Quotient of sum of exps

Percentage Accurate: 98.8% → 98.2%

Time: 8.2s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. a = 2.0910635163763607e+278
  2. b = -1.3799727760353565e+21

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: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999999934314354:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.9999999934314354)
   (/ (exp a) (+ (exp a) 1.0))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.9999999934314354) {
		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) <= 0.9999999934314354d0) 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) <= 0.9999999934314354) {
		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) <= 0.9999999934314354:
		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) <= 0.9999999934314354)
		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) <= 0.9999999934314354)
		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], 0.9999999934314354], 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) < 0.99999999343143542

   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 100.0%

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

   if 0.99999999343143542 < (exp.f64 a)

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

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

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

      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 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999999934314354:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

Alternative 3: 97.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999998:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.99999998) (/ (exp a) 2.0) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.99999998) {
		tmp = exp(a) / 2.0;
	} 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) <= 0.99999998d0) then
        tmp = exp(a) / 2.0d0
    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) <= 0.99999998) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.99999998:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.99999998)
		tmp = Float64(exp(a) / 2.0);
	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) <= 0.99999998)
		tmp = exp(a) / 2.0;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.99999998], N[(N[Exp[a], $MachinePrecision] / 2.0), $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) < 0.999999980000000011

   1. Initial program 98.7%

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{2}\right) \]
   7. Step-by-step derivation

   8. Simplified 96.4%

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

   if 0.999999980000000011 < (exp.f64 a)

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

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

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

      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 98.9%

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

   5. Simplified 98.9%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999998:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.2e+92)
   (/ (exp a) 2.0)
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.2e+92) {
		tmp = exp(a) / 2.0;
	} 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 <= 2.2d+92) then
        tmp = exp(a) / 2.0d0
    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 <= 2.2e+92) {
		tmp = Math.exp(a) / 2.0;
	} 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 <= 2.2e+92:
		tmp = math.exp(a) / 2.0
	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 <= 2.2e+92)
		tmp = Float64(exp(a) / 2.0);
	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 <= 2.2e+92)
		tmp = exp(a) / 2.0;
	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, 2.2e+92], N[(N[Exp[a], $MachinePrecision] / 2.0), $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 2 regimes

2. if b < 2.19999999999999992e92

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 76.4%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{2}\right) \]
   7. Step-by-step derivation

   8. Simplified 74.3%

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

   if 2.19999999999999992e92 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

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

      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 96.1%

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

   8. Simplified 96.1%

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

Alternative 5: 59.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-122}:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{2 + \frac{\left(b \cdot b\right) \cdot \left(0.25 \cdot \left(b \cdot b\right)\right) - b \cdot b}{b \cdot \left(b \cdot 0.5\right) - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.65e-122)
   (+
    0.5
    (*
     a
     (+
      0.25
      (*
       (* a a)
       (+ -0.020833333333333332 (* (* a a) 0.0020833333333333333))))))
   (if (<= b 1.35e+154)
     (/
      1.0
      (+
       2.0
       (/ (- (* (* b b) (* 0.25 (* b b))) (* b b)) (- (* b (* b 0.5)) b))))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.65e-122) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	} else if (b <= 1.35e+154) {
		tmp = 1.0 / (2.0 + ((((b * b) * (0.25 * (b * b))) - (b * b)) / ((b * (b * 0.5)) - b)));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.65d-122) then
        tmp = 0.5d0 + (a * (0.25d0 + ((a * a) * ((-0.020833333333333332d0) + ((a * a) * 0.0020833333333333333d0)))))
    else if (b <= 1.35d+154) then
        tmp = 1.0d0 / (2.0d0 + ((((b * b) * (0.25d0 * (b * b))) - (b * b)) / ((b * (b * 0.5d0)) - b)))
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.65e-122) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	} else if (b <= 1.35e+154) {
		tmp = 1.0 / (2.0 + ((((b * b) * (0.25 * (b * b))) - (b * b)) / ((b * (b * 0.5)) - b)));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.65e-122:
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))))
	elif b <= 1.35e+154:
		tmp = 1.0 / (2.0 + ((((b * b) * (0.25 * (b * b))) - (b * b)) / ((b * (b * 0.5)) - b)))
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.65e-122)
		tmp = Float64(0.5 + Float64(a * Float64(0.25 + Float64(Float64(a * a) * Float64(-0.020833333333333332 + Float64(Float64(a * a) * 0.0020833333333333333))))));
	elseif (b <= 1.35e+154)
		tmp = Float64(1.0 / Float64(2.0 + Float64(Float64(Float64(Float64(b * b) * Float64(0.25 * Float64(b * b))) - Float64(b * b)) / Float64(Float64(b * Float64(b * 0.5)) - b))));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.65e-122)
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	elseif (b <= 1.35e+154)
		tmp = 1.0 / (2.0 + ((((b * b) * (0.25 * (b * b))) - (b * b)) / ((b * (b * 0.5)) - b)));
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.65e-122], N[(0.5 + N[(a * N[(0.25 + N[(N[(a * a), $MachinePrecision] * N[(-0.020833333333333332 + N[(N[(a * a), $MachinePrecision] * 0.0020833333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(1.0 / N[(2.0 + N[(N[(N[(N[(b * b), $MachinePrecision] * N[(0.25 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.65e-122

