Quotient of sum of exps

Percentage Accurate: 99.0% → 99.4%

Time: 7.9s

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: 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: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return 1.0 / (1.0 + (exp(b) / exp(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) / exp(a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-/l* 100.0%

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

   3. remove-double-div 99.9%

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

   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 72.2%

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

   9. exp-neg 72.2%

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

   10. rgt-mult-inverse 100.0%

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

   11. prod-exp 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. Step-by-step derivation

   1. exp-diff 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 98.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -70:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -70.0) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -70.0) {
		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 (a <= (-70.0d0)) 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 (a <= -70.0) {
		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 a <= -70.0:
		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 (a <= -70.0)
		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 (a <= -70.0)
		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[a, -70.0], 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 a < -70

   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 2.7%

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

      9. exp-neg 2.7%

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

      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 0 100.0%

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

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   if -70 < a

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/l* 100.0%

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

      3. remove-double-div 99.9%

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

      4. exp-neg 99.9%

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

      5. associate-/r/ 99.9%

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

      6. /-rgt-identity 99.9%

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

      7. *-commutative 99.9%

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

      8. distribute-rgt-in 99.9%

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

      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 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 a around 0 99.0%

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

      1. +-commutative 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -70:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 3: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -70:\\ \;\;\;\;\frac{-e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -70.0) (/ (- (exp a)) a) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -70.0) {
		tmp = -exp(a) / a;
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-70.0d0)) then
        tmp = -exp(a) / a
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -70.0) {
		tmp = -Math.exp(a) / a;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -70.0:
		tmp = -math.exp(a) / a
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -70.0)
		tmp = Float64(Float64(-exp(a)) / a);
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -70.0)
		tmp = -exp(a) / a;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -70.0], N[((-N[Exp[a], $MachinePrecision]) / a), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -70

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in a around 0 97.3%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + a}} \]
   4. Step-by-step derivation

      1. +-commutative 97.3%

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

   5. Simplified 97.3%

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

   6. Taylor expanded in a around inf 97.3%

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

      1. frac-2neg 97.3%

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

      2. div-inv 97.3%

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

      3. add-sqr-sqrt 97.3%

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

      4. sqrt-unprod 97.3%

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

      5. sqr-neg 97.3%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 97.7%

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

   8. Applied egg-rr 97.7%

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

      1. associate-*r/ 97.7%

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

      2. *-rgt-identity 97.7%

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

   10. Simplified 97.7%

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

   if -70 < a

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/l* 100.0%

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

      3. remove-double-div 99.9%

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

      4. exp-neg 99.9%

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

      5. associate-/r/ 99.9%

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

      6. /-rgt-identity 99.9%

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

      7. *-commutative 99.9%

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

      8. distribute-rgt-in 99.9%

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

      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 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 a around 0 99.0%

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

      1. +-commutative 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -70:\\ \;\;\;\;\frac{-e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 4: 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]

Derivation

1. Initial program 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-/l* 100.0%

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

   3. remove-double-div 99.9%

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

   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 72.2%

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

   9. exp-neg 72.2%

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

   10. rgt-mult-inverse 100.0%

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

   11. prod-exp 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. Final simplification 99.9%

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

Alternative 5: 66.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -66:\\ \;\;\;\;\frac{-e^{a}}{a}\\ \mathbf{elif}\;a \leq -5.3 \cdot 10^{-20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -66.0)
   (/ (- (exp a)) a)
   (if (<= a -5.3e-20) 1.0 (+ 0.5 (* a 0.25)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -66.0) {
		tmp = -exp(a) / a;
	} else if (a <= -5.3e-20) {
		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 (a <= (-66.0d0)) then
        tmp = -exp(a) / a
    else if (a <= (-5.3d-20)) 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 (a <= -66.0) {
		tmp = -Math.exp(a) / a;
	} else if (a <= -5.3e-20) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -66.0:
		tmp = -math.exp(a) / a
	elif a <= -5.3e-20:
		tmp = 1.0
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -66.0)
		tmp = Float64(Float64(-exp(a)) / a);
	elseif (a <= -5.3e-20)
		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 (a <= -66.0)
		tmp = -exp(a) / a;
	elseif (a <= -5.3e-20)
		tmp = 1.0;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -66.0], N[((-N[Exp[a], $MachinePrecision]) / a), $MachinePrecision], If[LessEqual[a, -5.3e-20], 1.0, N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -66

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in a around 0 97.3%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + a}} \]
   4. Step-by-step derivation

      1. +-commutative 97.3%

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

   5. Simplified 97.3%

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

   6. Taylor expanded in a around inf 97.3%

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

      1. frac-2neg 97.3%

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

      2. div-inv 97.3%

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

      3. add-sqr-sqrt 97.3%

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

      4. sqrt-unprod 97.3%

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

      5. sqr-neg 97.3%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 97.7%

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

   8. Applied egg-rr 97.7%

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

      1. associate-*r/ 97.7%

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

      2. *-rgt-identity 97.7%

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

   10. Simplified 97.7%

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

   if -66 < a < -5.3000000000000002e-20

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

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

      2. associate-/l* 99.8%

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

      3. remove-double-div 99.3%

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

      4. exp-neg 99.5%

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

      5. associate-/r/ 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-rgt-in 99.5%

