Quotient of sum of exps

Percentage Accurate: 98.8% → 100.0%

Time: 6.5s

Alternatives: 11

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.8% 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 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%

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

   3. remove-double-div 100.0%

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

   4. exp-neg 100.0%

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

   5. associate-/r/ 100.0%

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

   6. /-rgt-identity 100.0%

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

   7. *-commutative 100.0%

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

   8. distribute-rgt-in 70.3%

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

   9. exp-neg 70.3%

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

   10. rgt-mult-inverse 100.0%

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

   11. prod-exp 100.0%

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

   12. unsub-neg 100.0%

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

3. Simplified 100.0%

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

   1. add-exp-log 100.0%

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

   2. log-rec 100.0%

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

   3. log1p-udef 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 92.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot -0.5\\ \mathbf{if}\;a \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-2 - a}{a \cdot a}\\ \mathbf{elif}\;a \leq -7.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{a - t_0}{a \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* a a) -0.5)))
   (if (<= a -1.35e+154)
     (/ (- -2.0 a) (* a a))
     (if (<= a -7.2e+102) (/ (- a t_0) (* a t_0)) (/ 1.0 (+ 1.0 (exp b)))))))

C

double code(double a, double b) {
	double t_0 = (a * a) * -0.5;
	double tmp;
	if (a <= -1.35e+154) {
		tmp = (-2.0 - a) / (a * a);
	} else if (a <= -7.2e+102) {
		tmp = (a - t_0) / (a * t_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) :: t_0
    real(8) :: tmp
    t_0 = (a * a) * (-0.5d0)
    if (a <= (-1.35d+154)) then
        tmp = ((-2.0d0) - a) / (a * a)
    else if (a <= (-7.2d+102)) then
        tmp = (a - t_0) / (a * t_0)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (a * a) * -0.5;
	double tmp;
	if (a <= -1.35e+154) {
		tmp = (-2.0 - a) / (a * a);
	} else if (a <= -7.2e+102) {
		tmp = (a - t_0) / (a * t_0);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (a * a) * -0.5
	tmp = 0
	if a <= -1.35e+154:
		tmp = (-2.0 - a) / (a * a)
	elif a <= -7.2e+102:
		tmp = (a - t_0) / (a * t_0)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(a * a) * -0.5)
	tmp = 0.0
	if (a <= -1.35e+154)
		tmp = Float64(Float64(-2.0 - a) / Float64(a * a));
	elseif (a <= -7.2e+102)
		tmp = Float64(Float64(a - t_0) / Float64(a * t_0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (a * a) * -0.5;
	tmp = 0.0;
	if (a <= -1.35e+154)
		tmp = (-2.0 - a) / (a * a);
	elseif (a <= -7.2e+102)
		tmp = (a - t_0) / (a * t_0);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[a, -1.35e+154], N[(N[(-2.0 - a), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -7.2e+102], N[(N[(a - t$95$0), $MachinePrecision] / N[(a * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.35000000000000003e154

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 0.0%

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

      9. exp-neg 0.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   5. Taylor expanded in a around 0 6.4%

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

      1. neg-mul-1 6.4%

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

      2. unsub-neg 6.4%

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

   7. Simplified 6.4%

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

   8. Taylor expanded in a around inf 6.4%

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

      1. distribute-neg-in 6.4%

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

      2. associate-*r/ 6.4%

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

      3. metadata-eval 6.4%

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

      4. distribute-neg-frac 6.4%

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

      5. metadata-eval 6.4%

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

      6. unpow2 6.4%

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

      7. distribute-neg-frac 6.4%

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

      8. metadata-eval 6.4%

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

   10. Simplified 6.4%

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

       1. associate-/r* 6.4%

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

       2. frac-add 100.0%

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

       3. metadata-eval 100.0%

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

       4. div-inv 100.0%

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

       5. fma-def 100.0%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-2}{a}, a, \frac{a}{-1}\right)}}{a \cdot a} \]

       6. div-inv 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-2}{a}, a, \color{blue}{a \cdot \frac{1}{-1}}\right)}{a \cdot a} \]

       7. metadata-eval 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-2}{a}, a, a \cdot \color{blue}{-1}\right)}{a \cdot a} \]

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-2}{a}, a, a \cdot -1\right)}{a \cdot a}} \]
   13. Step-by-step derivation

