Quotient of sum of exps

Percentage Accurate: 98.9% → 100.0%

Time: 6.8s

Alternatives: 12

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. *-lft-identity 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-/r/ 98.4%

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

   4. remove-double-neg 98.4%

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

   5. unsub-neg 98.4%

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

   6. div-sub 67.5%

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

   7. *-lft-identity 67.5%

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

   8. associate-*l/ 67.5%

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

   9. lft-mult-inverse 99.2%

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

   10. sub-neg 99.2%

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

   11. distribute-frac-neg 99.2%

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

   12. remove-double-neg 99.2%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

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

   1. div-exp 99.2%

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

   2. +-commutative 99.2%

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

   3. metadata-eval 99.2%

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

   4. sub-neg 99.2%

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

   5. add-exp-log 99.2%

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

   6. rec-exp 99.2%

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

   7. sub-neg 99.2%

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

   8. metadata-eval 99.2%

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

   9. +-commutative 99.2%

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

   10. log1p-define 99.2%

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

   11. div-exp 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 98.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.75 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{e^{-a} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.75e-13) (/ 1.0 (- (exp (- a)) -1.0)) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.75e-13) {
		tmp = 1.0 / (exp(-a) - -1.0);
	} else {
		tmp = 1.0 / (exp(b) - -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.75d-13)) then
        tmp = 1.0d0 / (exp(-a) - (-1.0d0))
    else
        tmp = 1.0d0 / (exp(b) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1.75e-13) {
		tmp = 1.0 / (Math.exp(-a) - -1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1.75e-13:
		tmp = 1.0 / (math.exp(-a) - -1.0)
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.75e-13)
		tmp = Float64(1.0 / Float64(exp(Float64(-a)) - -1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.75e-13)
		tmp = 1.0 / (exp(-a) - -1.0);
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.75e-13], N[(1.0 / N[(N[Exp[(-a)], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.7500000000000001e-13

   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%

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

      3. associate-/r/ 98.8%

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

      4. +-commutative 98.8%

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

      5. remove-double-neg 98.8%

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

      6. sub-neg 98.8%

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

      7. div-sub 6.9%

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

      8. neg-mul-1 6.9%

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

      9. *-commutative 6.9%

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

      10. associate-*r/ 6.9%

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

      11. metadata-eval 6.9%

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

      12. distribute-neg-frac 6.9%

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

      13. exp-neg 6.9%

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

      14. distribute-rgt-neg-out 6.9%

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

      15. exp-neg 6.9%

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

      16. rgt-mult-inverse 98.8%

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

      17. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in b around 0 98.7%

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

      1. rec-exp 98.6%

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

   7. Simplified 98.6%

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

   if -1.7500000000000001e-13 < a

   1. Initial program 98.2%

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

      1. *-lft-identity 98.2%

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

      2. associate-*l/ 98.2%

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

      3. associate-/r/ 98.2%

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

      4. +-commutative 98.2%

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

      5. remove-double-neg 98.2%

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

      6. sub-neg 98.2%

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

      7. div-sub 98.2%

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

      8. neg-mul-1 98.2%

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

      9. *-commutative 98.2%

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

      10. associate-*r/ 98.2%

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

      11. metadata-eval 98.2%

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

      12. distribute-neg-frac 98.2%

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

      13. exp-neg 98.2%

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

      14. distribute-rgt-neg-out 98.2%

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

      15. exp-neg 98.2%

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

      16. rgt-mult-inverse 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in a around 0 98.7%

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

Alternative 3: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3800:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3800.0) (/ (exp a) a) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3800.0) {
		tmp = exp(a) / a;
	} else {
		tmp = 1.0 / (exp(b) - -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-3800.0d0)) then
        tmp = exp(a) / a
    else
        tmp = 1.0d0 / (exp(b) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3800.0) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3800.0:
		tmp = math.exp(a) / a
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3800.0)
		tmp = Float64(exp(a) / a);
	else
		tmp = Float64(1.0 / Float64(exp(b) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3800.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3800.0], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3800

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in a around inf 100.0%

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

   if -3800 < 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%

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

      3. associate-/r/ 98.3%

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

      4. +-commutative 98.3%

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

      5. remove-double-neg 98.3%

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

      6. sub-neg 98.3%

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

      7. div-sub 97.7%

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

      8. neg-mul-1 97.7%

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

      9. *-commutative 97.7%

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

      10. associate-*r/ 97.7%

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

      11. metadata-eval 97.7%

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

      12. distribute-neg-frac 97.7%

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

      13. exp-neg 97.7%

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

      14. distribute-rgt-neg-out 97.7%

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

      15. exp-neg 97.7%

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

      16. rgt-mult-inverse 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in a around 0 97.2%