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

      9. exp-neg 99.8%

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

      10. rgt-mult-inverse 99.8%

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

      11. prod-exp 99.8%

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

      12. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

      1. exp-diff 99.8%

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

      2. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

   7. Applied egg-rr 68.6%

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

      1. *-inverses 69.3%

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

   9. Simplified 69.3%

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

   if -5.3000000000000002e-20 < a

   1. Initial program 99.9%

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

   2. Taylor expanded in b around 0 56.5%

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

   3. Taylor expanded in a around 0 56.5%

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

      1. *-commutative 56.5%

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

   5. Simplified 56.5%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -66:\\ \;\;\;\;\frac{-e^{a}}{a}\\ \mathbf{elif}\;a \leq -5.3 \cdot 10^{-20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 6: 66.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -750:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{elif}\;a \leq -3.1 \cdot 10^{-20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -750.0) (/ (exp a) a) (if (<= a -3.1e-20) 1.0 (+ 0.5 (* a 0.25)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -750.0) {
		tmp = exp(a) / a;
	} else if (a <= -3.1e-20) {
		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 (a <= (-750.0d0)) then
        tmp = exp(a) / a
    else if (a <= (-3.1d-20)) 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 (a <= -750.0) {
		tmp = Math.exp(a) / a;
	} else if (a <= -3.1e-20) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -750.0:
		tmp = math.exp(a) / a
	elif a <= -3.1e-20:
		tmp = 1.0
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -750.0)
		tmp = Float64(exp(a) / a);
	elseif (a <= -3.1e-20)
		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 (a <= -750.0)
		tmp = exp(a) / a;
	elseif (a <= -3.1e-20)
		tmp = 1.0;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -750.0], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], If[LessEqual[a, -3.1e-20], 1.0, N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -750

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in a around 0 100.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + a}} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around inf 100.0%

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

   if -750 < a < -3.1e-20

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/l* 99.7%

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

      3. remove-double-div 99.3%

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

      4. exp-neg 99.4%

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

      5. associate-/r/ 99.4%

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

      6. /-rgt-identity 99.4%

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

      7. *-commutative 99.4%

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

      8. distribute-rgt-in 99.4%

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

      9. exp-neg 99.7%

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

      10. rgt-mult-inverse 99.7%

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

      11. prod-exp 99.7%

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

      12. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

      1. exp-diff 99.7%

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

      2. clear-num 99.7%

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

   5. Applied egg-rr 99.7%

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

   7. Applied egg-rr 57.3%

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

      1. *-inverses 57.8%

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

   9. Simplified 57.8%

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

   if -3.1e-20 < a

   1. Initial program 99.9%

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

   2. Taylor expanded in b around 0 56.5%

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

   3. Taylor expanded in a around 0 56.5%

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

      1. *-commutative 56.5%

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

   5. Simplified 56.5%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -750:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{elif}\;a \leq -3.1 \cdot 10^{-20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 7: 54.4% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \end{array} \]

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (b * -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 <= (-2.0d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 + (b * (-0.25d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -2

   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 99.9%

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

      4. exp-neg 99.9%

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

      5. associate-/r/ 99.9%

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

      6. /-rgt-identity 99.9%

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

      7. *-commutative 99.9%

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

      8. distribute-rgt-in 99.9%

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

      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. exp-diff 100.0%

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

      2. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   9. Simplified 100.0%

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

   if -2 < b

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/l* 99.9%

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

      3. remove-double-div 99.9%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 65.2%

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

      9. exp-neg 65.2%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 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 a around 0 76.3%

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

      1. +-commutative 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in b around 0 46.0%

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

      1. *-commutative 46.0%

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

   9. Simplified 46.0%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \]

Alternative 8: 55.1% accurate, 43.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1

   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 99.9%

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

      4. exp-neg 99.9%

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

      5. associate-/r/ 99.9%

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

      6. /-rgt-identity 99.9%

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

      7. *-commutative 99.9%

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

      8. distribute-rgt-in 99.9%

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

      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. exp-diff 100.0%

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

      2. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   9. Simplified 100.0%

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

   if -1 < b

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/l* 99.9%

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

      3. remove-double-div 99.9%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 65.2%

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

      9. exp-neg 65.2%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 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 a around 0 76.3%

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

      1. +-commutative 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in b around 0 46.6%

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

      1. +-commutative 46.6%

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

   9. Simplified 46.6%

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

4. Final simplification 57.4%

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

Alternative 9: 54.2% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. remove-double-div 99.9%

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

      4. exp-neg 99.9%

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

      5. associate-/r/ 99.9%

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

      6. /-rgt-identity 99.9%

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

      7. *-commutative 99.9%

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

      8. distribute-rgt-in 99.9%

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

      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. exp-diff 100.0%

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

      2. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   9. Simplified 100.0%

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

   if -1.1000000000000001 < b

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/l* 99.9%

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

      3. remove-double-div 99.9%

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

      4. exp-neg 100.0%

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

      5. associate-/r/ 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-in 65.2%

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

      9. exp-neg 65.2%

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

      10. rgt-mult-inverse 100.0%

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

      11. prod-exp 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 a around 0 76.3%

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

      1. +-commutative 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in b around 0 45.0%

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

4. Final simplification 56.2%

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

Alternative 10: 39.6% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-/l* 100.0%

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

   3. remove-double-div 99.9%

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

   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 72.2%

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

   9. exp-neg 72.2%

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

   10. rgt-mult-inverse 100.0%

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

   11. prod-exp 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 a around 0 81.1%

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

   1. +-commutative 81.1%

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

6. Simplified 81.1%

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

7. Taylor expanded in b around 0 39.7%

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

8. Final simplification 39.7%

   \[\leadsto 0.5 \]

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 2023306 
(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))))