Quotient of sum of exps

Percentage Accurate: 99.0% → 100.0%

Time: 5.4s

Alternatives: 7

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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. remove-double-neg 99.6%

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

   5. unsub-neg 99.6%

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

   6. div-sub 72.2%

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

   7. *-lft-identity 72.2%

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

   8. associate-*l/ 72.2%

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

   9. lft-mult-inverse 100.0%

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

   10. sub-neg 100.0%

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

   11. distribute-frac-neg 100.0%

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

   12. remove-double-neg 100.0%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

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

   1. 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-define 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 98.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -710:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -710.0) 0.0 (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -710.0) {
		tmp = 0.0;
	} 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 <= (-710.0d0)) then
        tmp = 0.0d0
    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 <= -710.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -710.0:
		tmp = 0.0
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -710.0)
		tmp = 0.0;
	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 <= -710.0)
		tmp = 0.0;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -710.0], 0.0, N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -710

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 0.0%

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

      7. *-lft-identity 0.0%

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

      8. associate-*l/ 0.0%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

      1. 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-define 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in a around 0 43.3%

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

      1. log1p-define 43.3%

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

   9. Simplified 43.3%

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

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0} \]

   if -710 < a

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0 98.5%

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

Alternative 3: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. remove-double-neg 99.6%

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

   5. unsub-neg 99.6%

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

   6. div-sub 72.2%

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

   7. *-lft-identity 72.2%

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

   8. associate-*l/ 72.2%

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

   9. lft-mult-inverse 100.0%

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

   10. sub-neg 100.0%

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

   11. distribute-frac-neg 100.0%

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

   12. remove-double-neg 100.0%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 4: 80.0% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-85} \lor \neg \left(b \leq 2.1 \cdot 10^{-66}\right) \land b \leq 0.011:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -7.5e-7)
   1.0
   (if (or (<= b 1.95e-85) (and (not (<= b 2.1e-66)) (<= b 0.011)))
     (+ 0.5 (* a 0.25))
     0.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -7.5e-7) {
		tmp = 1.0;
	} else if ((b <= 1.95e-85) || (!(b <= 2.1e-66) && (b <= 0.011))) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-7.5d-7)) then
        tmp = 1.0d0
    else if ((b <= 1.95d-85) .or. (.not. (b <= 2.1d-66)) .and. (b <= 0.011d0)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -7.5e-7) {
		tmp = 1.0;
	} else if ((b <= 1.95e-85) || (!(b <= 2.1e-66) && (b <= 0.011))) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -7.5e-7:
		tmp = 1.0
	elif (b <= 1.95e-85) or (not (b <= 2.1e-66) and (b <= 0.011)):
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -7.5e-7)
		tmp = 1.0;
	elseif ((b <= 1.95e-85) || (!(b <= 2.1e-66) && (b <= 0.011)))
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -7.5e-7)
		tmp = 1.0;
	elseif ((b <= 1.95e-85) || (~((b <= 2.1e-66)) && (b <= 0.011)))
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -7.5e-7], 1.0, If[Or[LessEqual[b, 1.95e-85], And[N[Not[LessEqual[b, 2.1e-66]], $MachinePrecision], LessEqual[b, 0.011]]], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if b < -7.5000000000000002e-7

   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. remove-double-neg 97.7%

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

      5. unsub-neg 97.7%

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

      6. div-sub 93.3%

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

      7. *-lft-identity 93.3%

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

      8. associate-*l/ 93.3%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{b - a}\right)\right)}} \]

      2. expm1-undefine 99.9%

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(1 + e^{b - a}\right)} - 1}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(1 + e^{b - a}\right)} - 1}} \]
   7. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(1 + e^{b - a}\right)} + \left(-1\right)}} \]

      2. metadata-eval 99.9%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(1 + e^{b - a}\right)} + \color{blue}{-1}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{1}{\color{blue}{-1 + e^{\mathsf{log1p}\left(1 + e^{b - a}\right)}}} \]

      4. log1p-undefine 99.9%

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

      5. rem-exp-log 99.9%

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

      6. associate-+r+ 99.9%

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

      7. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

      1. add-exp-log 99.9%

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

      2. associate-+r+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. log1p-undefine 100.0%

         \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{log1p}\left(e^{b - a}\right)}}} \]

      5. add-sqr-sqrt 100.0%

         \[\leadsto \frac{1}{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b - a}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b - a}\right)}}}} \]

      6. sqrt-unprod 100.0%

         \[\leadsto \frac{1}{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b - a}\right) \cdot \mathsf{log1p}\left(e^{b - a}\right)}}}} \]

      7. sqr-neg 100.0%

         \[\leadsto \frac{1}{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b - a}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b - a}\right)\right)}}}} \]

      8. sqrt-unprod 88.9%

         \[\leadsto \frac{1}{e^{\color{blue}{\sqrt{-\mathsf{log1p}\left(e^{b - a}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b - a}\right)}}}} \]

