Quotient of sum of exps

Percentage Accurate: 98.9% → 100.0%

Time: 8.4s

Alternatives: 13

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. metadata-eval 99.6%

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

   5. *-rgt-identity 99.6%

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

   6. associate-*r/ 99.6%

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

   7. exp-neg 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. distribute-frac-neg2 99.5%

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

   10. distribute-lft-neg-out 99.5%

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

   11. +-commutative 99.5%

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

   12. distribute-neg-in 99.5%

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

   13. sub-neg 99.5%

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

   14. *-commutative 99.5%

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

   15. sub-neg 99.5%

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

   16. distribute-neg-in 99.5%

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

   17. +-commutative 99.5%

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

3. Simplified 100.0%

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

   1. add-exp-log 100.0%

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

   2. log-div 0.0%

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

   3. sub-neg 0.0%

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

   4. metadata-eval 0.0%

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

   5. distribute-neg-in 0.0%

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

   6. div-exp 0.0%

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

   7. log-div 100.0%

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

   8. metadata-eval 100.0%

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

   9. frac-2neg 100.0%

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

   10. div-exp 100.0%

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

   11. log-rec 100.0%

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

   12. log1p-define 100.0%

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

   13. div-exp 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. metadata-eval 99.6%

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

   5. *-rgt-identity 99.6%

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

   6. associate-*r/ 99.6%

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

   7. exp-neg 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. distribute-frac-neg2 99.5%

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

   10. distribute-lft-neg-out 99.5%

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

   11. +-commutative 99.5%

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

   12. distribute-neg-in 99.5%

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

   13. sub-neg 99.5%

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

   14. *-commutative 99.5%

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

   15. sub-neg 99.5%

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

   16. distribute-neg-in 99.5%

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

   17. +-commutative 99.5%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 3: 98.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4e-11) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4e-11) {
		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 <= (-4d-11)) 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 <= -4e-11) {
		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 <= -4e-11:
		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 <= -4e-11)
		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 <= -4e-11)
		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, -4e-11], 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 < -3.99999999999999976e-11

   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. metadata-eval 99.9%

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

      5. *-rgt-identity 99.9%

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

      6. associate-*r/ 99.9%

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

      7. exp-neg 99.9%

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

      8. distribute-neg-frac 99.9%

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

      9. distribute-frac-neg2 99.9%

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

      10. distribute-lft-neg-out 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. sub-neg 99.9%

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

      14. *-commutative 99.9%

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

      15. sub-neg 99.9%

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

      16. distribute-neg-in 99.9%

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

      17. +-commutative 99.9%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 97.4%

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

      1. rec-exp 97.4%

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

   7. Simplified 97.4%

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

   if -3.99999999999999976e-11 < a

   1. Initial program 99.4%

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

      1. *-lft-identity 99.4%

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

      2. associate-*l/ 99.4%

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

      3. associate-/r/ 99.4%

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

      4. metadata-eval 99.4%

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

      5. *-rgt-identity 99.4%

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

      6. associate-*r/ 99.4%

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

      7. exp-neg 99.4%

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

      8. distribute-neg-frac 99.4%

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

      9. distribute-frac-neg2 99.4%

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

      10. distribute-lft-neg-out 99.4%

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

      11. +-commutative 99.4%

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

      12. distribute-neg-in 99.4%

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

      13. sub-neg 99.4%

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

      14. *-commutative 99.4%

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

      15. sub-neg 99.4%

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

      16. distribute-neg-in 99.4%

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

      17. +-commutative 99.4%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 99.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -720:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -720.0) (/ (exp a) a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -720.0) {
		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 <= (-720.0d0)) 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 <= -720.0) {
		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 <= -720.0:
		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 <= -720.0)
		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 <= -720.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -720.0], 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 < -720