       1. fma-udef 100.0%

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

       2. *-commutative 100.0%

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

       3. *-commutative 100.0%

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

       4. neg-mul-1 100.0%

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

       5. +-commutative 100.0%

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

       6. remove-double-neg 100.0%

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

       7. distribute-lft-neg-out 100.0%

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

       8. distribute-rgt-neg-in 100.0%

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

       9. mul-1-neg 100.0%

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

       10. distribute-lft-neg-in 100.0%

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

       11. associate-*l* 100.0%

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

       12. unsub-neg 100.0%

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

       13. associate-*r/ 100.0%

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

       14. associate-*l* 100.0%

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

       15. metadata-eval 100.0%

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

       16. *-commutative 100.0%

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

       17. associate-/l* 100.0%

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

       18. *-inverses 100.0%

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

       19. metadata-eval 100.0%

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

       20. sub-neg 100.0%

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

       21. metadata-eval 100.0%

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

       22. +-commutative 100.0%

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

       23. sub-neg 100.0%

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

   14. Simplified 100.0%

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

   if -1.35000000000000003e154 < a < -7.2000000000000003e102

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 0.0%

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

      9. exp-neg 0.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

   5. Taylor expanded in a around 0 4.3%

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

      1. neg-mul-1 4.3%

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

      2. unsub-neg 4.3%

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

   7. Simplified 4.3%

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

   8. Taylor expanded in a around inf 4.3%

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

      1. distribute-neg-in 4.3%

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

      2. associate-*r/ 4.3%

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

      3. metadata-eval 4.3%

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

      4. distribute-neg-frac 4.3%

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

      5. metadata-eval 4.3%

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

      6. unpow2 4.3%

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

      7. distribute-neg-frac 4.3%

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

      8. metadata-eval 4.3%

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

   10. Simplified 4.3%

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

       1. clear-num 4.3%

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

       2. frac-add 100.0%

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

       3. *-un-lft-identity 100.0%

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

       4. div-inv 100.0%

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

       5. metadata-eval 100.0%

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

       6. div-inv 100.0%

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

       7. metadata-eval 100.0%

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

   12. Applied egg-rr 100.0%

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

   if -7.2000000000000003e102 < a

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 86.9%

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

      9. exp-neg 86.9%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 88.8%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-2 - a}{a \cdot a}\\ \mathbf{elif}\;a \leq -7.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{a - \left(a \cdot a\right) \cdot -0.5}{a \cdot \left(\left(a \cdot a\right) \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 3: 73.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+40}:\\ \;\;\;\;\frac{1}{a \cdot a + 4} \cdot \left(a + 2\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot -0.020833333333333332\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -9.5)
   1.0
   (if (<= b 2.4e+40)
     (* (/ 1.0 (+ (* a a) 4.0)) (+ a 2.0))
     (* (pow a 3.0) -0.020833333333333332))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -9.5) {
		tmp = 1.0;
	} else if (b <= 2.4e+40) {
		tmp = (1.0 / ((a * a) + 4.0)) * (a + 2.0);
	} else {
		tmp = pow(a, 3.0) * -0.020833333333333332;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-9.5d0)) then
        tmp = 1.0d0
    else if (b <= 2.4d+40) then
        tmp = (1.0d0 / ((a * a) + 4.0d0)) * (a + 2.0d0)
    else
        tmp = (a ** 3.0d0) * (-0.020833333333333332d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -9.5) {
		tmp = 1.0;
	} else if (b <= 2.4e+40) {
		tmp = (1.0 / ((a * a) + 4.0)) * (a + 2.0);
	} else {
		tmp = Math.pow(a, 3.0) * -0.020833333333333332;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -9.5:
		tmp = 1.0
	elif b <= 2.4e+40:
		tmp = (1.0 / ((a * a) + 4.0)) * (a + 2.0)
	else:
		tmp = math.pow(a, 3.0) * -0.020833333333333332
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -9.5)
		tmp = 1.0;
	elseif (b <= 2.4e+40)
		tmp = Float64(Float64(1.0 / Float64(Float64(a * a) + 4.0)) * Float64(a + 2.0));
	else
		tmp = Float64((a ^ 3.0) * -0.020833333333333332);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -9.5)
		tmp = 1.0;
	elseif (b <= 2.4e+40)
		tmp = (1.0 / ((a * a) + 4.0)) * (a + 2.0);
	else
		tmp = (a ^ 3.0) * -0.020833333333333332;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -9.5], 1.0, If[LessEqual[b, 2.4e+40], N[(N[(1.0 / N[(N[(a * a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] * N[(a + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 3.0], $MachinePrecision] * -0.020833333333333332), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.5