   1. Initial program 98.2%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 75.5%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \frac{-1}{48}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{-1}{48} + \color{blue}{\frac{1}{480} \cdot {a}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \color{blue}{\left(\frac{1}{480} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left({a}^{2} \cdot \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left(a \cdot a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 52.5%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 52.5%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)} \]

   if 1.65e-122 < b < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-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 85.1%

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

   5. Simplified 85.1%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 32.6%

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

   8. Simplified 32.6%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. flip-+ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. swap-sqr N/A

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

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

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. swap-sqr N/A

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

      14. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      19. --lowering--.f64 N/A

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

   10. Applied egg-rr 64.7%

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

   if 1.35000000000000003e154 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-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

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({b}^{2}\right)}\right) \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-122}:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{2 + \frac{\left(b \cdot b\right) \cdot \left(0.25 \cdot \left(b \cdot b\right)\right) - b \cdot b}{b \cdot \left(b \cdot 0.5\right) - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.2% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-10}:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\left(b \cdot b\right) \cdot \left(0.25 \cdot \left(b \cdot b\right)\right) - b \cdot b}{0.5 \cdot \left(b \cdot b\right) - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.4e-10)
   (+
    0.5
    (*
     a
     (+
      0.25
      (*
       (* a a)
       (+ -0.020833333333333332 (* (* a a) 0.0020833333333333333))))))
   (if (<= b 1.35e+154)
     (/ 1.0 (/ (- (* (* b b) (* 0.25 (* b b))) (* b b)) (- (* 0.5 (* b b)) b)))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.4e-10) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	} else if (b <= 1.35e+154) {
		tmp = 1.0 / ((((b * b) * (0.25 * (b * b))) - (b * b)) / ((0.5 * (b * b)) - b));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.4d-10) then
        tmp = 0.5d0 + (a * (0.25d0 + ((a * a) * ((-0.020833333333333332d0) + ((a * a) * 0.0020833333333333333d0)))))
    else if (b <= 1.35d+154) then
        tmp = 1.0d0 / ((((b * b) * (0.25d0 * (b * b))) - (b * b)) / ((0.5d0 * (b * b)) - b))
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.4e-10) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	} else if (b <= 1.35e+154) {
		tmp = 1.0 / ((((b * b) * (0.25 * (b * b))) - (b * b)) / ((0.5 * (b * b)) - b));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.4e-10:
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))))
	elif b <= 1.35e+154:
		tmp = 1.0 / ((((b * b) * (0.25 * (b * b))) - (b * b)) / ((0.5 * (b * b)) - b))
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.4e-10)
		tmp = Float64(0.5 + Float64(a * Float64(0.25 + Float64(Float64(a * a) * Float64(-0.020833333333333332 + Float64(Float64(a * a) * 0.0020833333333333333))))));
	elseif (b <= 1.35e+154)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(b * b) * Float64(0.25 * Float64(b * b))) - Float64(b * b)) / Float64(Float64(0.5 * Float64(b * b)) - b)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.4e-10)
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	elseif (b <= 1.35e+154)
		tmp = 1.0 / ((((b * b) * (0.25 * (b * b))) - (b * b)) / ((0.5 * (b * b)) - b));
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.4e-10], N[(0.5 + N[(a * N[(0.25 + N[(N[(a * a), $MachinePrecision] * N[(-0.020833333333333332 + N[(N[(a * a), $MachinePrecision] * 0.0020833333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(1.0 / N[(N[(N[(N[(b * b), $MachinePrecision] * N[(0.25 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.4e-10