   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 -720 < a

   1. Initial program 99.4%

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

      1. *-lft-identity 99.4%

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

      2. associate-*l/ 99.4%

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

      3. associate-/r/ 99.4%

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

      4. metadata-eval 99.4%

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

      5. *-rgt-identity 99.4%

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

      6. associate-*r/ 99.4%

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

      7. exp-neg 99.4%

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

      8. distribute-neg-frac 99.4%

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

      9. distribute-frac-neg2 99.4%

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

      10. distribute-lft-neg-out 99.4%

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

      11. +-commutative 99.4%

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

      12. distribute-neg-in 99.4%

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

      13. sub-neg 99.4%

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

      14. *-commutative 99.4%

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

      15. sub-neg 99.4%

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

      16. distribute-neg-in 99.4%

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

      17. +-commutative 99.4%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 97.5%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -720:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -16.5:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -16.5)
   (+ (exp b) 1.0)
   (if (<= b 2e+99)
     (/ 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 <= -16.5) {
		tmp = exp(b) + 1.0;
	} else if (b <= 2e+99) {
		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 <= (-16.5d0)) then
        tmp = exp(b) + 1.0d0
    else if (b <= 2d+99) 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 <= -16.5) {
		tmp = Math.exp(b) + 1.0;
	} else if (b <= 2e+99) {
		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 <= -16.5:
		tmp = math.exp(b) + 1.0
	elif b <= 2e+99:
		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 <= -16.5)
		tmp = Float64(exp(b) + 1.0);
	elseif (b <= 2e+99)
		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 <= -16.5)
		tmp = exp(b) + 1.0;
	elseif (b <= 2e+99)
		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, -16.5], N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[b, 2e+99], 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 < -16.5

   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. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-div 0.0%

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

      3. sub-neg 0.0%

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

      4. metadata-eval 0.0%

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

      5. distribute-neg-in 0.0%

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

      6. div-exp 0.0%

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

      7. log-div 100.0%

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

      8. metadata-eval 100.0%

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

      9. frac-2neg 100.0%

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

      10. div-exp 100.0%

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

      11. log-rec 100.0%

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

      12. log1p-define 100.0%

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

      13. div-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in a around 0 100.0%

      \[\leadsto e^{-\color{blue}{\log \left(1 + e^{b}\right)}} \]
   8. Step-by-step derivation

      1. log1p-define 100.0%

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

   9. Simplified 100.0%

      \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(e^{b}\right)}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 100.0%

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

       2. sqrt-unprod 100.0%

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

       3. sqr-neg 100.0%

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

       4. sqrt-unprod 100.0%

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

       5. add-sqr-sqrt 100.0%

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

       6. log1p-undefine 100.0%

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

       7. rem-exp-log 100.0%

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

       8. remove-double-div 100.0%

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

       9. exp-neg 100.0%

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

       10. +-commutative 100.0%

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

       11. exp-neg 100.0%

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

       12. remove-double-div 100.0%

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

   11. Applied egg-rr 100.0%

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

   if -16.5 < b < 1.9999999999999999e99

   1. Initial program 99.4%

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

      1. *-lft-identity 99.4%

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

      2. associate-*l/ 99.4%

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

      3. associate-/r/ 99.4%

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

      4. metadata-eval 99.4%

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

      5. *-rgt-identity 99.4%

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

      6. associate-*r/ 99.4%

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

      7. exp-neg 99.3%

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

      8. distribute-neg-frac 99.3%

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

      9. distribute-frac-neg2 99.3%

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

      10. distribute-lft-neg-out 99.3%

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

      11. +-commutative 99.3%

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

      12. distribute-neg-in 99.3%

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

      13. sub-neg 99.3%

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

      14. *-commutative 99.3%

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

      15. sub-neg 99.3%

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

      16. distribute-neg-in 99.3%

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

      17. +-commutative 99.3%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in b around 0 88.8%

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

      1. rec-exp 88.8%

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

   7. Simplified 88.8%

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

   8. Taylor expanded in a around 0 80.1%

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

   if 1.9999999999999999e99 < 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. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 97.7%

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

      1. *-commutative 97.7%

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

   8. Simplified 97.7%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -16.5:\\ \;\;\;\;e^{b} + 1\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return 1.0 / (exp((b - 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 - a)) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. remove-double-neg 99.6%

