Quotient of sum of exps

Percentage Accurate: 98.9% → 100.0%

Time: 9.2s

Alternatives: 10

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. clear-num 98.4%

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

   2. inv-pow 98.4%

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in a around inf 98.4%

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

   1. *-lft-identity 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-/r/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-*l/ 98.4%

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

   6. rec-exp 98.4%

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

   7. distribute-rgt-in 71.5%

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

   8. rec-exp 71.5%

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

   9. rgt-mult-inverse 99.2%

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

   10. rec-exp 99.2%

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

   11. associate-*r/ 99.2%

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

   12. *-rgt-identity 99.2%

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

7. Simplified 99.2%

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

   1. add-exp-log 99.2%

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

   2. log-rec 99.2%

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

   3. log1p-udef 99.2%

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

   4. div-exp 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 67.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1.0) {
		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 (exp(a) <= 1.0d0) 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 (Math.exp(a) <= 1.0) {
		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 math.exp(a) <= 1.0:
		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 (exp(a) <= 1.0)
		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 (exp(a) <= 1.0)
		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[N[Exp[a], $MachinePrecision], 1.0], 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 (exp.f64 a) < 1

   1. Initial program 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in a around inf 99.6%

      \[\leadsto \color{blue}{\frac{e^{a}}{e^{a} + e^{b}}} \]
   6. 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. *-lft-identity 99.6%

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

      5. associate-*l/ 99.6%

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

      6. rec-exp 99.6%

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

      7. distribute-rgt-in 71.6%

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

      8. rec-exp 71.6%

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

      9. rgt-mult-inverse 99.6%

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

      10. rec-exp 99.6%

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

      11. associate-*r/ 99.6%

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

      12. *-rgt-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in b around 0 65.7%

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

      1. rec-exp 65.7%

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

   10. Simplified 65.7%

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

   if 1 < (exp.f64 a)

   1. Initial program 66.7%

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

   3. Taylor expanded in a around 0 80.8%

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

4. Final simplification 66.2%

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

Alternative 3: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ (exp a) 2.0) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = exp(a) / 2.0d0
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 98.6%

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

   3. Taylor expanded in a around 0 98.6%

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

   4. Taylor expanded in b around 0 98.6%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0 97.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -3.1) 1.0 (/ (exp a) 2.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -3.1) {
		tmp = 1.0;
	} else {
		tmp = exp(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 <= (-3.1d0)) then
        tmp = 1.0d0
    else
        tmp = exp(a) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -3.1) {
		tmp = 1.0;
	} else {
		tmp = Math.exp(a) / 2.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -3.1:
		tmp = 1.0
	else:
		tmp = math.exp(a) / 2.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -3.1)
		tmp = 1.0;
	else
		tmp = Float64(exp(a) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -3.1)
		tmp = 1.0;
	else
		tmp = exp(a) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -3.1], 1.0, N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.10000000000000009

   1. Initial program 96.5%

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

      1. clear-num 96.5%

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

      2. inv-pow 96.5%

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in a around inf 96.5%

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

      1. *-lft-identity 96.5%

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

      2. associate-*l/ 96.5%

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

      3. associate-/r/ 96.5%

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

      4. *-lft-identity 96.5%

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

      5. associate-*l/ 96.5%

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

      6. rec-exp 96.5%

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

      7. distribute-rgt-in 96.5%

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

      8. rec-exp 96.5%

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

      9. rgt-mult-inverse 98.2%

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

      10. rec-exp 98.2%

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

      11. associate-*r/ 98.2%

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

      12. *-rgt-identity 98.2%

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

   7. Simplified 98.2%

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

      1. add-exp-log 98.2%

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

      2. log-rec 98.2%

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

      3. log1p-udef 98.2%

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

      4. div-exp 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. exp-neg 100.0%

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

       2. log1p-udef 100.0%

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

       3. add-exp-log 100.0%

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

       4. add-sqr-sqrt 100.0%

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

       5. associate-/r* 100.0%

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

   11. Applied egg-rr 98.7%

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

       1. *-inverses 98.7%

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

   13. Simplified 98.7%

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

   if -3.10000000000000009 < b

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0 97.4%

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

   4. Taylor expanded in b around 0 76.2%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. clear-num 98.4%

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

   2. inv-pow 98.4%

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in a around inf 98.4%

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

   1. *-lft-identity 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-/r/ 98.4%