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 78.2%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \frac{-1}{48}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{-1}{48} + \color{blue}{\frac{1}{480} \cdot {a}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \color{blue}{\left(\frac{1}{480} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left({a}^{2} \cdot \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left(a \cdot a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 54.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 54.7%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)} \]

   if 2.4e-10 < b < 1.35000000000000003e154

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-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 93.8%

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

   5. Simplified 93.8%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 5.7%

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

   8. Simplified 5.7%

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

   9. Taylor expanded in b around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. lft-mult-inverse N/A

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

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

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

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

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

       8. *-lowering-*.f64 5.7%

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

   11. Simplified 5.7%

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-lft-identity N/A

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

       4. flip-+ N/A

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

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

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

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

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

       7. *-commutative N/A

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

       8. *-commutative N/A

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

       9. swap-sqr N/A

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

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

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

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

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

       12. *-commutative N/A

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

       13. *-commutative N/A

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

       14. swap-sqr N/A

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

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

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

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

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

       17. metadata-eval N/A

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

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

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

       19. --lowering--.f64 N/A

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

   13. Applied egg-rr 59.5%

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

   if 1.35000000000000003e154 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-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

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({b}^{2}\right)}\right) \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-10}:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\left(b \cdot b\right) \cdot \left(0.25 \cdot \left(b \cdot b\right)\right) - b \cdot b}{0.5 \cdot \left(b \cdot b\right) - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.7% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-10}:\\ \;\;\;\;0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)\\ \mathbf{elif}\;b \leq 10^{+103}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.4e-10)
   (+
    0.5
    (*
     a
     (+
      0.25
      (*
       (* a a)
       (+ -0.020833333333333332 (* (* a a) 0.0020833333333333333))))))
   (if (<= b 1e+103)
     (* -0.020833333333333332 (* a (* a a)))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.4e-10) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	} else if (b <= 1e+103) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} 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 <= 2.4d-10) then
        tmp = 0.5d0 + (a * (0.25d0 + ((a * a) * ((-0.020833333333333332d0) + ((a * a) * 0.0020833333333333333d0)))))
    else if (b <= 1d+103) then
        tmp = (-0.020833333333333332d0) * (a * (a * a))
    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 <= 2.4e-10) {
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	} else if (b <= 1e+103) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} 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 <= 2.4e-10:
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))))
	elif b <= 1e+103:
		tmp = -0.020833333333333332 * (a * (a * a))
	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 <= 2.4e-10)
		tmp = Float64(0.5 + Float64(a * Float64(0.25 + Float64(Float64(a * a) * Float64(-0.020833333333333332 + Float64(Float64(a * a) * 0.0020833333333333333))))));
	elseif (b <= 1e+103)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	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 <= 2.4e-10)
		tmp = 0.5 + (a * (0.25 + ((a * a) * (-0.020833333333333332 + ((a * a) * 0.0020833333333333333)))));
	elseif (b <= 1e+103)
		tmp = -0.020833333333333332 * (a * (a * a));
	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, 2.4e-10], N[(0.5 + N[(a * N[(0.25 + N[(N[(a * a), $MachinePrecision] * N[(-0.020833333333333332 + N[(N[(a * a), $MachinePrecision] * 0.0020833333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+103], N[(-0.020833333333333332 * N[(a * N[(a * a), $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 3 regimes

2. if b < 2.4e-10

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 78.2%

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

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{2} + a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(a \cdot \left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{1}{4} + {a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{1}{480} \cdot {a}^{2} - \frac{1}{48}\right)}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{1}{480} \cdot {a}^{2}} - \frac{1}{48}\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{480} \cdot {a}^{2} + \frac{-1}{48}\right)\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{-1}{48} + \color{blue}{\frac{1}{480} \cdot {a}^{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \color{blue}{\left(\frac{1}{480} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \left({a}^{2} \cdot \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\frac{1}{480}}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\left(a \cdot a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 54.7%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{-1}{48}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \frac{1}{480}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 54.7%

      \[\leadsto \color{blue}{0.5 + a \cdot \left(0.25 + \left(a \cdot a\right) \cdot \left(-0.020833333333333332 + \left(a \cdot a\right) \cdot 0.0020833333333333333\right)\right)} \]

   if 2.4e-10 < b < 1e103

   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 51.6%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 2.5%