      9. add-sqr-sqrt 92.2%

         \[\leadsto \frac{1}{e^{\color{blue}{-\mathsf{log1p}\left(e^{b - a}\right)}}} \]

      10. exp-neg 92.2%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{\mathsf{log1p}\left(e^{b - a}\right)}}}} \]

      11. log1p-undefine 92.2%

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

      12. metadata-eval 92.2%

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

      13. associate-+r+ 92.2%

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

      14. add-exp-log 92.2%

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

      15. add-sqr-sqrt 92.2%

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

   10. Applied egg-rr 92.2%

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

       1. *-inverses 92.3%

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

   12. Simplified 92.3%

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

   if -7.5000000000000002e-7 < b < 1.94999999999999994e-85 or 2.1e-66 < b < 0.010999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0 99.0%

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

   4. Taylor expanded in a around 0 71.5%

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

      1. *-commutative 71.5%

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

   6. Simplified 71.5%

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

   if 1.94999999999999994e-85 < b < 2.1e-66 or 0.010999999999999999 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 61.5%

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

      7. *-lft-identity 61.5%

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

      8. associate-*l/ 61.5%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. 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-define 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in a around 0 93.6%

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

      1. log1p-define 93.6%

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

   9. Simplified 93.6%

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

   11. Applied egg-rr 98.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-85} \lor \neg \left(b \leq 2.1 \cdot 10^{-66}\right) \land b \leq 0.011:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.85:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-91}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-66}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-10}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.85)
   1.0
   (if (<= b 2e-91) 0.5 (if (<= b 2.1e-66) 0.0 (if (<= b 4e-10) 0.5 0.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.85) {
		tmp = 1.0;
	} else if (b <= 2e-91) {
		tmp = 0.5;
	} else if (b <= 2.1e-66) {
		tmp = 0.0;
	} else if (b <= 4e-10) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.85d0)) then
        tmp = 1.0d0
    else if (b <= 2d-91) then
        tmp = 0.5d0
    else if (b <= 2.1d-66) then
        tmp = 0.0d0
    else if (b <= 4d-10) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.85) {
		tmp = 1.0;
	} else if (b <= 2e-91) {
		tmp = 0.5;
	} else if (b <= 2.1e-66) {
		tmp = 0.0;
	} else if (b <= 4e-10) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.85:
		tmp = 1.0
	elif b <= 2e-91:
		tmp = 0.5
	elif b <= 2.1e-66:
		tmp = 0.0
	elif b <= 4e-10:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.85)
		tmp = 1.0;
	elseif (b <= 2e-91)
		tmp = 0.5;
	elseif (b <= 2.1e-66)
		tmp = 0.0;
	elseif (b <= 4e-10)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.85)
		tmp = 1.0;
	elseif (b <= 2e-91)
		tmp = 0.5;
	elseif (b <= 2.1e-66)
		tmp = 0.0;
	elseif (b <= 4e-10)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.85], 1.0, If[LessEqual[b, 2e-91], 0.5, If[LessEqual[b, 2.1e-66], 0.0, If[LessEqual[b, 4e-10], 0.5, 0.0]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.849999999999999978

   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/ 97.6%

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

      4. remove-double-neg 97.6%

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

      5. unsub-neg 97.6%

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

      6. div-sub 97.6%

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

      7. *-lft-identity 97.6%

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

      8. associate-*l/ 97.6%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{b - a}\right)\right)}} \]

      2. expm1-undefine 99.9%

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(1 + e^{b - a}\right)} - 1}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(1 + e^{b - a}\right)} - 1}} \]
   7. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(1 + e^{b - a}\right)} + \left(-1\right)}} \]

      2. metadata-eval 99.9%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(1 + e^{b - a}\right)} + \color{blue}{-1}} \]

      3. +-commutative 99.9%

         \[\leadsto \frac{1}{\color{blue}{-1 + e^{\mathsf{log1p}\left(1 + e^{b - a}\right)}}} \]

      4. log1p-undefine 99.9%

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

      5. rem-exp-log 99.9%

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

      6. associate-+r+ 99.9%

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

      7. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

      1. add-exp-log 99.9%

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

      2. associate-+r+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. log1p-undefine 100.0%

         \[\leadsto \frac{1}{e^{\color{blue}{\mathsf{log1p}\left(e^{b - a}\right)}}} \]

      5. add-sqr-sqrt 100.0%

         \[\leadsto \frac{1}{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b - a}\right)} \cdot \sqrt{\mathsf{log1p}\left(e^{b - a}\right)}}}} \]

      6. sqrt-unprod 100.0%

         \[\leadsto \frac{1}{e^{\color{blue}{\sqrt{\mathsf{log1p}\left(e^{b - a}\right) \cdot \mathsf{log1p}\left(e^{b - a}\right)}}}} \]

      7. sqr-neg 100.0%

         \[\leadsto \frac{1}{e^{\sqrt{\color{blue}{\left(-\mathsf{log1p}\left(e^{b - a}\right)\right) \cdot \left(-\mathsf{log1p}\left(e^{b - a}\right)\right)}}}} \]