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

   4. *-lft-identity 98.4%

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

   5. associate-*l/ 98.4%

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

   6. rec-exp 98.4%

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

   7. distribute-rgt-in 71.5%

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

   8. rec-exp 71.5%

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

   9. rgt-mult-inverse 99.2%

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

   10. rec-exp 99.2%

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

   11. associate-*r/ 99.2%

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

   12. *-rgt-identity 99.2%

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

7. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 99.2%

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

   3. log1p-udef 99.2%

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

   4. add-exp-log 99.2%

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

   5. +-commutative 99.2%

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

   6. div-exp 100.0%

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

9. Applied egg-rr 100.0%

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

    1. associate--l+ 100.0%

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

    2. metadata-eval 100.0%

       \[\leadsto \frac{1}{1 + \left(e^{b - a} + \color{blue}{0}\right)} \]

    3. +-rgt-identity 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 6: 54.3% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -4.8e-9) 1.0 (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -4.8e-9) {
		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 <= (-4.8d-9)) 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 <= -4.8e-9) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -4.8e-9:
		tmp = 1.0
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -4.8e-9)
		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 <= -4.8e-9)
		tmp = 1.0;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -4.8e-9], 1.0, N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.8e-9

   1. Initial program 96.6%

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

      1. clear-num 96.6%

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

      2. inv-pow 96.6%

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

   4. Applied egg-rr 96.6%

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

   5. Taylor expanded in a around inf 96.6%

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

      1. *-lft-identity 96.6%

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

      2. associate-*l/ 96.6%

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

      3. associate-/r/ 96.6%

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

      4. *-lft-identity 96.6%

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

      5. associate-*l/ 96.6%

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

      6. rec-exp 96.6%

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

      7. distribute-rgt-in 94.8%

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

      8. rec-exp 94.8%

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

      9. rgt-mult-inverse 98.3%

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

      10. rec-exp 98.3%

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

      11. associate-*r/ 98.3%

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

      12. *-rgt-identity 98.3%

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

   7. Simplified 98.3%

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

      1. add-exp-log 98.3%

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

      2. log-rec 98.3%

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

      3. log1p-udef 98.3%

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

      4. div-exp 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. exp-neg 100.0%

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

       2. log1p-udef 100.0%

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

       3. add-exp-log 100.0%

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

       4. add-sqr-sqrt 100.0%

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

       5. associate-/r* 100.0%

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

   11. Applied egg-rr 97.0%

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

       1. *-inverses 97.0%

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

   13. Simplified 97.0%

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

   if -4.8e-9 < b

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0 77.4%

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

   4. Taylor expanded in a around 0 43.2%

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

      1. *-commutative 43.2%

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

   6. Simplified 43.2%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.2% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.98:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.98) 1.0 (/ 1.0 (+ b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.98) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.98d0)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / (b + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.98) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.98:
		tmp = 1.0
	else:
		tmp = 1.0 / (b + 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.98)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / Float64(b + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.98)
		tmp = 1.0;
	else
		tmp = 1.0 / (b + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.98], 1.0, N[(1.0 / N[(b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.97999999999999998

   1. Initial program 96.5%

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

      1. clear-num 96.5%

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

      2. inv-pow 96.5%

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in a around inf 96.5%

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

      1. *-lft-identity 96.5%

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

      2. associate-*l/ 96.5%

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

      3. associate-/r/ 96.5%

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

      4. *-lft-identity 96.5%

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

      5. associate-*l/ 96.5%

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

      6. rec-exp 96.5%

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

      7. distribute-rgt-in 96.5%

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

      8. rec-exp 96.5%

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

      9. rgt-mult-inverse 98.2%

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

      10. rec-exp 98.2%

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

      11. associate-*r/ 98.2%

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

      12. *-rgt-identity 98.2%

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

   7. Simplified 98.2%

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

      1. add-exp-log 98.2%

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

      2. log-rec 98.2%

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

      3. log1p-udef 98.2%

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

      4. div-exp 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. exp-neg 100.0%

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

       2. log1p-udef 100.0%

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

       3. add-exp-log 100.0%

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

       4. add-sqr-sqrt 100.0%

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

       5. associate-/r* 100.0%

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

   11. Applied egg-rr 98.7%

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

       1. *-inverses 98.7%

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

   13. Simplified 98.7%

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

   if -0.97999999999999998 < b

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0 74.6%

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

   4. Taylor expanded in b around 0 43.7%

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

      1. +-commutative 43.7%

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

   6. Simplified 43.7%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.98:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.1% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.24:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.24) 1.0 (/ 1.0 (- 2.0 a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.24) {
		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 <= (-0.24d0)) 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 <= -0.24) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (2.0 - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.24:
		tmp = 1.0
	else:
		tmp = 1.0 / (2.0 - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.24)
		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 <= -0.24)
		tmp = 1.0;
	else
		tmp = 1.0 / (2.0 - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.24], 1.0, N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.23999999999999999

