Quotient of sum of exps

Percentage Accurate: 99.1% → 100.0%

Time: 7.5s

Alternatives: 11

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: 99.1% 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, 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 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. remove-double-neg 99.6%

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

   5. unsub-neg 99.6%

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

   6. div-sub 72.3%

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

   7. *-lft-identity 72.3%

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

   8. associate-*l/ 72.3%

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

   9. lft-mult-inverse 100.0%

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

   10. sub-neg 100.0%

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

   11. distribute-frac-neg 100.0%

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

   12. remove-double-neg 100.0%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 98.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0065:\\ \;\;\;\;\frac{1}{e^{-a} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.0065) (/ 1.0 (- (exp (- a)) -1.0)) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.0065) {
		tmp = 1.0 / (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 (a <= (-0.0065d0)) then
        tmp = 1.0d0 / (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 (a <= -0.0065) {
		tmp = 1.0 / (Math.exp(-a) - -1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.0065:
		tmp = 1.0 / (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 (a <= -0.0065)
		tmp = Float64(1.0 / Float64(exp(Float64(-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 (a <= -0.0065)
		tmp = 1.0 / (exp(-a) - -1.0);
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.0065], N[(1.0 / 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 a < -0.0064999999999999997

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. div-sub 2.8%

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

      8. neg-mul-1 2.8%

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

      9. *-commutative 2.8%

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

      10. associate-*r/ 2.8%

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

      11. metadata-eval 2.8%

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

      12. distribute-neg-frac 2.8%

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

      13. exp-neg 2.8%

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

      14. distribute-rgt-neg-out 2.8%

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

      15. exp-neg 2.8%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 100.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}}} - -1} \]
   6. Step-by-step derivation

      1. rec-exp 100.0%

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

   7. Simplified 100.0%

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

   if -0.0064999999999999997 < a

   1. Initial program 99.4%

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

      1. *-lft-identity 99.4%

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

      2. associate-*l/ 99.4%

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

      3. associate-/r/ 99.4%

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

      4. +-commutative 99.4%

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

      5. remove-double-neg 99.4%

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

      6. sub-neg 99.4%

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

      7. div-sub 99.4%

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

      8. neg-mul-1 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-*r/ 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. exp-neg 99.4%

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

      14. distribute-rgt-neg-out 99.4%

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

      15. exp-neg 99.4%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0065:\\ \;\;\;\;\frac{1}{e^{-a} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.5e+102)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
   (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.5e+102) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else {
		tmp = 1.0 / (exp(b) - -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-3.5d+102)) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    else
        tmp = 1.0d0 / (exp(b) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.5e+102) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.5e+102:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.5e+102)
		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(exp(b) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.5e+102)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.5e+102], 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[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.50000000000000011e102

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. div-sub 0.0%

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

      8. neg-mul-1 0.0%

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

      9. *-commutative 0.0%

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

      10. associate-*r/ 0.0%

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

      11. metadata-eval 0.0%

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

      12. distribute-neg-frac 0.0%

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

      13. exp-neg 0.0%

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

      14. distribute-rgt-neg-out 0.0%

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

      15. exp-neg 0.0%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 100.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}}} - -1} \]
   6. Step-by-step derivation

      1. rec-exp 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 98.1%

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

   if -3.50000000000000011e102 < a

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

      4. +-commutative 99.5%

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

      5. remove-double-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. div-sub 87.7%

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

      8. neg-mul-1 87.7%

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

      9. *-commutative 87.7%

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

      10. associate-*r/ 87.7%

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

      11. metadata-eval 87.7%

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

      12. distribute-neg-frac 87.7%

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

      13. exp-neg 87.7%

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

      14. distribute-rgt-neg-out 87.7%

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

      15. exp-neg 87.7%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 91.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot 0.16666666666666666\right)\\ \mathbf{if}\;b \leq 2.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \frac{\left(b \cdot 0.5\right) \cdot \left(b \cdot 0.5\right) - t\_0 \cdot t\_0}{b \cdot 0.5 - t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b 0.16666666666666666))))
   (if (<= b 2.8e+77)
     (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
     (if (<= b 5e+99)
       (/
        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 <= 2.8e+77) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else if (b <= 5e+99) {
		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 <= 2.8d+77) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    else if (b <= 5d+99) 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 <= 2.8e+77) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else if (b <= 5e+99) {
		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 <= 2.8e+77:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	elif b <= 5e+99:
		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 <= 2.8e+77)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))));
	elseif (b <= 5e+99)
		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 <= 2.8e+77)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	elseif (b <= 5e+99)
		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, 2.8e+77], 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, 5e+99], 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 < 2.8e77

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

      4. +-commutative 99.5%

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

      5. remove-double-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. div-sub 73.4%

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

      8. neg-mul-1 73.4%

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

      9. *-commutative 73.4%

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

      10. associate-*r/ 73.4%

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

      11. metadata-eval 73.4%

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

      12. distribute-neg-frac 73.4%

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

      13. exp-neg 73.4%

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

      14. distribute-rgt-neg-out 73.4%

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

      15. exp-neg 73.4%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 76.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}}} - -1} \]
   6. Step-by-step derivation

