Quotient of sum of exps

Percentage Accurate: 98.9% → 98.9%

Time: 8.6s

Alternatives: 15

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 69.9%

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

   8. neg-mul-1 69.9%

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

   9. *-commutative 69.9%

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

   10. associate-*r/ 69.9%

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

   11. metadata-eval 69.9%

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

   12. distribute-neg-frac 69.9%

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

   13. exp-neg 69.9%

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

   14. distribute-rgt-neg-out 69.9%

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

   15. exp-neg 69.9%

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

   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. Final simplification 100.0%

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

Alternative 3: 98.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.3e-6) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.3e-6) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.3d-6)) then
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -2.3e-6) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.3e-6:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.3e-6)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.3e-6)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.3e-6], N[(1.0 / N[(1.0 + N[Exp[(-a)], $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 < -2.3e-6

   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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 2.5%

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

      7. *-lft-identity 2.5%

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

      8. associate-*l/ 2.5%

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

      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. Taylor expanded in b around 0 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -2.3e-6 < a

   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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 99.9%

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

      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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 99.1%

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

      1. +-commutative 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.5e+48) (/ (exp a) a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.5e+48) {
		tmp = exp(a) / a;
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-6.5d+48)) then
        tmp = exp(a) / a
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6.5e+48) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.5e+48:
		tmp = math.exp(a) / a
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.5e+48)
		tmp = Float64(exp(a) / a);
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.5e+48)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.5e+48], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.49999999999999972e48

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around inf 100.0%

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

   if -6.49999999999999972e48 < a

   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/ 99.9%

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

      3. associate-/r/ 99.9%

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

      4. remove-double-neg 99.9%

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

      5. unsub-neg 99.9%

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

      6. div-sub 97.8%

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

      7. *-lft-identity 97.8%

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

      8. associate-*l/ 97.8%

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

      9. lft-mult-inverse 99.9%

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

      10. sub-neg 99.9%

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

      11. distribute-frac-neg 99.9%

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

      12. remove-double-neg 99.9%

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

      13. div-exp 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.5e-6) (/ (exp a) (+ a 2.0)) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.5e-6) {
		tmp = exp(a) / (a + 2.0);
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-3.5d-6)) then
        tmp = exp(a) / (a + 2.0d0)
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.5e-6) {
		tmp = Math.exp(a) / (a + 2.0);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.5e-6:
		tmp = math.exp(a) / (a + 2.0)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.5e-6)
		tmp = Float64(exp(a) / Float64(a + 2.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.5e-6)
		tmp = exp(a) / (a + 2.0);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.5e-6], N[(N[Exp[a], $MachinePrecision] / N[(a + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.49999999999999995e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0 100.0%

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

   4. Taylor expanded in a around 0 99.1%

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

      1. +-commutative 99.1%

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

   6. Simplified 99.1%

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

   if -3.49999999999999995e-6 < a

   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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 99.9%

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

      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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 99.1%

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

      1. +-commutative 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 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. remove-double-neg 100.0%

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

   5. unsub-neg 100.0%

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

   6. div-sub 69.9%

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

   7. *-lft-identity 69.9%

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

   8. associate-*l/ 69.9%

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

   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 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 7: 81.8% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -15.5:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -15.5)
   1.0
   (if (<= b 2.5e+125)
     (/ 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 <= -15.5) {
		tmp = 1.0;
	} else if (b <= 2.5e+125) {
		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 <= (-15.5d0)) then
        tmp = 1.0d0
    else if (b <= 2.5d+125) 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 <= -15.5) {
		tmp = 1.0;
	} else if (b <= 2.5e+125) {
		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 <= -15.5:
		tmp = 1.0
	elif b <= 2.5e+125:
		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 <= -15.5)
		tmp = 1.0;
	elseif (b <= 2.5e+125)
		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 <= -15.5)
		tmp = 1.0;
	elseif (b <= 2.5e+125)
		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, -15.5], 1.0, If[LessEqual[b, 2.5e+125], 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 3 regimes

2. if b < -15.5

   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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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. Step-by-step derivation

      1. div-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. add-exp-log 100.0%

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

      6. rec-exp 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. log1p-define 100.0%

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

      11. div-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   10. Simplified 100.0%