   1. Initial program 96.5%

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

      1. clear-num 96.5%

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

      2. inv-pow 96.5%

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in a around inf 96.5%

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

      1. *-lft-identity 96.5%

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

      2. associate-*l/ 96.5%

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

      3. associate-/r/ 96.5%

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

      4. *-lft-identity 96.5%

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

      5. associate-*l/ 96.5%

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

      6. rec-exp 96.5%

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

      7. distribute-rgt-in 96.5%

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

      8. rec-exp 96.5%

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

      9. rgt-mult-inverse 98.2%

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

      10. rec-exp 98.2%

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

      11. associate-*r/ 98.2%

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

      12. *-rgt-identity 98.2%

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

   7. Simplified 98.2%

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

      1. add-exp-log 98.2%

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

      2. log-rec 98.2%

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

      3. log1p-udef 98.2%

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

      4. div-exp 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. exp-neg 100.0%

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

       2. log1p-udef 100.0%

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

       3. add-exp-log 100.0%

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

       4. add-sqr-sqrt 100.0%

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

       5. associate-/r* 100.0%

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

   11. Applied egg-rr 98.7%

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

       1. *-inverses 98.7%

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

   13. Simplified 98.7%

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

   if -0.23999999999999999 < b

   1. Initial program 99.0%

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

      1. clear-num 99.0%

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

      2. inv-pow 99.0%

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in a around inf 99.0%

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

      1. *-lft-identity 99.0%

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

      2. associate-*l/ 99.0%

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

      3. associate-/r/ 99.0%

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

      4. *-lft-identity 99.0%

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

      5. associate-*l/ 99.0%

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

      6. rec-exp 99.0%

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

      7. distribute-rgt-in 64.3%

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

      8. rec-exp 64.3%

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

      9. rgt-mult-inverse 99.5%

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

      10. rec-exp 99.5%

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

      11. associate-*r/ 99.5%

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

      12. *-rgt-identity 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in b around 0 78.1%

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

      1. rec-exp 78.1%

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

   10. Simplified 78.1%

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

   11. Taylor expanded in a around 0 43.7%

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

       1. neg-mul-1 43.7%

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

       2. unsub-neg 43.7%

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

   13. Simplified 43.7%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.24:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.2% accurate, 50.7× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.0) {
		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 <= (-1.0d0)) 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 <= -1.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1

   1. Initial program 96.5%

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

      1. clear-num 96.5%

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

      2. inv-pow 96.5%

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in a around inf 96.5%

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

      1. *-lft-identity 96.5%

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

      2. associate-*l/ 96.5%

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

      3. associate-/r/ 96.5%

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

      4. *-lft-identity 96.5%

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

      5. associate-*l/ 96.5%

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

      6. rec-exp 96.5%

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

      7. distribute-rgt-in 96.5%

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

      8. rec-exp 96.5%

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

      9. rgt-mult-inverse 98.2%

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

      10. rec-exp 98.2%

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

      11. associate-*r/ 98.2%

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

      12. *-rgt-identity 98.2%

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

   7. Simplified 98.2%

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

      1. add-exp-log 98.2%

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

      2. log-rec 98.2%

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

      3. log1p-udef 98.2%

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

      4. div-exp 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. exp-neg 100.0%

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

       2. log1p-udef 100.0%

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

       3. add-exp-log 100.0%

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

       4. add-sqr-sqrt 100.0%

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

       5. associate-/r* 100.0%

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

   11. Applied egg-rr 98.7%

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

       1. *-inverses 98.7%

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

   13. Simplified 98.7%

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

   if -1 < b

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0 74.6%

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

   4. Taylor expanded in b around 0 42.8%

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

4. Final simplification 55.2%

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

Alternative 10: 39.1% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in a around 0 80.3%

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

4. Taylor expanded in b around 0 37.4%

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

5. Final simplification 37.4%

   \[\leadsto 0.5 \]
6. 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 2024021 
(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))))