      1. rec-exp 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in a around 0 65.8%

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

   if 2.8e77 < b < 5.00000000000000008e99

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. div-sub 37.5%

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

      8. neg-mul-1 37.5%

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

      9. *-commutative 37.5%

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

      10. associate-*r/ 37.5%

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

      11. metadata-eval 37.5%

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

      12. distribute-neg-frac 37.5%

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

      13. exp-neg 37.5%

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

      14. distribute-rgt-neg-out 37.5%

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

      15. exp-neg 37.5%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 8.6%

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

      1. *-commutative 8.6%

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

   8. Simplified 8.6%

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

      1. distribute-lft-in 8.6%

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

      2. flip-+ 100.0%

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

   10. Applied egg-rr 100.0%

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

   if 5.00000000000000008e99 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. div-sub 73.2%

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

      8. neg-mul-1 73.2%

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

      9. *-commutative 73.2%

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

      10. associate-*r/ 73.2%

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

      11. metadata-eval 73.2%

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

      12. distribute-neg-frac 73.2%

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

      13. exp-neg 73.2%

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

      14. distribute-rgt-neg-out 73.2%

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

      15. exp-neg 73.2%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   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 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \frac{\left(b \cdot 0.5\right) \cdot \left(b \cdot 0.5\right) - \left(b \cdot \left(b \cdot 0.16666666666666666\right)\right) \cdot \left(b \cdot \left(b \cdot 0.16666666666666666\right)\right)}{b \cdot 0.5 - b \cdot \left(b \cdot 0.16666666666666666\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.5e+124)
   (/ 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 <= 6.5e+124) {
		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 <= 6.5d+124) 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 <= 6.5e+124) {
		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 <= 6.5e+124:
		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 <= 6.5e+124)
		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 <= 6.5e+124)
		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, 6.5e+124], 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 < 6.50000000000000008e124

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

      4. +-commutative 99.5%

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

      5. remove-double-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. div-sub 70.6%

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

      8. neg-mul-1 70.6%

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

      9. *-commutative 70.6%

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

      10. associate-*r/ 70.6%

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

      11. metadata-eval 70.6%

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

      12. distribute-neg-frac 70.6%

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

      13. exp-neg 70.6%

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

      14. distribute-rgt-neg-out 70.6%

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

      15. exp-neg 70.6%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 76.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}}} - -1} \]
   6. Step-by-step derivation

      1. rec-exp 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in a around 0 64.5%

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

   if 6.50000000000000008e124 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. div-sub 78.8%

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

      8. neg-mul-1 78.8%

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

      9. *-commutative 78.8%

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

      10. associate-*r/ 78.8%

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

      11. metadata-eval 78.8%

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

      12. distribute-neg-frac 78.8%

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

      13. exp-neg 78.8%

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

      14. distribute-rgt-neg-out 78.8%

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

      15. exp-neg 78.8%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 84.3%

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

      1. *-commutative 84.3%

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

   8. Simplified 84.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.0% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.2e+101)
   (/ 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 <= 6.2e+101) {
		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 <= 6.2d+101) 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 <= 6.2e+101) {
		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 <= 6.2e+101:
		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 <= 6.2e+101)
		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 <= 6.2e+101)
		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, 6.2e+101], 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 < 6.19999999999999998e101

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

      4. +-commutative 99.5%

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

      5. remove-double-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. div-sub 72.0%

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

      8. neg-mul-1 72.0%

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

      9. *-commutative 72.0%

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

      10. associate-*r/ 72.0%

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

      11. metadata-eval 72.0%

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

      12. distribute-neg-frac 72.0%

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

      13. exp-neg 72.0%

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

      14. distribute-rgt-neg-out 72.0%

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

      15. exp-neg 72.0%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 76.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}}} - -1} \]
   6. Step-by-step derivation

      1. rec-exp 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in a around 0 64.7%

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

   if 6.19999999999999998e101 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. div-sub 73.2%

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

      8. neg-mul-1 73.2%

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

      9. *-commutative 73.2%

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

      10. associate-*r/ 73.2%

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

      11. metadata-eval 73.2%

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

      12. distribute-neg-frac 73.2%

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

      13. exp-neg 73.2%

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

      14. distribute-rgt-neg-out 73.2%

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

      15. exp-neg 73.2%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   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 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+116}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.15e+116)
   (/ 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 <= 1.15e+116) {
		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 <= 1.15d+116) 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 <= 1.15e+116) {
		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 <= 1.15e+116:
		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 <= 1.15e+116)
		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 <= 1.15e+116)
		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, 1.15e+116], 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 < 1.14999999999999997e116

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

      4. +-commutative 99.5%

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

      5. remove-double-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. div-sub 70.9%

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

      8. neg-mul-1 70.9%

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

      9. *-commutative 70.9%

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

      10. associate-*r/ 70.9%

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

      11. metadata-eval 70.9%

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

      12. distribute-neg-frac 70.9%

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

      13. exp-neg 70.9%

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

      14. distribute-rgt-neg-out 70.9%

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

      15. exp-neg 70.9%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 76.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}}} - -1} \]
   6. Step-by-step derivation

