Quotient of sum of exps

Percentage Accurate: 99.1% → 100.0%

Time: 8.9s

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: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 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 98.0%

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

   1. *-lft-identity 98.0%

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

   2. associate-*l/ 98.0%

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

   3. associate-/r/ 98.0%

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

   4. metadata-eval 98.0%

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

   5. *-rgt-identity 98.0%

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

   6. associate-*r/ 98.0%

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

   7. exp-neg 98.0%

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

   8. distribute-neg-frac 98.0%

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

   9. distribute-frac-neg2 98.0%

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

   10. distribute-lft-neg-out 98.0%

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

   11. +-commutative 98.0%

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

   12. distribute-neg-in 98.0%

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

   13. sub-neg 98.0%

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

   14. *-commutative 98.0%

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

   15. sub-neg 98.0%

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

   16. distribute-neg-in 98.0%

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

   17. +-commutative 98.0%

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

3. Simplified 99.2%

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

   1. add-exp-log 99.2%

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

   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 99.2%

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

   11. log-rec 99.2%

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

   12. log1p-define 99.2%

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

   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: 98.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -88000000:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -88000000.0)
   (/ (exp a) (+ a 2.0))
   (if (<= a 1.75e-15) (/ 1.0 (+ 1.0 (exp b))) (/ 1.0 (+ 1.0 (exp (- a)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -88000000.0) {
		tmp = exp(a) / (a + 2.0);
	} else if (a <= 1.75e-15) {
		tmp = 1.0 / (1.0 + exp(b));
	} else {
		tmp = 1.0 / (1.0 + exp(-a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-88000000.0d0)) then
        tmp = exp(a) / (a + 2.0d0)
    else if (a <= 1.75d-15) then
        tmp = 1.0d0 / (1.0d0 + exp(b))
    else
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -88000000.0) {
		tmp = Math.exp(a) / (a + 2.0);
	} else if (a <= 1.75e-15) {
		tmp = 1.0 / (1.0 + Math.exp(b));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -88000000.0:
		tmp = math.exp(a) / (a + 2.0)
	elif a <= 1.75e-15:
		tmp = 1.0 / (1.0 + math.exp(b))
	else:
		tmp = 1.0 / (1.0 + math.exp(-a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -88000000.0)
		tmp = Float64(exp(a) / Float64(a + 2.0));
	elseif (a <= 1.75e-15)
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -88000000.0)
		tmp = exp(a) / (a + 2.0);
	elseif (a <= 1.75e-15)
		tmp = 1.0 / (1.0 + exp(b));
	else
		tmp = 1.0 / (1.0 + exp(-a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -88000000.0], N[(N[Exp[a], $MachinePrecision] / N[(a + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.75e-15], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.8e7

   1. Initial program 98.4%

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

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

   if -8.8e7 < a < 1.75e-15

   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 99.4%

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

   5. Taylor expanded in a around 0 99.2%

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

   if 1.75e-15 < a

   1. Initial program 66.7%

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

      1. *-lft-identity 66.7%

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

      2. associate-*l/ 66.7%

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

      3. associate-/r/ 66.7%

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

      4. metadata-eval 66.7%

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

      5. *-rgt-identity 66.7%

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

      6. associate-*r/ 66.7%

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

      7. exp-neg 66.5%

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

      8. distribute-neg-frac 66.5%

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

      9. distribute-frac-neg2 66.5%

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

      10. distribute-lft-neg-out 66.5%

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

      11. +-commutative 66.5%

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

      12. distribute-neg-in 66.5%

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

      13. sub-neg 66.5%

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

      14. *-commutative 66.5%

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

      15. sub-neg 66.5%

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

      16. distribute-neg-in 66.5%

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

      17. +-commutative 66.5%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in b around 0 99.8%

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

      1. rec-exp 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -88000000:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.3e+95)
   (/ 1.0 (+ 2.0 (* a (+ (* a (+ 0.5 (* a -0.16666666666666666))) -1.0))))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.3e+95) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-3.3d+95)) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * (-0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.3e+95) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.3e+95:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.3e+95)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.3e+95)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.3e+95], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.2999999999999998e95

   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 b around 0 100.0%

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

      1. rec-exp 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in a around 0 98.1%