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

   5. unsub-neg 99.6%

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

   6. div-sub 76.1%

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

   7. *-lft-identity 76.1%

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

   8. associate-*l/ 76.1%

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

   9. lft-mult-inverse 100.0%

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

   10. sub-neg 100.0%

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

   11. distribute-frac-neg 100.0%

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

   12. remove-double-neg 100.0%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 7: 66.8% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{+127}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.25e+127)
   (/ 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 <= 1.25e+127) {
		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 <= 1.25d+127) 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 <= 1.25e+127) {
		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 <= 1.25e+127:
		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 <= 1.25e+127)
		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 <= 1.25e+127)
		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, 1.25e+127], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2500000000000001e127

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

      4. metadata-eval 99.5%

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

      5. *-rgt-identity 99.5%

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

      6. associate-*r/ 99.5%

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

      7. exp-neg 99.5%

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

      8. distribute-neg-frac 99.5%

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

      9. distribute-frac-neg2 99.5%

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

      10. distribute-lft-neg-out 99.5%

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

      11. +-commutative 99.5%

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

      12. distribute-neg-in 99.5%

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

      13. sub-neg 99.5%

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

      14. *-commutative 99.5%

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

      15. sub-neg 99.5%

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

      16. distribute-neg-in 99.5%

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

      17. +-commutative 99.5%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 74.6%

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

      1. rec-exp 74.5%

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

   7. Simplified 74.5%

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

   8. Taylor expanded in a around 0 67.7%

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

   if 1.2500000000000001e127 < 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. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 92.2%

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

      1. *-commutative 92.2%

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

   8. Simplified 92.2%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{+127}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.5% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.7e+99)
   (/ 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 <= 1.7e+99) {
		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 <= 1.7d+99) 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 <= 1.7e+99) {
		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 <= 1.7e+99:
		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 <= 1.7e+99)
		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 <= 1.7e+99)
		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, 1.7e+99], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.69999999999999992e99

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

      4. metadata-eval 99.5%

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

      5. *-rgt-identity 99.5%

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

      6. associate-*r/ 99.5%

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

      7. exp-neg 99.5%

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

      8. distribute-neg-frac 99.5%

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

      9. distribute-frac-neg2 99.5%

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

      10. distribute-lft-neg-out 99.5%

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

      11. +-commutative 99.5%

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

      12. distribute-neg-in 99.5%

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

      13. sub-neg 99.5%

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

      14. *-commutative 99.5%

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

      15. sub-neg 99.5%

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

      16. distribute-neg-in 99.5%

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

      17. +-commutative 99.5%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 74.6%

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

      1. rec-exp 74.6%

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

   7. Simplified 74.6%

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

   8. Taylor expanded in a around 0 67.7%

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

   if 1.69999999999999992e99 < 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. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 97.7%

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

      1. *-commutative 97.7%

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

   8. Simplified 97.7%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.4% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{+126}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.7e+126)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a 0.5)))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.7e+126) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.7d+126) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * 0.5d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.7e+126) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.7e+126:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.7e+126)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * 0.5)))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.7e+126)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.7e+126], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.69999999999999995e126

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*l/ 99.5%

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

      3. associate-/r/ 99.5%

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

      4. metadata-eval 99.5%

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

      5. *-rgt-identity 99.5%

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

      6. associate-*r/ 99.5%

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

      7. exp-neg 99.5%

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

      8. distribute-neg-frac 99.5%

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

      9. distribute-frac-neg2 99.5%

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

      10. distribute-lft-neg-out 99.5%

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

      11. +-commutative 99.5%

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

      12. distribute-neg-in 99.5%

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

      13. sub-neg 99.5%

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

      14. *-commutative 99.5%

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

      15. sub-neg 99.5%

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

      16. distribute-neg-in 99.5%

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

      17. +-commutative 99.5%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in b around 0 74.6%

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

      1. rec-exp 74.5%

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

   7. Simplified 74.5%

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

   8. Taylor expanded in a around 0 63.0%

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

   if 1.69999999999999995e126 < 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. metadata-eval 100.0%