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. exp-neg 100.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(e^{b}, e^{a}, 1\right)}}{\sqrt{\mathsf{fma}\left(e^{b}, e^{a}, 1\right)}}} \]
   8. Step-by-step derivation

      1. *-inverses 100.0%

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

   9. Simplified 100.0%

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

   if -9.5 < b < 2.4e40

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/l* 100.0%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 63.3%

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

      9. exp-neg 63.3%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 96.1%

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

   5. Taylor expanded in a around 0 59.0%

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

      1. neg-mul-1 59.0%

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

      2. unsub-neg 59.0%

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

   7. Simplified 59.0%

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

      1. flip-- 74.2%

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

      2. associate-/r/ 74.2%

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

      3. sub-neg 74.2%

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

      4. metadata-eval 74.2%

         \[\leadsto \frac{1}{\color{blue}{4} + \left(-a \cdot a\right)} \cdot \left(2 + a\right) \]

      5. add-sqr-sqrt 25.7%

         \[\leadsto \frac{1}{4 + \left(-a \cdot \color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)}\right)} \cdot \left(2 + a\right) \]

      6. sqrt-unprod 74.2%

         \[\leadsto \frac{1}{4 + \left(-a \cdot \color{blue}{\sqrt{a \cdot a}}\right)} \cdot \left(2 + a\right) \]

      7. sqr-neg 74.2%

         \[\leadsto \frac{1}{4 + \left(-a \cdot \sqrt{\color{blue}{\left(-a\right) \cdot \left(-a\right)}}\right)} \cdot \left(2 + a\right) \]

      8. sqrt-unprod 48.5%

         \[\leadsto \frac{1}{4 + \left(-a \cdot \color{blue}{\left(\sqrt{-a} \cdot \sqrt{-a}\right)}\right)} \cdot \left(2 + a\right) \]

      9. add-sqr-sqrt 74.2%

         \[\leadsto \frac{1}{4 + \left(-a \cdot \color{blue}{\left(-a\right)}\right)} \cdot \left(2 + a\right) \]

      10. distribute-lft-neg-out 74.2%

          \[\leadsto \frac{1}{4 + \color{blue}{\left(-a\right) \cdot \left(-a\right)}} \cdot \left(2 + a\right) \]

      11. sqr-neg 74.2%

          \[\leadsto \frac{1}{4 + \color{blue}{a \cdot a}} \cdot \left(2 + a\right) \]

      12. +-commutative 74.2%

          \[\leadsto \frac{1}{4 + a \cdot a} \cdot \color{blue}{\left(a + 2\right)} \]

   9. Applied egg-rr 74.2%

      \[\leadsto \color{blue}{\frac{1}{4 + a \cdot a} \cdot \left(a + 2\right)} \]

   if 2.4e40 < 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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 66.2%

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

      9. exp-neg 66.2%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 35.9%

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

   5. Taylor expanded in a around 0 2.7%

      \[\leadsto \color{blue}{-0.020833333333333332 \cdot {a}^{3} + \left(0.5 + 0.25 \cdot a\right)} \]

   6. Taylor expanded in a around inf 41.3%

      \[\leadsto \color{blue}{-0.020833333333333332 \cdot {a}^{3}} \]
   7. Step-by-step derivation

      1. *-commutative 41.3%

         \[\leadsto \color{blue}{{a}^{3} \cdot -0.020833333333333332} \]

   8. Simplified 41.3%

      \[\leadsto \color{blue}{{a}^{3} \cdot -0.020833333333333332} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+40}:\\ \;\;\;\;\frac{1}{a \cdot a + 4} \cdot \left(a + 2\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{3} \cdot -0.020833333333333332\\ \end{array} \]

Alternative 4: 98.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8.6e-5) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8.6e-5) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-8.6d-5)) then
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8.6e-5) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8.6e-5:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8.6e-5)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8.6e-5)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8.6e-5], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.6000000000000003e-5