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

   if -3.2999999999999998e95 < a

   1. Initial program 97.6%

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

      1. *-lft-identity 97.6%

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

      2. associate-*l/ 97.6%

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

      3. associate-/r/ 97.5%

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

      4. metadata-eval 97.5%

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

      5. *-rgt-identity 97.5%

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

      6. associate-*r/ 97.5%

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

      7. exp-neg 97.6%

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

      8. distribute-neg-frac 97.6%

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

      9. distribute-frac-neg2 97.6%

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

      10. distribute-lft-neg-out 97.6%

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

      11. +-commutative 97.6%

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

      12. distribute-neg-in 97.6%

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

      13. sub-neg 97.6%

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

      14. *-commutative 97.6%

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

      15. sub-neg 97.6%

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

      16. distribute-neg-in 97.6%

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

      17. +-commutative 97.6%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in a around 0 92.1%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -88000000:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -88000000.0) (/ (exp a) (+ a 2.0)) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -88000000.0) {
		tmp = exp(a) / (a + 2.0);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-88000000.0d0)) then
        tmp = exp(a) / (a + 2.0d0)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -88000000.0) {
		tmp = Math.exp(a) / (a + 2.0);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -88000000.0:
		tmp = math.exp(a) / (a + 2.0)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -88000000.0)
		tmp = Float64(exp(a) / Float64(a + 2.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -88000000.0)
		tmp = exp(a) / (a + 2.0);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -88000000.0], N[(N[Exp[a], $MachinePrecision] / N[(a + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.8e7

   1. Initial program 98.4%

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

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

   if -8.8e7 < a

   1. Initial program 97.9%

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

      1. *-lft-identity 97.9%

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

      2. associate-*l/ 97.9%

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

      3. associate-/r/ 97.9%

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

      4. metadata-eval 97.9%

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

      5. *-rgt-identity 97.9%

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

      6. associate-*r/ 97.9%

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

      7. exp-neg 97.9%

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

      8. distribute-neg-frac 97.9%

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

      9. distribute-frac-neg2 97.9%

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

      10. distribute-lft-neg-out 97.9%

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

      11. +-commutative 97.9%

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

      12. distribute-neg-in 97.9%

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

      13. sub-neg 97.9%

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

      14. *-commutative 97.9%

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

      15. sub-neg 97.9%

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

      16. distribute-neg-in 97.9%

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

      17. +-commutative 97.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in a around 0 97.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -88000000:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 98.0%

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

   1. *-lft-identity 98.0%

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

   2. associate-*l/ 98.0%

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

   3. associate-/r/ 98.0%

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

   4. remove-double-neg 98.0%

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

   5. unsub-neg 98.0%

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

   6. div-sub 73.0%

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

   7. *-lft-identity 73.0%

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

   8. associate-*l/ 73.0%

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

   9. lft-mult-inverse 99.2%

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

   10. sub-neg 99.2%

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

   11. distribute-frac-neg 99.2%

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

   12. remove-double-neg 99.2%

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

   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 6: 67.8% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.4e+154)
   (/ 1.0 (+ 2.0 (* a (+ (* a (+ 0.5 (* a -0.16666666666666666))) -1.0))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.4e+154) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} 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 <= 1.4d+154) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * (-0.16666666666666666d0)))) + (-1.0d0))))
    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 <= 1.4e+154) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.4e+154:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.4e+154)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))) + -1.0))));
	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 <= 1.4e+154)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.4e+154], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 < 1.4e154

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-*l/ 97.8%

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

      3. associate-/r/ 97.8%

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

      4. metadata-eval 97.8%

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

      5. *-rgt-identity 97.8%

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

      6. associate-*r/ 97.8%

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

      7. exp-neg 97.8%

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

      8. distribute-neg-frac 97.8%

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

      9. distribute-frac-neg2 97.8%

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

      10. distribute-lft-neg-out 97.8%

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

      11. +-commutative 97.8%

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

      12. distribute-neg-in 97.8%

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

      13. sub-neg 97.8%

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

      14. *-commutative 97.8%

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

      15. sub-neg 97.8%

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

      16. distribute-neg-in 97.8%

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

      17. +-commutative 97.8%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in b around 0 74.5%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{e^{a}}}} \]
   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 66.0%

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

   if 1.4e154 < 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 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.2% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.05e+103)
   (/ 1.0 (+ 2.0 (* a (+ (* a (+ 0.5 (* a -0.16666666666666666))) -1.0))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.05e+103) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.05d+103) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (0.5d0 + (a * (-0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.05e+103) {
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.05e+103:
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.05e+103)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.05e+103)
		tmp = 1.0 / (2.0 + (a * ((a * (0.5 + (a * -0.16666666666666666))) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.05e+103], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 < 1.0500000000000001e103

