Quotient of sum of exps

Percentage Accurate: 98.9% → 100.0%

Time: 5.7s

Alternatives: 12

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{b - a}\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (exp (- (log1p (exp (- b a))))))

C

double code(double a, double b) {
	return exp(-log1p(exp((b - a))));
}

Java

public static double code(double a, double b) {
	return Math.exp(-Math.log1p(Math.exp((b - a))));
}

Python

def code(a, b):
	return math.exp(-math.log1p(math.exp((b - a))))

Julia

function code(a, b)
	return exp(Float64(-log1p(exp(Float64(b - a)))))
end

Wolfram

code[a_, b_] := N[Exp[(-N[Log[1 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Simplified 100.0%

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

   1. add-exp-log 100.0%

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

   2. frac-2neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. log-rec 100.0%

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

   5. sub-neg 100.0%

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

   6. neg-mul-1 100.0%

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

   7. distribute-neg-in 100.0%

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

   8. metadata-eval 100.0%

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

   9. log1p-define 100.0%

      \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(--1 \cdot e^{b - a}\right)}} \]

   10. neg-mul-1 100.0%

       \[\leadsto e^{-\mathsf{log1p}\left(-\color{blue}{\left(-e^{b - a}\right)}\right)} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(-\left(-e^{b - a}\right)\right)}} \]

7. Final simplification 100.0%

   \[\leadsto e^{-\mathsf{log1p}\left(e^{b - a}\right)} \]
8. Add Preprocessing

Alternative 2: 98.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6200000:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6200000.0) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6200000.0) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-6200000.0d0)) then
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6200000.0) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6200000.0:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6200000.0)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6200000.0)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6200000.0], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.2e6

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 100.0%

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

   if -6.2e6 < a

   1. Initial program 98.9%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 99.0%

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

Alternative 3: 93.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{+100}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.4e+100)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.4e+100) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-6.4d+100)) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6.4e+100) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.4e+100:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.4e+100)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.4e+100)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.4e+100], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.3999999999999998e100

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 100.0%

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

   6. Taylor expanded in a around 0 98.2%

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

   if -6.3999999999999998e100 < a

   1. Initial program 99.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 92.5%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.4 \cdot 10^{+100}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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]

Derivation

1. Initial program 99.2%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 5: 72.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot 0.16666666666666666\right)\\ \mathbf{if}\;b \leq 6.1 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 10^{+94}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \frac{\left(b \cdot 0.5\right) \cdot \left(b \cdot 0.5\right) - t\_0 \cdot t\_0}{b \cdot 0.5 - t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b 0.16666666666666666))))
   (if (<= b 6.1e+69)
     (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
     (if (<= b 1e+94)
       (/
        1.0
        (+
         2.0
         (*
          b
          (+
           1.0
           (/ (- (* (* b 0.5) (* b 0.5)) (* t_0 t_0)) (- (* b 0.5) t_0))))))
       (/
        1.0
        (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))))

C

double code(double a, double b) {
	double t_0 = b * (b * 0.16666666666666666);
	double tmp;
	if (b <= 6.1e+69) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else if (b <= 1e+94) {
		tmp = 1.0 / (2.0 + (b * (1.0 + ((((b * 0.5) * (b * 0.5)) - (t_0 * t_0)) / ((b * 0.5) - t_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) :: t_0
    real(8) :: tmp
    t_0 = b * (b * 0.16666666666666666d0)
    if (b <= 6.1d+69) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    else if (b <= 1d+94) then
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + ((((b * 0.5d0) * (b * 0.5d0)) - (t_0 * t_0)) / ((b * 0.5d0) - t_0)))))
    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 t_0 = b * (b * 0.16666666666666666);
	double tmp;
	if (b <= 6.1e+69) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else if (b <= 1e+94) {
		tmp = 1.0 / (2.0 + (b * (1.0 + ((((b * 0.5) * (b * 0.5)) - (t_0 * t_0)) / ((b * 0.5) - t_0)))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (b * 0.16666666666666666)
	tmp = 0
	if b <= 6.1e+69:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	elif b <= 1e+94:
		tmp = 1.0 / (2.0 + (b * (1.0 + ((((b * 0.5) * (b * 0.5)) - (t_0 * t_0)) / ((b * 0.5) - t_0)))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(b * 0.16666666666666666))
	tmp = 0.0
	if (b <= 6.1e+69)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))));
	elseif (b <= 1e+94)
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(Float64(Float64(Float64(b * 0.5) * Float64(b * 0.5)) - Float64(t_0 * t_0)) / Float64(Float64(b * 0.5) - t_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)
	t_0 = b * (b * 0.16666666666666666);
	tmp = 0.0;
	if (b <= 6.1e+69)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	elseif (b <= 1e+94)
		tmp = 1.0 / (2.0 + (b * (1.0 + ((((b * 0.5) * (b * 0.5)) - (t_0 * t_0)) / ((b * 0.5) - t_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_] := Block[{t$95$0 = N[(b * N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 6.1e+69], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+94], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(N[(N[(N[(b * 0.5), $MachinePrecision] * N[(b * 0.5), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(b * 0.5), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 6.1000000000000001e69

