Quotient of sum of exps

Percentage Accurate: 99.0% → 100.0%

Time: 5.8s

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: 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 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-/l* 98.8%

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

   3. remove-double-div 98.8%

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

   4. exp-neg 98.8%

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

   5. associate-/r/ 98.8%

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

   6. /-rgt-identity 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-in 74.6%

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

   9. exp-neg 74.6%

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

   10. rgt-mult-inverse 99.6%

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

   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: 98.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a) / b;
	} 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) / b
    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) / b;
	} 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) / b
	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) / b);
	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) / b;
	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] / b), $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 98.4%

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

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

      3. remove-double-div 98.4%

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

      4. exp-neg 98.4%

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

      5. associate-/r/ 98.4%

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

      6. /-rgt-identity 98.4%

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

      7. *-commutative 98.4%

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

      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 98.4%

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

      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 73.0%

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

      1. distribute-rgt1-in 100.0%

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

      2. rec-exp 100.0%

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

      3. associate-*r/ 100.0%

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

      4. +-commutative 100.0%

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

      5. *-rgt-identity 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.9%

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

      1. *-lft-identity 98.9%

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

      2. associate-/l* 98.9%

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

      3. remove-double-div 98.9%

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

      4. exp-neg 98.9%

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

      5. associate-/r/ 98.9%

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

      6. /-rgt-identity 98.9%

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

      7. *-commutative 98.9%

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

      8. distribute-rgt-in 98.9%

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

      9. exp-neg 98.9%

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

      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 97.7%

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

4. Final simplification 98.2%

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

Alternative 3: 65.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.8:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.8) (/ (exp a) b) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.8) {
		tmp = exp(a) / b;
	} 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 (exp(a) <= 0.8d0) then
        tmp = exp(a) / b
    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 (Math.exp(a) <= 0.8) {
		tmp = Math.exp(a) / b;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.8:
		tmp = math.exp(a) / b
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.8)
		tmp = Float64(exp(a) / b);
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.8)
		tmp = exp(a) / b;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.8], N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.80000000000000004

   1. Initial program 98.4%

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

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

      3. remove-double-div 98.4%

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

      4. exp-neg 98.4%

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

      5. associate-/r/ 98.4%

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

      6. /-rgt-identity 98.4%

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

      7. *-commutative 98.4%

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

      8. distribute-rgt-in 1.6%

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

      9. exp-neg 1.6%

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

      10. rgt-mult-inverse 98.4%

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

      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 71.9%

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

      1. distribute-rgt1-in 98.5%

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

      2. rec-exp 98.5%

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

      3. associate-*r/ 98.5%

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

      4. +-commutative 98.5%

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

      5. *-rgt-identity 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in b around inf 98.5%

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

   if 0.80000000000000004 < (exp.f64 a)

   1. Initial program 98.9%

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

      1. *-lft-identity 98.9%

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

      2. associate-/l* 98.9%

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

      3. remove-double-div 98.9%

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

      4. exp-neg 98.9%

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

      5. associate-/r/ 98.9%

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

      6. /-rgt-identity 98.9%

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

      7. *-commutative 98.9%

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

      8. distribute-rgt-in 98.9%

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

      9. exp-neg 98.9%

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

      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 b around 0 59.9%

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

   5. Taylor expanded in a around 0 58.0%

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

      1. *-commutative 58.0%

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

   7. Simplified 58.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.8:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 4: 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 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-/l* 98.8%