   1. Initial program 97.7%

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

      1. *-lft-identity 97.7%

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

      2. associate-*l/ 97.7%

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

      3. associate-/r/ 97.7%

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

      4. metadata-eval 97.7%

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

      5. *-rgt-identity 97.7%

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

      6. associate-*r/ 97.7%

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

      7. exp-neg 97.7%

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

      8. distribute-neg-frac 97.7%

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

      9. distribute-frac-neg2 97.7%

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

      10. distribute-lft-neg-out 97.7%

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

      11. +-commutative 97.7%

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

      12. distribute-neg-in 97.7%

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

      13. sub-neg 97.7%

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

      14. *-commutative 97.7%

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

      15. sub-neg 97.7%

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

      16. distribute-neg-in 97.7%

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

      17. +-commutative 97.7%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in b around 0 76.3%

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

      1. rec-exp 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in a around 0 67.8%

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

   if 1.0500000000000001e103 < 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 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(0.5 + a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.3% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 4.7e+153)
   (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.7e+153) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} 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 <= 4.7d+153) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * 0.5d0) + (-1.0d0))))
    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 <= 4.7e+153) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.7e+153:
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.7e+153)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * 0.5) + -1.0))));
	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 <= 4.7e+153)
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.7e+153], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $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 < 4.69999999999999968e153

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-*l/ 97.8%

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

      3. associate-/r/ 97.8%

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

      4. metadata-eval 97.8%

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

      5. *-rgt-identity 97.8%

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

      6. associate-*r/ 97.8%

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

      7. exp-neg 97.8%

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

      8. distribute-neg-frac 97.8%

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

      9. distribute-frac-neg2 97.8%

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

      10. distribute-lft-neg-out 97.8%

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

      11. +-commutative 97.8%

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

      12. distribute-neg-in 97.8%

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

      13. sub-neg 97.8%

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

      14. *-commutative 97.8%

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

      15. sub-neg 97.8%

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

      16. distribute-neg-in 97.8%

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

      17. +-commutative 97.8%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in b around 0 74.5%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{e^{a}}}} \]
   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 62.7%

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

   if 4.69999999999999968e153 < 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 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.3% accurate, 19.0× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1e+154)
   (/ 1.0 (+ 2.0 (- (* a (* a 0.5)) a)))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1e+154) {
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a));
	} 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 <= 1d+154) then
        tmp = 1.0d0 / (2.0d0 + ((a * (a * 0.5d0)) - a))
    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 <= 1e+154) {
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1e+154:
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1e+154)
		tmp = Float64(1.0 / Float64(2.0 + Float64(Float64(a * Float64(a * 0.5)) - a)));
	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 <= 1e+154)
		tmp = 1.0 / (2.0 + ((a * (a * 0.5)) - a));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1e+154], N[(1.0 / N[(2.0 + N[(N[(a * N[(a * 0.5), $MachinePrecision]), $MachinePrecision] - a), $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 < 1.00000000000000004e154

   1. Initial program 97.8%

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

      1. *-lft-identity 97.8%

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

      2. associate-*l/ 97.8%

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

      3. associate-/r/ 97.8%

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

      4. metadata-eval 97.8%

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

      5. *-rgt-identity 97.8%

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

      6. associate-*r/ 97.8%

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

      7. exp-neg 97.8%

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

      8. distribute-neg-frac 97.8%

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

      9. distribute-frac-neg2 97.8%

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

      10. distribute-lft-neg-out 97.8%

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

      11. +-commutative 97.8%

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

      12. distribute-neg-in 97.8%

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

      13. sub-neg 97.8%

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

      14. *-commutative 97.8%

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

      15. sub-neg 97.8%

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

      16. distribute-neg-in 97.8%

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

      17. +-commutative 97.8%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in b around 0 74.5%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{e^{a}}}} \]
   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 62.7%

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

      1. sub-neg 62.7%

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

      2. metadata-eval 62.7%

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

      3. distribute-rgt-in 62.7%

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

      4. *-commutative 62.7%

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

      5. neg-mul-1 62.7%

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

   10. Applied egg-rr 62.7%

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

   if 1.00000000000000004e154 < 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 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 66.6%

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

Alternative 10: 53.0% accurate, 27.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