      1. rec-exp 76.6%

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

   7. Simplified 76.6%

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

   8. Taylor expanded in a around 0 59.2%

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

   if 1.14999999999999997e116 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. +-commutative 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sub-neg 100.0%

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

      7. div-sub 77.4%

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

      8. neg-mul-1 77.4%

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

      9. *-commutative 77.4%

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

      10. associate-*r/ 77.4%

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

      11. metadata-eval 77.4%

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

      12. distribute-neg-frac 77.4%

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

      13. exp-neg 77.4%

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

      14. distribute-rgt-neg-out 77.4%

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

      15. exp-neg 77.4%

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

      16. rgt-mult-inverse 100.0%

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

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 82.8%

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

      1. *-commutative 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+116}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.8% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a 0.5))))))

C

double code(double a, double b) {
	return 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * 0.5d0))))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
}

Python

def code(a, b):
	return 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))))

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * 0.5)))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. +-commutative 99.6%

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

   5. remove-double-neg 99.6%

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

   6. sub-neg 99.6%

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

   7. div-sub 72.3%

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

   8. neg-mul-1 72.3%

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

   9. *-commutative 72.3%

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

   10. associate-*r/ 72.3%

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

   11. metadata-eval 72.3%

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

   12. distribute-neg-frac 72.3%

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

   13. exp-neg 72.3%

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

   14. distribute-rgt-neg-out 72.3%

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

   15. exp-neg 72.3%

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

   16. rgt-mult-inverse 100.0%

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

   17. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in b around 0 65.9%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}}} - -1} \]
6. Step-by-step derivation

   1. rec-exp 65.9%

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

7. Simplified 65.9%

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

8. Taylor expanded in a around 0 50.3%

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

9. Final simplification 50.3%

   \[\leadsto \frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)} \]
10. Add Preprocessing

Alternative 9: 39.2% 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 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. +-commutative 99.6%

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

   5. remove-double-neg 99.6%

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

   6. sub-neg 99.6%

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

   7. div-sub 72.3%

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

   8. neg-mul-1 72.3%

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

   9. *-commutative 72.3%

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

   10. associate-*r/ 72.3%

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

   11. metadata-eval 72.3%

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

   12. distribute-neg-frac 72.3%

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

   13. exp-neg 72.3%

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

   14. distribute-rgt-neg-out 72.3%

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

   15. exp-neg 72.3%

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

   16. rgt-mult-inverse 100.0%

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

   17. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in b around 0 65.9%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}}} - -1} \]
6. Step-by-step derivation

   1. rec-exp 65.9%

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

7. Simplified 65.9%

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

8. Taylor expanded in a around 0 38.0%

   \[\leadsto \color{blue}{0.5 + 0.25 \cdot a} \]
9. Step-by-step derivation

   1. *-commutative 38.0%

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

10. Simplified 38.0%

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

11. Final simplification 38.0%

    \[\leadsto 0.5 + a \cdot 0.25 \]
12. Add Preprocessing

Alternative 10: 39.8% 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 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. +-commutative 99.6%

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

   5. remove-double-neg 99.6%

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

   6. sub-neg 99.6%

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

   7. div-sub 72.3%

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

   8. neg-mul-1 72.3%

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

   9. *-commutative 72.3%

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

   10. associate-*r/ 72.3%

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

   11. metadata-eval 72.3%

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

   12. distribute-neg-frac 72.3%

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

   13. exp-neg 72.3%

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

   14. distribute-rgt-neg-out 72.3%

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

   15. exp-neg 72.3%

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

   16. rgt-mult-inverse 100.0%

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

   17. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in b around 0 65.9%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}}} - -1} \]
6. Step-by-step derivation

   1. rec-exp 65.9%

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

7. Simplified 65.9%

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

8. Taylor expanded in a around 0 38.7%

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

   1. neg-mul-1 38.7%

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

   2. unsub-neg 38.7%

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

10. Simplified 38.7%

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

11. Final simplification 38.7%

    \[\leadsto \frac{1}{2 - a} \]
12. Add Preprocessing

Alternative 11: 39.0% 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 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. +-commutative 99.6%

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

   5. remove-double-neg 99.6%

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

   6. sub-neg 99.6%

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

   7. div-sub 72.3%

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

   8. neg-mul-1 72.3%

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

   9. *-commutative 72.3%

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

   10. associate-*r/ 72.3%

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

   11. metadata-eval 72.3%

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

   12. distribute-neg-frac 72.3%

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

   13. exp-neg 72.3%

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

   14. distribute-rgt-neg-out 72.3%

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

   15. exp-neg 72.3%

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

   16. rgt-mult-inverse 100.0%

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

   17. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in a around 0 81.8%

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

6. Taylor expanded in b around 0 37.9%

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

7. Final simplification 37.9%

   \[\leadsto 0.5 \]
8. 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 2024071 
(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))))