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

   3. remove-double-div 98.8%

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

   4. exp-neg 98.8%

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

   5. associate-/r/ 98.8%

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

   6. /-rgt-identity 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-in 74.6%

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

   9. exp-neg 74.6%

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

   10. rgt-mult-inverse 99.6%

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

   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 5: 60.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -16500000:\\ \;\;\;\;1 + e^{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + a \cdot \left(-1 - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -16500000.0)
   (+ 1.0 (exp b))
   (/ 1.0 (+ (+ b 2.0) (* a (- -1.0 b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -16500000.0) {
		tmp = 1.0 + exp(b);
	} else {
		tmp = 1.0 / ((b + 2.0) + (a * (-1.0 - b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-16500000.0d0)) then
        tmp = 1.0d0 + exp(b)
    else
        tmp = 1.0d0 / ((b + 2.0d0) + (a * ((-1.0d0) - b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -16500000.0) {
		tmp = 1.0 + Math.exp(b);
	} else {
		tmp = 1.0 / ((b + 2.0) + (a * (-1.0 - b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -16500000.0:
		tmp = 1.0 + math.exp(b)
	else:
		tmp = 1.0 / ((b + 2.0) + (a * (-1.0 - b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -16500000.0)
		tmp = Float64(1.0 + exp(b));
	else
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) + Float64(a * Float64(-1.0 - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -16500000.0)
		tmp = 1.0 + exp(b);
	else
		tmp = 1.0 / ((b + 2.0) + (a * (-1.0 - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -16500000.0], N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] + N[(a * N[(-1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.65e7

   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 99.9%

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

      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%

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

   6. Taylor expanded in a around 0 100.0%

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

      1. log1p-def 100.0%

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

   8. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. sqr-neg 100.0%

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

      4. sqrt-unprod 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. log1p-udef 100.0%

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

      7. rem-exp-log 100.0%

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

      8. +-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   if -1.65e7 < b

   1. Initial program 98.6%

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

      1. *-lft-identity 98.6%

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

      2. associate-/l* 98.6%

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

      3. remove-double-div 98.6%

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

      4. exp-neg 98.6%

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

      5. associate-/r/ 98.6%

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

      6. /-rgt-identity 98.6%

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

      7. *-commutative 98.6%

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

      8. distribute-rgt-in 69.5%

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

      9. exp-neg 69.5%

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

      10. rgt-mult-inverse 99.5%

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

      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 72.5%

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

      1. distribute-rgt1-in 80.5%

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

      2. rec-exp 80.5%

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

      3. associate-*r/ 80.5%

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

      4. +-commutative 80.5%

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

      5. *-rgt-identity 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in a around 0 54.7%

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

      1. associate-+r+ 54.7%

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

      2. +-commutative 54.7%

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

      3. mul-1-neg 54.7%

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

      4. distribute-rgt-neg-in 54.7%

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

      5. distribute-neg-in 54.7%

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

      6. metadata-eval 54.7%

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

      7. unsub-neg 54.7%

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

   9. Simplified 54.7%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -16500000:\\ \;\;\;\;1 + e^{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + a \cdot \left(-1 - b\right)}\\ \end{array} \]

Alternative 6: 45.2% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-151}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + a \cdot \left(-1 - b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1e-151) (+ 0.5 (* a 0.25)) (/ 1.0 (+ (+ b 2.0) (* a (- -1.0 b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1e-151) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0 / ((b + 2.0) + (a * (-1.0 - b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1d-151)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 1.0d0 / ((b + 2.0d0) + (a * ((-1.0d0) - b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1e-151) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0 / ((b + 2.0) + (a * (-1.0 - b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1e-151:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 1.0 / ((b + 2.0) + (a * (-1.0 - b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1e-151)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) + Float64(a * Float64(-1.0 - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1e-151)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 1.0 / ((b + 2.0) + (a * (-1.0 - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1e-151], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] + N[(a * N[(-1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.9999999999999994e-152

