Quotient of sum of exps

Percentage Accurate: 98.8% → 100.0%

Time: 5.8s

Alternatives: 10

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 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 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. *-lft-identity 98.8%

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

   4. metadata-eval 98.8%

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

   5. times-frac 98.8%

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

   6. *-commutative 98.8%

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

   7. times-frac 98.8%

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

   8. exp-neg 98.8%

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

   9. /-rgt-identity 98.8%

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

   10. distribute-lft-in 68.7%

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

   11. exp-neg 68.7%

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

   12. lft-mult-inverse 99.6%

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

   13. prod-exp 100.0%

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

   14. +-commutative 100.0%

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

   15. 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%

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

Alternative 2: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6e-8) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6e-8) {
		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 <= (-6d-8)) 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 <= -6e-8) {
		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 <= -6e-8:
		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 <= -6e-8)
		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 <= -6e-8)
		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, -6e-8], 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 < -5.99999999999999946e-8

   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. *-lft-identity 100.0%

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

      4. metadata-eval 100.0%

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

      5. times-frac 100.0%

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

      6. *-commutative 100.0%

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

      7. times-frac 100.0%

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

      8. exp-neg 99.9%

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

      9. /-rgt-identity 99.9%

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

      10. distribute-lft-in 4.9%

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

      11. exp-neg 4.9%

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

      12. lft-mult-inverse 100.0%

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

      13. prod-exp 100.0%

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

      14. +-commutative 100.0%

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

      15. 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 99.0%

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

   if -5.99999999999999946e-8 < a

   1. Initial program 98.3%

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

      1. *-lft-identity 98.3%

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

      2. associate-/l* 98.3%

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

      3. *-lft-identity 98.3%

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

      4. metadata-eval 98.3%

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

      5. times-frac 98.3%

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

      6. *-commutative 98.3%

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

      7. times-frac 98.3%

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

      8. exp-neg 98.3%

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

      9. /-rgt-identity 98.3%

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

      10. distribute-lft-in 98.3%

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

      11. exp-neg 98.3%

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

      12. lft-mult-inverse 99.4%

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

      13. prod-exp 100.0%

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

      14. +-commutative 100.0%

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

      15. 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 98.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 3: 98.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -720:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -720.0) (/ (exp a) a) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -720.0) {
		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 <= (-720.0d0)) 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 <= -720.0) {
		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 <= -720.0:
		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 <= -720.0)
		tmp = 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 <= -720.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -720.0], 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 < -720

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   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 -720 < a

   1. Initial program 98.3%

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

      1. *-lft-identity 98.3%

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

      2. associate-/l* 98.3%

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

      3. *-lft-identity 98.3%

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

      4. metadata-eval 98.3%

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

      5. times-frac 98.3%

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

      6. *-commutative 98.3%

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

      7. times-frac 98.3%

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

      8. exp-neg 98.3%

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

      9. /-rgt-identity 98.3%

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

      10. distribute-lft-in 98.3%

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

      11. exp-neg 98.3%

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

      12. lft-mult-inverse 99.4%

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

      13. prod-exp 100.0%

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

      14. +-commutative 100.0%

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

      15. 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 97.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -720:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 4: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.52:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.52) (/ (exp a) (+ a 2.0)) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.52) {
		tmp = exp(a) / (a + 2.0);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.52d0)) then
        tmp = exp(a) / (a + 2.0d0)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.52) {
		tmp = Math.exp(a) / (a + 2.0);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.52:
		tmp = math.exp(a) / (a + 2.0)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.52)
		tmp = Float64(exp(a) / Float64(a + 2.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.52)
		tmp = exp(a) / (a + 2.0);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.52], N[(N[Exp[a], $MachinePrecision] / N[(a + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.52000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in a around 0 99.0%