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/l* 99.9%

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

      3. remove-double-div 99.9%

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

      4. exp-neg 99.9%

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

      5. associate-/r/ 99.9%

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

      6. /-rgt-identity 99.9%

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

      7. *-commutative 99.9%

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

      8. distribute-rgt-in 6.1%

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

      9. exp-neg 6.1%

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

      10. rgt-mult-inverse 99.9%

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

      11. prod-exp 99.9%

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

      12. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in b around 0 99.9%

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

   if -8.6000000000000003e-5 < a

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. exp-neg 100.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a around 0 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.6 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

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 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%

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

   3. remove-double-div 100.0%

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

   4. exp-neg 100.0%

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

   5. associate-/r/ 100.0%

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

   6. /-rgt-identity 100.0%

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

   7. *-commutative 100.0%

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

   8. distribute-rgt-in 70.3%

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

   9. exp-neg 70.3%

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

   10. rgt-mult-inverse 100.0%

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

   11. prod-exp 100.0%

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

   12. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 6: 67.2% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.6:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot a + 4} \cdot \left(a + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -5.6) 1.0 (* (/ 1.0 (+ (* a a) 4.0)) (+ a 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -5.6) {
		tmp = 1.0;
	} else {
		tmp = (1.0 / ((a * a) + 4.0)) * (a + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-5.6d0)) then
        tmp = 1.0d0
    else
        tmp = (1.0d0 / ((a * a) + 4.0d0)) * (a + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -5.6) {
		tmp = 1.0;
	} else {
		tmp = (1.0 / ((a * a) + 4.0)) * (a + 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -5.6:
		tmp = 1.0
	else:
		tmp = (1.0 / ((a * a) + 4.0)) * (a + 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -5.6)
		tmp = 1.0;
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(a * a) + 4.0)) * Float64(a + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -5.6)
		tmp = 1.0;
	else
		tmp = (1.0 / ((a * a) + 4.0)) * (a + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -5.6], 1.0, N[(N[(1.0 / N[(N[(a * a), $MachinePrecision] + 4.0), $MachinePrecision]), $MachinePrecision] * N[(a + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.5999999999999996

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. exp-neg 100.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(e^{b}, e^{a}, 1\right)}}{\sqrt{\mathsf{fma}\left(e^{b}, e^{a}, 1\right)}}} \]
   8. Step-by-step derivation

      1. *-inverses 100.0%

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

   9. Simplified 100.0%

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

   if -5.5999999999999996 < 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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 64.3%

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

      9. exp-neg 64.3%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 76.0%

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

   5. Taylor expanded in a around 0 40.6%

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

      1. neg-mul-1 40.6%

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

      2. unsub-neg 40.6%

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

   7. Simplified 40.6%

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

      1. flip-- 56.0%

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

      2. associate-/r/ 56.0%

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

      3. sub-neg 56.0%

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

      4. metadata-eval 56.0%

         \[\leadsto \frac{1}{\color{blue}{4} + \left(-a \cdot a\right)} \cdot \left(2 + a\right) \]

      5. add-sqr-sqrt 17.5%

         \[\leadsto \frac{1}{4 + \left(-a \cdot \color{blue}{\left(\sqrt{a} \cdot \sqrt{a}\right)}\right)} \cdot \left(2 + a\right) \]

      6. sqrt-unprod 56.0%

         \[\leadsto \frac{1}{4 + \left(-a \cdot \color{blue}{\sqrt{a \cdot a}}\right)} \cdot \left(2 + a\right) \]

      7. sqr-neg 56.0%

         \[\leadsto \frac{1}{4 + \left(-a \cdot \sqrt{\color{blue}{\left(-a\right) \cdot \left(-a\right)}}\right)} \cdot \left(2 + a\right) \]

      8. sqrt-unprod 38.5%

         \[\leadsto \frac{1}{4 + \left(-a \cdot \color{blue}{\left(\sqrt{-a} \cdot \sqrt{-a}\right)}\right)} \cdot \left(2 + a\right) \]

      9. add-sqr-sqrt 56.0%

         \[\leadsto \frac{1}{4 + \left(-a \cdot \color{blue}{\left(-a\right)}\right)} \cdot \left(2 + a\right) \]