   1. Initial program 98.6%

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

      1. *-lft-identity 98.6%

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

      2. associate-/l* 98.6%

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

      3. remove-double-div 98.6%

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

      4. exp-neg 98.6%

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

      5. associate-/r/ 98.6%

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

      6. /-rgt-identity 98.6%

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

      7. *-commutative 98.6%

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

      8. distribute-rgt-in 93.4%

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

      9. exp-neg 93.3%

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

      10. rgt-mult-inverse 98.6%

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

      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 53.0%

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

   5. Taylor expanded in a around 0 45.1%

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

      1. *-commutative 45.1%

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

   7. Simplified 45.1%

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

   if -9.9999999999999994e-152 < b

   1. Initial program 98.9%

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

      1. *-lft-identity 98.9%

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

      2. associate-/l* 98.9%

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

      3. remove-double-div 98.9%

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

      4. exp-neg 98.9%

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

      5. associate-/r/ 98.9%

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

      6. /-rgt-identity 98.9%

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

      7. *-commutative 98.9%

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

      8. distribute-rgt-in 66.7%

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

      9. exp-neg 66.7%

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

      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. Taylor expanded in b around 0 71.0%

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

      1. distribute-rgt1-in 77.7%

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

      2. rec-exp 77.7%

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

      3. associate-*r/ 77.7%

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

      4. +-commutative 77.7%

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

      5. *-rgt-identity 77.7%

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

   6. Simplified 77.7%

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

   7. Taylor expanded in a around 0 50.4%

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

      1. associate-+r+ 50.4%

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

      2. +-commutative 50.4%

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

      3. mul-1-neg 50.4%

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

      4. distribute-rgt-neg-in 50.4%

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

      5. distribute-neg-in 50.4%

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

      6. metadata-eval 50.4%

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

      7. unsub-neg 50.4%

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

   9. Simplified 50.4%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-151}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + a \cdot \left(-1 - b\right)}\\ \end{array} \]

Alternative 7: 44.3% accurate, 33.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 75000000000000:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(b + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 75000000000000.0) (+ 0.5 (* a 0.25)) (/ -1.0 (* a (+ b 1.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 75000000000000.0) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = -1.0 / (a * (b + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 75000000000000.0d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = (-1.0d0) / (a * (b + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 75000000000000.0) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = -1.0 / (a * (b + 1.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 75000000000000.0:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = -1.0 / (a * (b + 1.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 75000000000000.0)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(-1.0 / Float64(a * Float64(b + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 75000000000000.0)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = -1.0 / (a * (b + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 75000000000000.0], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(a * N[(b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.5e13

   1. Initial program 98.4%

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

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

      3. remove-double-div 98.4%

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

      4. exp-neg 98.4%

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

      5. associate-/r/ 98.4%

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

      6. /-rgt-identity 98.4%

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

      7. *-commutative 98.4%

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

      8. distribute-rgt-in 78.7%

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

      9. exp-neg 78.7%

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

      10. rgt-mult-inverse 99.4%

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

      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 79.2%

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

   5. Taylor expanded in a around 0 57.6%

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

      1. *-commutative 57.6%

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

   7. Simplified 57.6%

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

   if 7.5e13 < 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 61.9%

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

      9. exp-neg 61.9%

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

      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 b around 0 41.8%

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

      1. distribute-rgt1-in 41.8%

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

      2. rec-exp 41.8%

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

      3. associate-*r/ 41.8%

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

      4. +-commutative 41.8%

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

      5. *-rgt-identity 41.8%

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

   6. Simplified 41.8%

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

   7. Taylor expanded in a around 0 19.4%

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

      1. associate-+r+ 19.4%

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

      2. +-commutative 19.4%

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

      3. mul-1-neg 19.4%

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

      4. distribute-rgt-neg-in 19.4%

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

      5. distribute-neg-in 19.4%

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

      6. metadata-eval 19.4%

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

      7. unsub-neg 19.4%

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

   9. Simplified 19.4%

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

   10. Taylor expanded in a around inf 17.9%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 75000000000000:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(b + 1\right)}\\ \end{array} \]