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

   8. Simplified 2.5%

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

   9. Taylor expanded in a around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 27.5%

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

   11. Simplified 27.5%

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

   if 1e103 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

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

      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 100.0%

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

   8. Simplified 100.0%

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

Alternative 8: 61.0% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 400:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 10^{+103}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 400.0)
   (+ 0.5 (* a 0.25))
   (if (<= b 1e+103)
     (* -0.020833333333333332 (* a (* a a)))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 400.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1e+103) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} 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 <= 400.0d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= 1d+103) then
        tmp = (-0.020833333333333332d0) * (a * (a * a))
    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 <= 400.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1e+103) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} 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 <= 400.0:
		tmp = 0.5 + (a * 0.25)
	elif b <= 1e+103:
		tmp = -0.020833333333333332 * (a * (a * a))
	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 <= 400.0)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= 1e+103)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	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 <= 400.0)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= 1e+103)
		tmp = -0.020833333333333332 * (a * (a * a));
	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, 400.0], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+103], N[(-0.020833333333333332 * N[(a * N[(a * a), $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 3 regimes

2. if b < 400

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 78.4%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 53.8%

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

   if 400 < b < 1e103

   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 46.2%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 2.5%

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

   8. Simplified 2.5%

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

   9. Taylor expanded in a around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 30.4%

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

   11. Simplified 30.4%

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

   if 1e103 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

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

      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 100.0%

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

   8. Simplified 100.0%

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

Alternative 9: 58.8% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 350:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;-0.020833333333333332 \cdot \left(a \cdot \left(a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 350.0)
   (+ 0.5 (* a 0.25))
   (if (<= b 1.35e+154)
     (* -0.020833333333333332 (* a (* a a)))
     (/ 2.0 (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 350.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1.35e+154) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 350.0d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= 1.35d+154) then
        tmp = (-0.020833333333333332d0) * (a * (a * a))
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 350.0) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1.35e+154) {
		tmp = -0.020833333333333332 * (a * (a * a));
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 350.0:
		tmp = 0.5 + (a * 0.25)
	elif b <= 1.35e+154:
		tmp = -0.020833333333333332 * (a * (a * a))
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 350.0)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= 1.35e+154)
		tmp = Float64(-0.020833333333333332 * Float64(a * Float64(a * a)));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 350.0)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= 1.35e+154)
		tmp = -0.020833333333333332 * (a * (a * a));
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 350.0], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(-0.020833333333333332 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 350

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 78.4%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 53.8%

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

   if 350 < b < 1.35000000000000003e154

   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 39.9%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 2.6%

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

   8. Simplified 2.6%

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

   9. Taylor expanded in a around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 40.0%

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

   11. Simplified 40.0%

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

   if 1.35000000000000003e154 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. /-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

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({b}^{2}\right)}\right) \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 100.0%

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

   11. Simplified 100.0%

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

Alternative 10: 53.1% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-10}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.4e-10) (+ 0.5 (* a 0.25)) (/ 2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.4e-10) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.4d-10) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.4e-10) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.4e-10:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.4e-10)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.4e-10)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.4e-10], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.4e-10

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 78.2%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 54.3%

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

   if 2.4e-10 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

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

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

      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 97.0%

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

   5. Simplified 97.0%

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

   6. Taylor expanded in b around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 54.3%

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

   8. Simplified 54.3%

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

   9. Taylor expanded in b around inf

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

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

          \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({b}^{2}\right)}\right) \]

       2. unpow2 N/A

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

       3. *-lowering-*.f64 54.3%

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

   11. Simplified 54.3%

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

Alternative 11: 39.3% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + a \cdot 0.25 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ 0.5 (* a 0.25)))

C

double code(double a, double b) {
	return 0.5 + (a * 0.25);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0 + (a * 0.25d0)
end function

Java

public static double code(double a, double b) {
	return 0.5 + (a * 0.25);
}

Python

def code(a, b):
	return 0.5 + (a * 0.25)

Julia

function code(a, b)
	return Float64(0.5 + Float64(a * 0.25))
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5 + (a * 0.25);
end

Wolfram

code[a_, b_] := N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

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 70.4%

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

6. Taylor expanded in a around 0

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

8. Simplified 41.5%

   \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
9. Add Preprocessing

Alternative 12: 39.1% 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 98.8%

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

3. Taylor expanded in a around 0

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

   1. /-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 80.3%

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

5. Simplified 80.3%

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

6. Taylor expanded in b around 0

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

8. Simplified 40.9%

   \[\leadsto \color{blue}{0.5} \]
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 2024191 
(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))))