      10. distribute-lft-neg-out 56.0%

          \[\leadsto \frac{1}{4 + \color{blue}{\left(-a\right) \cdot \left(-a\right)}} \cdot \left(2 + a\right) \]

      11. sqr-neg 56.0%

          \[\leadsto \frac{1}{4 + \color{blue}{a \cdot a}} \cdot \left(2 + a\right) \]

      12. +-commutative 56.0%

          \[\leadsto \frac{1}{4 + a \cdot a} \cdot \color{blue}{\left(a + 2\right)} \]

   9. Applied egg-rr 56.0%

      \[\leadsto \color{blue}{\frac{1}{4 + a \cdot a} \cdot \left(a + 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.6:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot a + 4} \cdot \left(a + 2\right)\\ \end{array} \]

Alternative 7: 58.1% accurate, 27.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 10^{+15}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 - a}{a \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.4)
   1.0
   (if (<= b 1e+15) (/ 1.0 (- 2.0 a)) (/ (- -2.0 a) (* a a)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.4) {
		tmp = 1.0;
	} else if (b <= 1e+15) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = (-2.0 - a) / (a * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.4d0)) then
        tmp = 1.0d0
    else if (b <= 1d+15) then
        tmp = 1.0d0 / (2.0d0 - a)
    else
        tmp = ((-2.0d0) - a) / (a * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.4) {
		tmp = 1.0;
	} else if (b <= 1e+15) {
		tmp = 1.0 / (2.0 - a);
	} else {
		tmp = (-2.0 - a) / (a * a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.4:
		tmp = 1.0
	elif b <= 1e+15:
		tmp = 1.0 / (2.0 - a)
	else:
		tmp = (-2.0 - a) / (a * a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.4)
		tmp = 1.0;
	elseif (b <= 1e+15)
		tmp = Float64(1.0 / Float64(2.0 - a));
	else
		tmp = Float64(Float64(-2.0 - a) / Float64(a * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.4)
		tmp = 1.0;
	elseif (b <= 1e+15)
		tmp = 1.0 / (2.0 - a);
	else
		tmp = (-2.0 - a) / (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.4], 1.0, If[LessEqual[b, 1e+15], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 - a), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.3999999999999999

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. exp-neg 100.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(e^{b}, e^{a}, 1\right)}}{\sqrt{\mathsf{fma}\left(e^{b}, e^{a}, 1\right)}}} \]
   8. Step-by-step derivation

      1. *-inverses 100.0%

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

   9. Simplified 100.0%

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

   if -1.3999999999999999 < b < 1e15

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/l* 100.0%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 65.4%

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

      9. exp-neg 65.4%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 96.7%

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

   5. Taylor expanded in a around 0 61.4%

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

      1. neg-mul-1 61.4%

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

      2. unsub-neg 61.4%

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

   7. Simplified 61.4%

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

   if 1e15 < 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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 62.3%

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

      9. exp-neg 62.3%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 39.6%

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

   5. Taylor expanded in a around 0 4.0%

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

      1. neg-mul-1 4.0%

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

      2. unsub-neg 4.0%

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

   7. Simplified 4.0%

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

   8. Taylor expanded in a around inf 3.3%

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

      1. distribute-neg-in 3.3%

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

      2. associate-*r/ 3.3%

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

      3. metadata-eval 3.3%

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

      4. distribute-neg-frac 3.3%

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

      5. metadata-eval 3.3%

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

      6. unpow2 3.3%

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

      7. distribute-neg-frac 3.3%

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

      8. metadata-eval 3.3%

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

   10. Simplified 3.3%

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

       1. associate-/r* 3.3%

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

       2. frac-add 22.7%

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

       3. metadata-eval 22.7%

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

       4. div-inv 22.7%

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

       5. fma-def 22.7%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-2}{a}, a, \frac{a}{-1}\right)}}{a \cdot a} \]

       6. div-inv 22.7%

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-2}{a}, a, \color{blue}{a \cdot \frac{1}{-1}}\right)}{a \cdot a} \]

       7. metadata-eval 22.7%

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-2}{a}, a, a \cdot \color{blue}{-1}\right)}{a \cdot a} \]

   12. Applied egg-rr 22.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{-2}{a}, a, a \cdot -1\right)}{a \cdot a}} \]
   13. Step-by-step derivation