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

      5. *-rgt-identity 100.0%

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

      6. associate-*r/ 100.0%

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

      7. exp-neg 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. distribute-lft-neg-out 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. sub-neg 100.0%

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

      14. *-commutative 100.0%

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

      15. sub-neg 100.0%

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

      16. distribute-neg-in 100.0%

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

      17. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 92.2%

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

      1. *-commutative 92.2%

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

   8. Simplified 92.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{+126}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.1% accurate, 27.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. metadata-eval 99.6%

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

   5. *-rgt-identity 99.6%

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

   6. associate-*r/ 99.6%

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

   7. exp-neg 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. distribute-frac-neg2 99.5%

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

   10. distribute-lft-neg-out 99.5%

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

   11. +-commutative 99.5%

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

   12. distribute-neg-in 99.5%

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

   13. sub-neg 99.5%

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

   14. *-commutative 99.5%

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

   15. sub-neg 99.5%

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

   16. distribute-neg-in 99.5%

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

   17. +-commutative 99.5%

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

3. Simplified 100.0%

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

5. Taylor expanded in b around 0 68.2%

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

   1. rec-exp 68.2%

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

7. Simplified 68.2%

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

8. Taylor expanded in a around 0 56.0%

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

9. Final simplification 56.0%

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

Alternative 11: 39.0% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. metadata-eval 99.6%

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

   5. *-rgt-identity 99.6%

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

   6. associate-*r/ 99.6%

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

   7. exp-neg 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. distribute-frac-neg2 99.5%

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

   10. distribute-lft-neg-out 99.5%

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

   11. +-commutative 99.5%

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

   12. distribute-neg-in 99.5%

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

   13. sub-neg 99.5%

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

   14. *-commutative 99.5%

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

   15. sub-neg 99.5%

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

   16. distribute-neg-in 99.5%

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

   17. +-commutative 99.5%

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

3. Simplified 100.0%

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

5. Taylor expanded in b around 0 68.2%

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

   1. rec-exp 68.2%

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

7. Simplified 68.2%

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

8. Taylor expanded in a around 0 44.5%

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

   1. *-commutative 44.5%

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

10. Simplified 44.5%

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

11. Final simplification 44.5%

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

Alternative 12: 39.7% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. metadata-eval 99.6%

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

   5. *-rgt-identity 99.6%

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

   6. associate-*r/ 99.6%

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

   7. exp-neg 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. distribute-frac-neg2 99.5%

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

   10. distribute-lft-neg-out 99.5%

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

   11. +-commutative 99.5%

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

   12. distribute-neg-in 99.5%

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

   13. sub-neg 99.5%

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

   14. *-commutative 99.5%

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

   15. sub-neg 99.5%

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

   16. distribute-neg-in 99.5%

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

   17. +-commutative 99.5%

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

3. Simplified 100.0%

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

5. Taylor expanded in b around 0 68.2%

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

   1. rec-exp 68.2%

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

7. Simplified 68.2%

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

8. Taylor expanded in a around 0 45.1%

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

   1. neg-mul-1 45.1%

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

   2. unsub-neg 45.1%

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

10. Simplified 45.1%

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

11. Final simplification 45.1%

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

Alternative 13: 38.9% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. metadata-eval 99.6%

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

   5. *-rgt-identity 99.6%

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

   6. associate-*r/ 99.6%

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

   7. exp-neg 99.5%

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

   8. distribute-neg-frac 99.5%

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

   9. distribute-frac-neg2 99.5%

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

   10. distribute-lft-neg-out 99.5%

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

   11. +-commutative 99.5%

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

   12. distribute-neg-in 99.5%

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

   13. sub-neg 99.5%

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

   14. *-commutative 99.5%

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

   15. sub-neg 99.5%

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

   16. distribute-neg-in 99.5%

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

   17. +-commutative 99.5%

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

3. Simplified 100.0%

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

5. Taylor expanded in a around 0 83.0%

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

6. Taylor expanded in b around 0 43.7%

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

7. Final simplification 43.7%

   \[\leadsto 0.5 \]
8. Add Preprocessing

Developer target: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024112 
(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))))