Alternative 8: 44.6% accurate, 33.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{b}}{1 - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.4) (+ 0.5 (* a 0.25)) (/ (/ 1.0 b) (- 1.0 a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.4) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = (1.0 / b) / (1.0 - 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 = 0.5d0 + (a * 0.25d0)
    else
        tmp = (1.0d0 / b) / (1.0d0 - a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.4) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = (1.0 / b) / (1.0 - a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.4:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = (1.0 / b) / (1.0 - a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(Float64(1.0 / b) / Float64(1.0 - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.4)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = (1.0 / b) / (1.0 - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.4], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / b), $MachinePrecision] / N[(1.0 - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 98.4%

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

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

      3. remove-double-div 98.4%

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

      4. exp-neg 98.4%

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

      5. associate-/r/ 98.4%

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

      6. /-rgt-identity 98.4%

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

      7. *-commutative 98.4%

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

      8. distribute-rgt-in 78.8%

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

      9. exp-neg 78.8%

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

      10. rgt-mult-inverse 99.4%

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

      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 80.3%

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

   5. Taylor expanded in a around 0 58.7%

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

      1. *-commutative 58.7%

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

   7. Simplified 58.7%

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

   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 62.7%

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

      9. exp-neg 62.7%

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

      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. Taylor expanded in b around 0 41.0%

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

      1. distribute-rgt1-in 41.0%

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

      2. rec-exp 41.0%

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

      3. associate-*r/ 41.0%

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

      4. +-commutative 41.0%

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

      5. *-rgt-identity 41.0%

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

   6. Simplified 41.0%

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

   7. Taylor expanded in a around 0 18.4%

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

      1. associate-+r+ 18.4%

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

      2. +-commutative 18.4%

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

      3. mul-1-neg 18.4%

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

      4. distribute-rgt-neg-in 18.4%

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

      5. distribute-neg-in 18.4%

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

      6. metadata-eval 18.4%

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

      7. unsub-neg 18.4%

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

   9. Simplified 18.4%

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

   10. Taylor expanded in b around inf 18.4%

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

       1. associate-/r* 18.4%

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

       2. mul-1-neg 18.4%

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

       3. unsub-neg 18.4%

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

   12. Simplified 18.4%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{b}}{1 - a}\\ \end{array} \]

Alternative 9: 39.9% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-/l* 98.8%

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

   3. remove-double-div 98.8%

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

   4. exp-neg 98.8%

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

   5. associate-/r/ 98.8%

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

   6. /-rgt-identity 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-in 74.6%

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

   9. exp-neg 74.6%

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

   10. rgt-mult-inverse 99.6%

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

   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 69.5%

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

5. Taylor expanded in a around 0 44.1%

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

   1. *-commutative 44.1%

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

7. Simplified 44.1%

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

8. Final simplification 44.1%

   \[\leadsto 0.5 + a \cdot 0.25 \]

Alternative 10: 40.6% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-/l* 98.8%

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

   3. remove-double-div 98.8%

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

   4. exp-neg 98.8%

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

   5. associate-/r/ 98.8%

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

   6. /-rgt-identity 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-in 74.6%

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

   9. exp-neg 74.6%

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

   10. rgt-mult-inverse 99.6%

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

   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 69.5%

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

5. Taylor expanded in a around 0 44.3%

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

   1. neg-mul-1 44.3%

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

   2. unsub-neg 44.3%

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

7. Simplified 44.3%

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

8. Final simplification 44.3%

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

Alternative 11: 39.7% 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 98.8%

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

   1. *-lft-identity 98.8%

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

   2. associate-/l* 98.8%

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

   3. remove-double-div 98.8%

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

   4. exp-neg 98.8%

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

   5. associate-/r/ 98.8%

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

   6. /-rgt-identity 98.8%

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

   7. *-commutative 98.8%

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

   8. distribute-rgt-in 74.6%

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

   9. exp-neg 74.6%

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

   10. rgt-mult-inverse 99.6%

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

   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 83.5%

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

5. Taylor expanded in b around 0 43.7%

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

6. Final simplification 43.7%

   \[\leadsto 0.5 \]

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 2023309 
(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))))