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

      1. +-commutative 99.0%

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

   5. Simplified 99.0%

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

   if -0.52000000000000002 < a

   1. Initial program 98.3%

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

      1. *-lft-identity 98.3%

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

      2. associate-/l* 98.3%

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

      3. *-lft-identity 98.3%

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

      4. metadata-eval 98.3%

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

      5. times-frac 98.3%

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

      6. *-commutative 98.3%

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

      7. times-frac 98.3%

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

      8. exp-neg 98.3%

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

      9. /-rgt-identity 98.3%

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

      10. distribute-lft-in 98.3%

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

      11. exp-neg 98.3%

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

      12. lft-mult-inverse 99.4%

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

      13. prod-exp 100.0%

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

      14. +-commutative 100.0%

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

      15. 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 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.52:\\ \;\;\;\;\frac{e^{a}}{a + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 5: 65.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.2:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.2) (/ (exp a) a) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.2) {
		tmp = exp(a) / a;
	} 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 <= (-2.2d0)) then
        tmp = exp(a) / a
    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 <= -2.2) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.2:
		tmp = math.exp(a) / a
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.2)
		tmp = Float64(exp(a) / a);
	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 <= -2.2)
		tmp = exp(a) / a;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.2], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.2000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   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 -2.2000000000000002 < a

   1. Initial program 98.3%

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

      1. *-lft-identity 98.3%

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

      2. associate-/l* 98.3%

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

      3. *-lft-identity 98.3%

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

      4. metadata-eval 98.3%

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

      5. times-frac 98.3%

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

      6. *-commutative 98.3%

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

      7. times-frac 98.3%

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

      8. exp-neg 98.3%

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

      9. /-rgt-identity 98.3%

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

      10. distribute-lft-in 98.3%

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

      11. exp-neg 98.3%

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

      12. lft-mult-inverse 99.4%

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

      13. prod-exp 100.0%

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

      14. +-commutative 100.0%

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

      15. 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 57.1%

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

   5. Taylor expanded in a around 0 55.2%

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

      1. *-commutative 55.2%

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

   7. Simplified 55.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.2:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 6: 53.8% accurate, 43.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := If[LessEqual[b, -1.0], 1.0, N[(0.5 + N[(a * 0.25), $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. *-lft-identity 100.0%

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

      4. metadata-eval 100.0%

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

      5. times-frac 100.0%

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

      6. *-commutative 100.0%

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

      7. times-frac 100.0%

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

      8. exp-neg 100.0%

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

      9. /-rgt-identity 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. exp-neg 100.0%

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

      12. lft-mult-inverse 100.0%

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

      13. prod-exp 100.0%

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

      14. +-commutative 100.0%

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

      15. 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%

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

      2. rec-exp 100.0%

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

      3. log1p-def 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. Step-by-step derivation

      1. *-un-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   9. Simplified 100.0%

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

   if -1 < 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. *-lft-identity 98.6%

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

      4. metadata-eval 98.6%

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

      5. times-frac 98.6%

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

      6. *-commutative 98.6%

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

      7. times-frac 98.6%

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

      8. exp-neg 98.6%

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

      9. /-rgt-identity 98.6%

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

      10. distribute-lft-in 64.1%

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

      11. exp-neg 64.1%

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

      12. lft-mult-inverse 99.5%

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

      13. prod-exp 100.0%

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

      14. +-commutative 100.0%

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

      15. 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 77.6%

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

   5. Taylor expanded in a around 0 42.3%

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

      1. *-commutative 42.3%

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

   7. Simplified 42.3%

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

4. Final simplification 49.8%

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

Alternative 7: 53.9% 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. *-lft-identity 100.0%

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

      4. metadata-eval 100.0%

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

      5. times-frac 100.0%

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

      6. *-commutative 100.0%

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

      7. times-frac 100.0%

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

      8. exp-neg 100.0%

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

      9. /-rgt-identity 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. exp-neg 100.0%

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

      12. lft-mult-inverse 100.0%

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

      13. prod-exp 100.0%

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

      14. +-commutative 100.0%

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

      15. 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%

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

      2. rec-exp 100.0%

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

      3. log1p-def 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. Step-by-step derivation

      1. *-un-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   9. Simplified 100.0%

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

   if -2 < 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. *-lft-identity 98.6%