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

Alternative 4: 75.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -720:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -720.0)
   (/ (exp a) a)
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -720.0) {
		tmp = exp(a) / a;
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	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 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    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 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -720.0:
		tmp = math.exp(a) / a
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	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(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	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 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	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[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -720

   1. Initial program 98.8%

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

   3. Taylor expanded in b around 0 100.0%

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

   4. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. 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%

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

      3. associate-/r/ 98.3%

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

      4. +-commutative 98.3%

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

      5. remove-double-neg 98.3%

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

      6. sub-neg 98.3%

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

      7. div-sub 98.2%

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

      8. neg-mul-1 98.2%

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

      9. *-commutative 98.2%

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

      10. associate-*r/ 98.2%

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

      11. metadata-eval 98.2%

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

      12. distribute-neg-frac 98.2%

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

      13. exp-neg 98.2%

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

      14. distribute-rgt-neg-out 98.2%

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

      15. exp-neg 98.2%

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

      16. rgt-mult-inverse 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in a around 0 97.2%

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

   6. Taylor expanded in b around 0 68.4%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + 0.16666666666666666 \cdot b\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 68.4%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + \color{blue}{b \cdot 0.16666666666666666}\right)\right)} \]

   8. Simplified 68.4%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. *-lft-identity 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-/r/ 98.4%

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

   4. remove-double-neg 98.4%

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

   5. unsub-neg 98.4%

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

   6. div-sub 67.5%

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

   7. *-lft-identity 67.5%

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

   8. associate-*l/ 67.5%

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

   9. lft-mult-inverse 99.2%

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

   10. sub-neg 99.2%

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

   11. distribute-frac-neg 99.2%

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

   12. remove-double-neg 99.2%

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

   13. div-exp 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 6: 70.8% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.36 \cdot 10^{+70}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.36e+70)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.36e+70) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.36d+70) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.36e+70) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.36e+70:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.36e+70)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.36e+70)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.36e+70], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35999999999999995e70

   1. Initial program 98.5%

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

      1. *-lft-identity 98.5%

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

      2. associate-*l/ 98.4%

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

      3. associate-/r/ 98.4%

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

      4. +-commutative 98.4%

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

      5. remove-double-neg 98.4%

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

      6. sub-neg 98.4%

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

      7. div-sub 69.0%

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

      8. neg-mul-1 69.0%

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

      9. *-commutative 69.0%

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

      10. associate-*r/ 69.0%

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

      11. metadata-eval 69.0%

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

      12. distribute-neg-frac 69.0%

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

      13. exp-neg 69.0%

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

      14. distribute-rgt-neg-out 69.0%

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

      15. exp-neg 69.0%

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

      16. rgt-mult-inverse 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in b around 0 78.4%

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

      1. rec-exp 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in a around 0 66.9%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(0.5 + -0.16666666666666666 \cdot a\right) - 1\right)}} \]

   if 1.35999999999999995e70 < b

   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%

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

      3. associate-/r/ 98.3%

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

      4. +-commutative 98.3%

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

      5. remove-double-neg 98.3%

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

      6. sub-neg 98.3%

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

      7. div-sub 62.7%

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

      8. neg-mul-1 62.7%

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

      9. *-commutative 62.7%

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

      10. associate-*r/ 62.7%

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

      11. metadata-eval 62.7%

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

      12. distribute-neg-frac 62.7%

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

      13. exp-neg 62.7%

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

      14. distribute-rgt-neg-out 62.7%

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

      15. exp-neg 62.7%

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

      16. rgt-mult-inverse 98.3%

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

      17. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 92.3%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + 0.16666666666666666 \cdot b\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 92.3%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + \color{blue}{b \cdot 0.16666666666666666}\right)\right)} \]

   8. Simplified 92.3%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.36 \cdot 10^{+70}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7e+147)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a (+ 0.5 (* a -0.16666666666666666)))))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7e+147) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 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 <= 7d+147) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * (0.5d0 + (a * (-0.16666666666666666d0)))))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 7e+147) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7e+147:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7e+147)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * Float64(0.5 + Float64(a * -0.16666666666666666)))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7e+147)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * (0.5 + (a * -0.16666666666666666))))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7e+147], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * N[(0.5 + N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.99999999999999949e147