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

   if -15.5 < b < 2.49999999999999981e125

   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/ 99.9%

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

      3. associate-/r/ 99.9%

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

      4. remove-double-neg 99.9%

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

      5. unsub-neg 99.9%

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

      6. div-sub 66.0%

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

      7. *-lft-identity 66.0%

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

      8. associate-*l/ 66.0%

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

      9. lft-mult-inverse 99.9%

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

      10. sub-neg 99.9%

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

      11. distribute-frac-neg 99.9%

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

      12. remove-double-neg 99.9%

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

      13. div-exp 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 86.9%

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

      1. +-commutative 86.9%

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

   7. Simplified 86.9%

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

   8. Taylor expanded in a around 0 79.7%

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

   if 2.49999999999999981e125 < 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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 55.3%

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

      7. *-lft-identity 55.3%

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

      8. associate-*l/ 55.3%

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

      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. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around 0 86.6%

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

      1. *-commutative 86.6%

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

   10. Simplified 86.6%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -15.5:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -7.2)
   1.0
   (if (<= b 7.6e+64)
     (/ 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 <= -7.2) {
		tmp = 1.0;
	} else if (b <= 7.6e+64) {
		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 <= (-7.2d0)) then
        tmp = 1.0d0
    else if (b <= 7.6d+64) 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 <= -7.2) {
		tmp = 1.0;
	} else if (b <= 7.6e+64) {
		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 <= -7.2:
		tmp = 1.0
	elif b <= 7.6e+64:
		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 <= -7.2)
		tmp = 1.0;
	elseif (b <= 7.6e+64)
		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 <= -7.2)
		tmp = 1.0;
	elseif (b <= 7.6e+64)
		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, -7.2], 1.0, If[LessEqual[b, 7.6e+64], 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 3 regimes

2. if b < -7.20000000000000018

   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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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. Step-by-step derivation

      1. div-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. add-exp-log 100.0%

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

      6. rec-exp 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. log1p-define 100.0%

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

      11. div-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   10. Simplified 100.0%

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

   if -7.20000000000000018 < b < 7.6000000000000002e64

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-/r/ 99.9%

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

      4. remove-double-neg 99.9%

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

      5. unsub-neg 99.9%

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

      6. div-sub 64.8%

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

      7. *-lft-identity 64.8%

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

      8. associate-*l/ 64.8%

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

      9. lft-mult-inverse 99.9%

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

      10. sub-neg 99.9%

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

      11. distribute-frac-neg 99.9%

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

      12. remove-double-neg 99.9%

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

      13. div-exp 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 92.8%

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

      1. +-commutative 92.8%

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

   7. Simplified 92.8%

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

   8. Taylor expanded in a around 0 85.5%

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

   if 7.6000000000000002e64 < 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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 60.7%

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

      7. *-lft-identity 60.7%

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

      8. associate-*l/ 60.7%

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

      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. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around 0 91.0%

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

      1. *-commutative 91.0%

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

   10. Simplified 91.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.5% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -25:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -25.0)
   1.0
   (if (<= b 2.5e+125)
     (/ 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 <= -25.0) {
		tmp = 1.0;
	} else if (b <= 2.5e+125) {
		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 <= (-25.0d0)) then
        tmp = 1.0d0
    else if (b <= 2.5d+125) 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 <= -25.0) {
		tmp = 1.0;
	} else if (b <= 2.5e+125) {
		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 <= -25.0:
		tmp = 1.0
	elif b <= 2.5e+125:
		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 <= -25.0)
		tmp = 1.0;
	elseif (b <= 2.5e+125)
		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 <= -25.0)
		tmp = 1.0;
	elseif (b <= 2.5e+125)
		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, -25.0], 1.0, If[LessEqual[b, 2.5e+125], 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 3 regimes

2. if b < -25

   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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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. Step-by-step derivation

      1. div-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. add-exp-log 100.0%

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

      6. rec-exp 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. log1p-define 100.0%

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

      11. div-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   10. Simplified 100.0%

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

   if -25 < b < 2.49999999999999981e125

   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/ 99.9%

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

      3. associate-/r/ 99.9%

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

      4. remove-double-neg 99.9%

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

      5. unsub-neg 99.9%

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

      6. div-sub 66.0%

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

      7. *-lft-identity 66.0%

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

      8. associate-*l/ 66.0%

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

      9. lft-mult-inverse 99.9%

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

      10. sub-neg 99.9%

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

      11. distribute-frac-neg 99.9%

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

      12. remove-double-neg 99.9%

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

      13. div-exp 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 86.9%

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

      1. +-commutative 86.9%

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

   7. Simplified 86.9%

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

   8. Taylor expanded in a around 0 74.3%

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

   if 2.49999999999999981e125 < 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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 55.3%

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

      7. *-lft-identity 55.3%

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

      8. associate-*l/ 55.3%

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

      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. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around 0 86.6%