   1. Initial program 99.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 74.3%

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

   6. Taylor expanded in a around 0 66.3%

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

   if 6.1000000000000001e69 < b < 1e94

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 7.1%

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

      1. *-commutative 7.1%

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

   8. Simplified 7.1%

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

      1. distribute-lft-in 7.1%

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

      2. flip-+ 86.6%

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

   10. Applied egg-rr 86.6%

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

   if 1e94 < b

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. *-commutative 100.0%

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

   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. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.1 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 10^{+94}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \frac{\left(b \cdot 0.5\right) \cdot \left(b \cdot 0.5\right) - \left(b \cdot \left(b \cdot 0.16666666666666666\right)\right) \cdot \left(b \cdot \left(b \cdot 0.16666666666666666\right)\right)}{b \cdot 0.5 - b \cdot \left(b \cdot 0.16666666666666666\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.5% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+84}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3e+84)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3e+84) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} 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 <= 3d+84) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    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 <= 3e+84) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} 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 <= 3e+84:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	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 <= 3e+84)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))));
	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 <= 3e+84)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	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, 3e+84], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 2 regimes

2. if b < 2.99999999999999996e84

   1. Initial program 99.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 74.6%

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

   6. Taylor expanded in a around 0 66.1%

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

   if 2.99999999999999996e84 < b

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 89.0%

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

      1. *-commutative 89.0%

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

   8. Simplified 89.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+84}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{+135}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.8e+135)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.8e+135) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 8.8d+135) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 8.8e+135) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 8.8e+135:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8.8e+135)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 8.8e+135)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8.8e+135], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.7999999999999998e135

   1. Initial program 99.1%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 71.6%

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

   6. Taylor expanded in a around 0 62.9%

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

   if 8.7999999999999998e135 < b

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 79.8%

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

      1. *-commutative 79.8%

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

   8. Simplified 79.8%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{+135}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.0% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.6e+136)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a 0.5)))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.6e+136) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.6d+136) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * 0.5d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.6e+136) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.6e+136:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.6e+136)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * 0.5)))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.6e+136)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.6e+136], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6000000000000001e136

   1. Initial program 99.1%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 71.6%

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

   6. Taylor expanded in a around 0 58.0%

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

   if 2.6000000000000001e136 < b

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 79.8%

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

      1. *-commutative 79.8%

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

   8. Simplified 79.8%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.8% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.9e+136)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a 0.5)))))
   (/ (/ -2.0 b) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.9e+136) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else {
		tmp = (-2.0 / b) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 3.9d+136) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * 0.5d0))))
    else
        tmp = ((-2.0d0) / b) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.9e+136) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else {
		tmp = (-2.0 / b) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.9e+136:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))))
	else:
		tmp = (-2.0 / b) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.9e+136)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * 0.5)))));
	else
		tmp = Float64(Float64(-2.0 / b) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.9e+136)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	else
		tmp = (-2.0 / b) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.9e+136], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / b), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.90000000000000019e136

   1. Initial program 99.1%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 71.6%

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

   6. Taylor expanded in a around 0 58.0%

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

   if 3.90000000000000019e136 < b

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 5.8%

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

      1. +-commutative 5.8%

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

   8. Simplified 5.8%

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

   9. Taylor expanded in b around inf 5.8%

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

       1. associate-*r/ 5.8%

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

       2. metadata-eval 5.8%

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

   11. Simplified 5.8%

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

   12. Taylor expanded in b around 0 78.8%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{b}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.3% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.4e+41) (/ 1.0 (- 2.0 a)) (/ (/ -2.0 b) b)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.4e+41) {
		tmp = 1.0 / (2.0 - 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 <= 2.4d+41) then
        tmp = 1.0d0 / (2.0d0 - 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 <= 2.4e+41) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = (-2.0 / b) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.4e+41:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = (-2.0 / b) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.4e+41)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(Float64(-2.0 / b) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.4e+41)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = (-2.0 / b) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.4e+41], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / b), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.4000000000000002e41

   1. Initial program 99.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 76.6%

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

   6. Taylor expanded in a around 0 49.8%

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

      1. neg-mul-1 49.8%

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

      2. unsub-neg 49.8%

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

   8. Simplified 49.8%

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

   if 2.4000000000000002e41 < b

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 4.9%

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

      1. +-commutative 4.9%

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

   8. Simplified 4.9%

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

   9. Taylor expanded in b around inf 4.9%

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

       1. associate-*r/ 4.9%

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

       2. metadata-eval 4.9%

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

   11. Simplified 4.9%

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

   12. Taylor expanded in b around 0 43.6%

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

Alternative 11: 40.8% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (- 2.0 a)))

C

double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 - a)
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Python

def code(a, b):
	return 1.0 / (2.0 - a)

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 - a))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 - a);
end

Wolfram

code[a_, b_] := N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
4. Add Preprocessing

5. Taylor expanded in b around 0 67.4%

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

6. Taylor expanded in a around 0 40.4%

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

   1. neg-mul-1 40.4%

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

   2. unsub-neg 40.4%

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

8. Simplified 40.4%

   \[\leadsto \frac{1}{\color{blue}{2 - a}} \]
9. Add Preprocessing

Alternative 12: 39.9% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 99.2%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{-1}{-1 - e^{b - a}}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 80.3%

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

6. Taylor expanded in b around 0 39.6%

   \[\leadsto \color{blue}{0.5} \]
7. Add Preprocessing

Developer target: 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 2024107 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :alt
  (/ 1.0 (+ 1.0 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))