   1. Initial program 98.5%

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

      1. *-lft-identity 98.5%

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

      2. associate-*l/ 98.5%

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

      3. associate-/r/ 98.5%

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

      4. +-commutative 98.5%

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

      5. remove-double-neg 98.5%

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

      6. sub-neg 98.5%

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

      7. div-sub 67.9%

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

      8. neg-mul-1 67.9%

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

      9. *-commutative 67.9%

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

      10. associate-*r/ 67.9%

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

      11. metadata-eval 67.9%

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

      12. distribute-neg-frac 67.9%

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

      13. exp-neg 67.9%

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

      14. distribute-rgt-neg-out 67.9%

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

      15. exp-neg 67.9%

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

      16. rgt-mult-inverse 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in b around 0 76.9%

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

      1. rec-exp 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in a around 0 64.2%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(a \cdot \left(0.5 + -0.16666666666666666 \cdot a\right) - 1\right)}} \]

   if 6.99999999999999949e147 < b

   1. Initial program 97.9%

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

      1. *-lft-identity 97.9%

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

      2. associate-*l/ 97.9%

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

      3. associate-/r/ 97.9%

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

      4. +-commutative 97.9%

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

      5. remove-double-neg 97.9%

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

      6. sub-neg 97.9%

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

      7. div-sub 66.0%

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

      8. neg-mul-1 66.0%

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

      9. *-commutative 66.0%

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

      10. associate-*r/ 66.0%

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

      11. metadata-eval 66.0%

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

      12. distribute-neg-frac 66.0%

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

      13. exp-neg 66.0%

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

      14. distribute-rgt-neg-out 66.0%

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

      15. exp-neg 66.0%

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

      16. rgt-mult-inverse 97.9%

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

      17. metadata-eval 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 98.1%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 98.1%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + \color{blue}{b \cdot 0.5}\right)} \]

   8. Simplified 98.1%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot \left(0.5 + a \cdot -0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.6% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7e+147)
   (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a 0.5)))))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7e+147) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 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 <= 7d+147) then
        tmp = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * 0.5d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 7e+147) {
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 7e+147:
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7e+147)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * 0.5)))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 7e+147)
		tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7e+147], N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.99999999999999949e147

   1. Initial program 98.5%

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

      1. *-lft-identity 98.5%

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

      2. associate-*l/ 98.5%

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

      3. associate-/r/ 98.5%

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

      4. +-commutative 98.5%

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

      5. remove-double-neg 98.5%

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

      6. sub-neg 98.5%

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

      7. div-sub 67.9%

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

      8. neg-mul-1 67.9%

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

      9. *-commutative 67.9%

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

      10. associate-*r/ 67.9%

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

      11. metadata-eval 67.9%

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

      12. distribute-neg-frac 67.9%

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

      13. exp-neg 67.9%

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

      14. distribute-rgt-neg-out 67.9%

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

      15. exp-neg 67.9%

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

      16. rgt-mult-inverse 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in b around 0 76.9%

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

      1. rec-exp 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in a around 0 59.9%

      \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(0.5 \cdot a - 1\right)}} \]

   if 6.99999999999999949e147 < b

   1. Initial program 97.9%

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

      1. *-lft-identity 97.9%

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

      2. associate-*l/ 97.9%

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

      3. associate-/r/ 97.9%

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

      4. +-commutative 97.9%

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

      5. remove-double-neg 97.9%

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

      6. sub-neg 97.9%

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

      7. div-sub 66.0%

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

      8. neg-mul-1 66.0%

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

      9. *-commutative 66.0%

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

      10. associate-*r/ 66.0%

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

      11. metadata-eval 66.0%

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

      12. distribute-neg-frac 66.0%

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

      13. exp-neg 66.0%

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

      14. distribute-rgt-neg-out 66.0%

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

      15. exp-neg 66.0%

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

      16. rgt-mult-inverse 97.9%

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

      17. metadata-eval 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in a around 0 100.0%

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

   6. Taylor expanded in b around 0 98.1%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + 0.5 \cdot b\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 98.1%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + \color{blue}{b \cdot 0.5}\right)} \]

   8. Simplified 98.1%

      \[\leadsto \frac{1}{\color{blue}{2 + b \cdot \left(1 + b \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.7% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 2.0 (* a (+ -1.0 (* a 0.5))))))

