^{a} + e^{b}\right), -1\right), a\right)\right) \]

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

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(e^{a} + e^{b}\right)\right), -1\right), a\right)\right) \]

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

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \left(e^{b}\right)\right)\right), -1\right), a\right)\right) \]

      12. exp-lowering-exp.f64 97.9%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{exp.f64}\left(b\right)\right)\right), -1\right), a\right)\right) \]

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in a around inf

      \[\leadsto \mathsf{exp.f64}\left(\color{blue}{a}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.6%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 100.0%

       \[\leadsto \color{blue}{1} \]

   if -1.1000000000000001 < b

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 71.4%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 71.4%

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

   6. Taylor expanded in b around 0

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

   8. Simplified 40.0%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 38.8% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in a around 0

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

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + e^{b}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{b}\right)}\right)\right) \]

   3. exp-lowering-exp.f64 76.6%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(b\right)\right)\right) \]

5. Simplified 76.6%

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

6. Taylor expanded in b around 0

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

8. Simplified 36.1%

   \[\leadsto \color{blue}{0.5} \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.9× 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 2024152 
(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))))