   1. *-lft-identity 98.0%

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

   2. associate-*l/ 98.0%

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

   3. associate-/r/ 98.0%

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

   4. metadata-eval 98.0%

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

   5. *-rgt-identity 98.0%

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

   6. associate-*r/ 98.0%

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

   7. exp-neg 98.0%

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

   8. distribute-neg-frac 98.0%

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

   9. distribute-frac-neg2 98.0%

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

   10. distribute-lft-neg-out 98.0%

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

   11. +-commutative 98.0%

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

   12. distribute-neg-in 98.0%

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

   13. sub-neg 98.0%

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

   14. *-commutative 98.0%

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

   15. sub-neg 98.0%

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

   16. distribute-neg-in 98.0%

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

   17. +-commutative 98.0%

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

3. Simplified 99.2%

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

5. Taylor expanded in b around 0 70.4%

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

   1. rec-exp 70.4%

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

7. Simplified 70.4%

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

8. Taylor expanded in a around 0 58.7%

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

9. Final simplification 58.7%

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

Alternative 11: 40.0% 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 98.0%

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

   1. *-lft-identity 98.0%

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

   2. associate-*l/ 98.0%

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

   3. associate-/r/ 98.0%

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

   4. metadata-eval 98.0%

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

   5. *-rgt-identity 98.0%

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

   6. associate-*r/ 98.0%

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

   7. exp-neg 98.0%

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

   8. distribute-neg-frac 98.0%

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

   9. distribute-frac-neg2 98.0%

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

   10. distribute-lft-neg-out 98.0%

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

   11. +-commutative 98.0%

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

   12. distribute-neg-in 98.0%

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

   13. sub-neg 98.0%

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

   14. *-commutative 98.0%

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

   15. sub-neg 98.0%

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

   16. distribute-neg-in 98.0%

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

   17. +-commutative 98.0%

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

3. Simplified 99.2%

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

5. Taylor expanded in b around 0 70.4%

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

   1. rec-exp 70.4%

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

7. Simplified 70.4%

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

8. Taylor expanded in a around 0 43.5%

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

   1. *-commutative 43.5%

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

10. Simplified 43.5%

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

11. Final simplification 43.5%

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

Alternative 12: 40.7% 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 98.0%

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

   1. *-lft-identity 98.0%

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

   2. associate-*l/ 98.0%

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

   3. associate-/r/ 98.0%

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

   4. metadata-eval 98.0%

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

   5. *-rgt-identity 98.0%

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

   6. associate-*r/ 98.0%

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

   7. exp-neg 98.0%

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

   8. distribute-neg-frac 98.0%

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

   9. distribute-frac-neg2 98.0%

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

   10. distribute-lft-neg-out 98.0%

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

   11. +-commutative 98.0%

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

   12. distribute-neg-in 98.0%

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

   13. sub-neg 98.0%

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

   14. *-commutative 98.0%

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

   15. sub-neg 98.0%

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

   16. distribute-neg-in 98.0%

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

   17. +-commutative 98.0%

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

3. Simplified 99.2%

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

5. Taylor expanded in b around 0 70.4%

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

   1. rec-exp 70.4%

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

7. Simplified 70.4%

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

8. Taylor expanded in a around 0 44.2%

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

   1. neg-mul-1 44.2%

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

   2. unsub-neg 44.2%

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

10. Simplified 44.2%

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

11. Final simplification 44.2%

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

Alternative 13: 39.9% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.0%

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

   1. *-lft-identity 98.0%

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

   2. associate-*l/ 98.0%

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

   3. associate-/r/ 98.0%

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

   4. metadata-eval 98.0%

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

   5. *-rgt-identity 98.0%

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

   6. associate-*r/ 98.0%

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

   7. exp-neg 98.0%

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

   8. distribute-neg-frac 98.0%

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

   9. distribute-frac-neg2 98.0%

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

   10. distribute-lft-neg-out 98.0%

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

   11. +-commutative 98.0%

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

   12. distribute-neg-in 98.0%

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

   13. sub-neg 98.0%

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

   14. *-commutative 98.0%

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

   15. sub-neg 98.0%

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

   16. distribute-neg-in 98.0%

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

   17. +-commutative 98.0%

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

3. Simplified 99.2%

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

5. Taylor expanded in a around 0 80.7%

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

6. Taylor expanded in b around 0 43.3%

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

7. Final simplification 43.3%

   \[\leadsto 0.5 \]
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 2024095 
(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))))