      8. sqrt-unprod 93.0%

         \[\leadsto \frac{1}{e^{\color{blue}{\sqrt{-\mathsf{log1p}\left(e^{b - a}\right)} \cdot \sqrt{-\mathsf{log1p}\left(e^{b - a}\right)}}}} \]

      9. add-sqr-sqrt 96.4%

         \[\leadsto \frac{1}{e^{\color{blue}{-\mathsf{log1p}\left(e^{b - a}\right)}}} \]

      10. exp-neg 96.4%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{\mathsf{log1p}\left(e^{b - a}\right)}}}} \]

      11. log1p-undefine 96.4%

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

      12. metadata-eval 96.4%

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

      13. associate-+r+ 96.4%

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

      14. add-exp-log 96.4%

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

      15. add-sqr-sqrt 96.4%

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

   10. Applied egg-rr 96.4%

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

       1. *-inverses 96.4%

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

   12. Simplified 96.4%

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

   if -0.849999999999999978 < b < 2.00000000000000004e-91 or 2.1e-66 < b < 4.00000000000000015e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in b around 0 99.0%

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

   4. Taylor expanded in a around 0 69.1%

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

   if 2.00000000000000004e-91 < b < 2.1e-66 or 4.00000000000000015e-10 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 61.5%

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

      7. *-lft-identity 61.5%

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

      8. associate-*l/ 61.5%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. 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-define 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in a around 0 93.6%

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

      1. log1p-define 93.6%

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

   9. Simplified 93.6%

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

   11. Applied egg-rr 98.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.85:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-91}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-66}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-10}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.05 \cdot 10^{-85}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-66}:\\ \;\;\;\;0\\ \mathbf{elif}\;b \leq 0.07:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.05e-85) 0.5 (if (<= b 2.1e-66) 0.0 (if (<= b 0.07) 0.5 0.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.05e-85) {
		tmp = 0.5;
	} else if (b <= 2.1e-66) {
		tmp = 0.0;
	} else if (b <= 0.07) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.05d-85) then
        tmp = 0.5d0
    else if (b <= 2.1d-66) then
        tmp = 0.0d0
    else if (b <= 0.07d0) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.05e-85) {
		tmp = 0.5;
	} else if (b <= 2.1e-66) {
		tmp = 0.0;
	} else if (b <= 0.07) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.05e-85:
		tmp = 0.5
	elif b <= 2.1e-66:
		tmp = 0.0
	elif b <= 0.07:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.05e-85)
		tmp = 0.5;
	elseif (b <= 2.1e-66)
		tmp = 0.0;
	elseif (b <= 0.07)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.05e-85)
		tmp = 0.5;
	elseif (b <= 2.1e-66)
		tmp = 0.0;
	elseif (b <= 0.07)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.05e-85], 0.5, If[LessEqual[b, 2.1e-66], 0.0, If[LessEqual[b, 0.07], 0.5, 0.0]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.04999999999999997e-85 or 2.1e-66 < b < 0.070000000000000007

   1. Initial program 99.3%

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

   3. Taylor expanded in b around 0 78.0%

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

   4. Taylor expanded in a around 0 56.0%

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

   if 2.04999999999999997e-85 < b < 2.1e-66 or 0.070000000000000007 < b

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r/ 100.0%

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

      4. remove-double-neg 100.0%

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

      5. unsub-neg 100.0%

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

      6. div-sub 61.5%

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

      7. *-lft-identity 61.5%

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

      8. associate-*l/ 61.5%

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

      9. lft-mult-inverse 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-frac-neg 100.0%

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

      12. remove-double-neg 100.0%

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

      13. div-exp 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. 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-define 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in a around 0 93.6%

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

      1. log1p-define 93.6%

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

   9. Simplified 93.6%

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

   11. Applied egg-rr 98.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 46.8% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.0)

C

double code(double a, double b) {
	return 0.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.0d0
end function

Java

public static double code(double a, double b) {
	return 0.0;
}

Python

def code(a, b):
	return 0.0

Julia

function code(a, b)
	return 0.0
end

MATLAB

function tmp = code(a, b)
	tmp = 0.0;
end

Wolfram

code[a_, b_] := 0.0

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-/r/ 99.6%

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

   4. remove-double-neg 99.6%

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

   5. unsub-neg 99.6%

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

   6. div-sub 72.2%

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

   7. *-lft-identity 72.2%

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

   8. associate-*l/ 72.2%

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

   9. lft-mult-inverse 100.0%

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

   10. sub-neg 100.0%

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

   11. distribute-frac-neg 100.0%

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

   12. remove-double-neg 100.0%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

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

   1. 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-define 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in a around 0 83.4%

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

   1. log1p-define 83.4%

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

9. Simplified 83.4%

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

11. Applied egg-rr 50.4%

    \[\leadsto \color{blue}{0} \]
12. 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 2024111 
(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))))