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

      4. metadata-eval 98.6%

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

      5. times-frac 98.6%

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

      6. *-commutative 98.6%

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

      7. times-frac 98.6%

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

      8. exp-neg 98.6%

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

      9. /-rgt-identity 98.6%

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

      10. distribute-lft-in 64.1%

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

      11. exp-neg 64.1%

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

      12. lft-mult-inverse 99.5%

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

      13. prod-exp 100.0%

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

      14. +-commutative 100.0%

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

      15. 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 76.5%

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

   5. Taylor expanded in b around 0 43.2%

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

      1. *-commutative 43.2%

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

   7. Simplified 43.2%

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

4. Final simplification 50.5%

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

Alternative 8: 54.7% 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. *-lft-identity 100.0%

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

      4. metadata-eval 100.0%

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

      5. times-frac 100.0%

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

      6. *-commutative 100.0%

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

      7. times-frac 100.0%

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

      8. exp-neg 100.0%

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

      9. /-rgt-identity 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. exp-neg 100.0%

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

      12. lft-mult-inverse 100.0%

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

      13. prod-exp 100.0%

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

      14. +-commutative 100.0%

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

      15. 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%

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

      2. rec-exp 100.0%

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

      3. log1p-def 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. Step-by-step derivation

      1. *-un-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   9. Simplified 100.0%

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

   if -1 < 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. *-lft-identity 98.6%

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

      4. metadata-eval 98.6%

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

      5. times-frac 98.6%

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

      6. *-commutative 98.6%

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

      7. times-frac 98.6%

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

      8. exp-neg 98.6%

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

      9. /-rgt-identity 98.6%

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

      10. distribute-lft-in 64.1%

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

      11. exp-neg 64.1%

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

      12. lft-mult-inverse 99.5%

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

      13. prod-exp 100.0%

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

      14. +-commutative 100.0%

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

      15. 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 76.5%

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

   5. Taylor expanded in b around 0 43.8%

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

      1. +-commutative 43.8%

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

   7. Simplified 43.8%

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

4. Final simplification 51.1%

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

Alternative 9: 53.6% accurate, 99.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1

   1. Initial program 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. *-lft-identity 100.0%

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

      4. metadata-eval 100.0%

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

      5. times-frac 100.0%

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

      6. *-commutative 100.0%

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

      7. times-frac 100.0%

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

      8. exp-neg 100.0%

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

      9. /-rgt-identity 100.0%

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

      10. distribute-lft-in 100.0%

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

      11. exp-neg 100.0%

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

      12. lft-mult-inverse 100.0%

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

      13. prod-exp 100.0%

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

      14. +-commutative 100.0%

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

      15. 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%

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

      2. rec-exp 100.0%

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

      3. log1p-def 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. Step-by-step derivation

      1. *-un-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-inverses 100.0%

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

   9. Simplified 100.0%

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

   if -1 < 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. *-lft-identity 98.6%

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

      4. metadata-eval 98.6%

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

      5. times-frac 98.6%

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

      6. *-commutative 98.6%

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

      7. times-frac 98.6%

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

      8. exp-neg 98.6%

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

      9. /-rgt-identity 98.6%

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

      10. distribute-lft-in 64.1%

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

      11. exp-neg 64.1%

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

      12. lft-mult-inverse 99.5%

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

      13. prod-exp 100.0%

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

      14. +-commutative 100.0%

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

      15. 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 76.5%

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

   5. Taylor expanded in b around 0 42.0%

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

4. Final simplification 49.5%

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

Alternative 10: 39.0% 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. *-lft-identity 98.8%

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

   4. metadata-eval 98.8%

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

   5. times-frac 98.8%

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

   6. *-commutative 98.8%

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

   7. times-frac 98.8%

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

   8. exp-neg 98.8%

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

   9. /-rgt-identity 98.8%

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

   10. distribute-lft-in 68.7%

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

   11. exp-neg 68.7%

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

   12. lft-mult-inverse 99.6%

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

   13. prod-exp 100.0%

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

   14. +-commutative 100.0%

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

   15. 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 79.6%

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

5. Taylor expanded in b around 0 39.0%

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

6. Final simplification 39.0%

   \[\leadsto 0.5 \]

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 2023305 
(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))))