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

      1. *-commutative 86.6%

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

   10. Simplified 86.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -25:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -7.2) 1.0 (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -7.2) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 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 <= (-7.2d0)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -7.2) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -7.2:
		tmp = 1.0
	else:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -7.2)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -7.2)
		tmp = 1.0;
	else
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -7.2], 1.0, N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.20000000000000018

   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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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. Step-by-step derivation

      1. div-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. add-exp-log 100.0%

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

      6. rec-exp 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. log1p-define 100.0%

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

      11. div-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   10. Simplified 100.0%

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

   if -7.20000000000000018 < 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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 63.6%

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

      7. *-lft-identity 63.6%

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

      8. associate-*l/ 63.6%

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

      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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 77.9%

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

      1. +-commutative 77.9%

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

   7. Simplified 77.9%

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

   8. Taylor expanded in a around 0 64.5%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.3% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -1.05) 1.0 (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.05) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.05d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 + (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.05) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.05:
		tmp = 1.0
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.05)
		tmp = 1.0;
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.05)
		tmp = 1.0;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.05], 1.0, N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.05000000000000004

   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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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. Step-by-step derivation

      1. div-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. add-exp-log 100.0%

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

      6. rec-exp 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. log1p-define 100.0%

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

      11. div-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   10. Simplified 100.0%

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

   if -1.05000000000000004 < 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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 63.6%

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

      7. *-lft-identity 63.6%

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

      8. associate-*l/ 63.6%

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

      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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 77.9%

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

      1. +-commutative 77.9%

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

   7. Simplified 77.9%

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

   8. Taylor expanded in a around 0 42.0%

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

      1. *-commutative 42.0%

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

   10. Simplified 42.0%

       \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.4% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -2.0) 1.0 (+ 0.5 (* b -0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (b * -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-2.0d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 + (b * (-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -2.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (b * -0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -2.0:
		tmp = 1.0
	else:
		tmp = 0.5 + (b * -0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -2.0)
		tmp = 1.0;
	else
		tmp = Float64(0.5 + Float64(b * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -2.0)
		tmp = 1.0;
	else
		tmp = 0.5 + (b * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -2.0], 1.0, N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2

   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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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. Step-by-step derivation

      1. div-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. add-exp-log 100.0%

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

      6. rec-exp 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. log1p-define 100.0%

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

      11. div-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   10. Simplified 100.0%

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

   if -2 < 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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 63.6%

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

      7. *-lft-identity 63.6%

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

      8. associate-*l/ 63.6%

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

      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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 76.3%

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

      1. +-commutative 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in b around 0 42.2%

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

      1. *-commutative 42.2%

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

   10. Simplified 42.2%

       \[\leadsto \color{blue}{0.5 + b \cdot -0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.1% accurate, 30.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1

   1. Initial program 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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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. Step-by-step derivation

      1. div-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. add-exp-log 100.0%

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

      6. rec-exp 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. log1p-define 100.0%

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

      11. div-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   10. Simplified 100.0%

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

   if -1 < 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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 63.6%

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

      7. *-lft-identity 63.6%

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

      8. associate-*l/ 63.6%

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

      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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 76.3%

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

      1. +-commutative 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in b around 0 43.2%

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

      1. +-commutative 43.2%

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

   10. Simplified 43.2%

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

4. Final simplification 53.0%

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

Alternative 14: 54.1% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -1.1) 1.0 0.5))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.1d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.1:
		tmp = 1.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.1)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.1)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.1], 1.0, 0.5]

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001

   1. Initial program 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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 100.0%

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

      7. *-lft-identity 100.0%

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

      8. associate-*l/ 100.0%

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

      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. Step-by-step derivation

      1. div-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. add-exp-log 100.0%

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

      6. rec-exp 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. log1p-define 100.0%

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

      11. div-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   10. Simplified 100.0%

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

   if -1.1000000000000001 < 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. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 63.6%

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

      7. *-lft-identity 63.6%

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

      8. associate-*l/ 63.6%

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

      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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in a around 0 76.3%

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

      1. +-commutative 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in b around 0 41.5%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.6% 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 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. remove-double-neg 100.0%

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

   5. unsub-neg 100.0%

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

   6. div-sub 69.9%

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

   7. *-lft-identity 69.9%

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

   8. associate-*l/ 69.9%

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

   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 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in a around 0 80.4%

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

   1. +-commutative 80.4%

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

7. Simplified 80.4%

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

8. Taylor expanded in b around 0 37.6%

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

9. Final simplification 37.6%

   \[\leadsto 0.5 \]
10. 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 2024081 
(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))))