       1. fma-udef 22.7%

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

       2. *-commutative 22.7%

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

       3. *-commutative 22.7%

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

       4. neg-mul-1 22.7%

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

       5. +-commutative 22.7%

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

       6. remove-double-neg 22.7%

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

       7. distribute-lft-neg-out 22.7%

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

       8. distribute-rgt-neg-in 22.7%

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

       9. mul-1-neg 22.7%

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

       10. distribute-lft-neg-in 22.7%

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

       11. associate-*l* 22.7%

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

       12. unsub-neg 22.7%

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

       13. associate-*r/ 22.7%

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

       14. associate-*l* 22.7%

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

       15. metadata-eval 22.7%

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

       16. *-commutative 22.7%

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

       17. associate-/l* 22.7%

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

       18. *-inverses 22.7%

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

       19. metadata-eval 22.7%

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

       20. sub-neg 22.7%

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

       21. metadata-eval 22.7%

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

       22. +-commutative 22.7%

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

       23. sub-neg 22.7%

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

   14. Simplified 22.7%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 10^{+15}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 - a}{a \cdot a}\\ \end{array} \]

Alternative 8: 53.6% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00019:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.00019) 1.0 (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.00019) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.00019d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 + (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.00019) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.00019:
		tmp = 1.0
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.00019)
		tmp = 1.0;
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.00019)
		tmp = 1.0;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.00019], 1.0, N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.9000000000000001e-4

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 97.7%

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

      9. exp-neg 97.7%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(e^{b}, e^{a}, 1\right)}}{\sqrt{\mathsf{fma}\left(e^{b}, e^{a}, 1\right)}}} \]
   8. Step-by-step derivation

      1. *-inverses 97.8%

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

   9. Simplified 97.8%

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

   if -1.9000000000000001e-4 < 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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 64.6%

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

      9. exp-neg 64.6%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 75.9%

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

   5. Taylor expanded in a around 0 40.3%

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

      1. *-commutative 40.3%

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

   7. Simplified 40.3%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00019:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 9: 54.3% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -1.4) 1.0 (/ 1.0 (- 2.0 a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.4) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (2.0 - a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.4d0)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / (2.0d0 - a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.4) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (2.0 - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.4:
		tmp = 1.0
	else:
		tmp = 1.0 / (2.0 - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.4)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / Float64(2.0 - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.4)
		tmp = 1.0;
	else
		tmp = 1.0 / (2.0 - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.4], 1.0, N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3999999999999999

   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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. exp-neg 100.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(e^{b}, e^{a}, 1\right)}}{\sqrt{\mathsf{fma}\left(e^{b}, e^{a}, 1\right)}}} \]
   8. Step-by-step derivation

      1. *-inverses 100.0%

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

   9. Simplified 100.0%

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

   if -1.3999999999999999 < 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%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 64.3%

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

      9. exp-neg 64.3%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 76.0%

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

   5. Taylor expanded in a around 0 40.6%

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

      1. neg-mul-1 40.6%

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

      2. unsub-neg 40.6%

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

   7. Simplified 40.6%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - a}\\ \end{array} \]

Alternative 10: 53.4% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. exp-neg 100.0%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(e^{b}, e^{a}, 1\right)}}{\sqrt{\mathsf{fma}\left(e^{b}, e^{a}, 1\right)}}} \]
   8. Step-by-step derivation

      1. *-inverses 100.0%

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

   9. Simplified 100.0%

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

   if -1.1000000000000001 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-div 100.0%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 64.3%

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

      9. exp-neg 64.3%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 100.0%

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

      12. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in b around 0 76.0%

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

   5. Taylor expanded in a around 0 39.3%

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

4. Final simplification 49.5%

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

Alternative 11: 38.8% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-/l* 100.0%

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

   3. remove-double-div 100.0%

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

   4. exp-neg 100.0%

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

   5. associate-/r/ 100.0%

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

   6. /-rgt-identity 100.0%

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

   7. *-commutative 100.0%

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

   8. distribute-rgt-in 70.3%

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

   9. exp-neg 70.3%

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

   10. rgt-mult-inverse 100.0%

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

   11. prod-exp 100.0%

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

   12. unsub-neg 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in b around 0 66.4%

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

5. Taylor expanded in a around 0 35.8%

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

6. Final simplification 35.8%

   \[\leadsto 0.5 \]

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 2023272 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

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