C

double code(double a, double b) {
	return 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 + (a * ((-1.0d0) + (a * 0.5d0))))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
}

Python

def code(a, b):
	return 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))))

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 + Float64(a * Float64(-1.0 + Float64(a * 0.5)))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 + (a * (-1.0 + (a * 0.5))));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(2.0 + N[(a * N[(-1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

   1. *-lft-identity 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-/r/ 98.4%

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

   4. +-commutative 98.4%

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

   5. remove-double-neg 98.4%

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

   6. sub-neg 98.4%

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

   7. div-sub 67.5%

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

   8. neg-mul-1 67.5%

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

   9. *-commutative 67.5%

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

   10. associate-*r/ 67.5%

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

   11. metadata-eval 67.5%

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

   12. distribute-neg-frac 67.5%

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

   13. exp-neg 67.5%

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

   14. distribute-rgt-neg-out 67.5%

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

   15. exp-neg 67.5%

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

   16. rgt-mult-inverse 99.2%

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

   17. metadata-eval 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in b around 0 69.0%

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

   1. rec-exp 69.0%

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

7. Simplified 69.0%

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

8. Taylor expanded in a around 0 52.6%

   \[\leadsto \frac{1}{\color{blue}{2 + a \cdot \left(0.5 \cdot a - 1\right)}} \]

9. Final simplification 52.6%

   \[\leadsto \frac{1}{2 + a \cdot \left(-1 + a \cdot 0.5\right)} \]
10. Add Preprocessing

Alternative 10: 40.8% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. *-lft-identity 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-/r/ 98.4%

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

   4. +-commutative 98.4%

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

   5. remove-double-neg 98.4%

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

   6. sub-neg 98.4%

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

   7. div-sub 67.5%

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

   8. neg-mul-1 67.5%

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

   9. *-commutative 67.5%

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

   10. associate-*r/ 67.5%

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

   11. metadata-eval 67.5%

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

   12. distribute-neg-frac 67.5%

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

   13. exp-neg 67.5%

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

   14. distribute-rgt-neg-out 67.5%

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

   15. exp-neg 67.5%

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

   16. rgt-mult-inverse 99.2%

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

   17. metadata-eval 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in b around 0 69.0%

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

   1. rec-exp 69.0%

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

7. Simplified 69.0%

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

8. Taylor expanded in a around 0 37.8%

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

   1. neg-mul-1 37.8%

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

   2. unsub-neg 37.8%

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

10. Simplified 37.8%

    \[\leadsto \frac{1}{\color{blue}{2 - a}} \]
11. Add Preprocessing

Alternative 11: 40.2% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. *-lft-identity 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-/r/ 98.4%

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

   4. +-commutative 98.4%

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

   5. remove-double-neg 98.4%

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

   6. sub-neg 98.4%

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

   7. div-sub 67.5%

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

   8. neg-mul-1 67.5%

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

   9. *-commutative 67.5%

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

   10. associate-*r/ 67.5%

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

   11. metadata-eval 67.5%

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

   12. distribute-neg-frac 67.5%

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

   13. exp-neg 67.5%

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

   14. distribute-rgt-neg-out 67.5%

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

   15. exp-neg 67.5%

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

   16. rgt-mult-inverse 99.2%

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

   17. metadata-eval 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in b around 0 69.0%

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

   1. rec-exp 69.0%

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

7. Simplified 69.0%

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

8. Taylor expanded in a around 0 37.3%

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

   1. *-commutative 37.3%

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

10. Simplified 37.3%

    \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
11. Add Preprocessing

Alternative 12: 40.0% 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 98.4%

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

   1. *-lft-identity 98.4%

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

   2. associate-*l/ 98.4%

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

   3. associate-/r/ 98.4%

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

   4. +-commutative 98.4%

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

   5. remove-double-neg 98.4%

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

   6. sub-neg 98.4%

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

   7. div-sub 67.5%

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

   8. neg-mul-1 67.5%

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

   9. *-commutative 67.5%

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

   10. associate-*r/ 67.5%

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

   11. metadata-eval 67.5%

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

   12. distribute-neg-frac 67.5%

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

   13. exp-neg 67.5%

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

   14. distribute-rgt-neg-out 67.5%

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

   15. exp-neg 67.5%

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

   16. rgt-mult-inverse 99.2%

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

   17. metadata-eval 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in a around 0 79.1%

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

6. Taylor expanded in b around 0 36.9%

   \[\leadsto \color{blue}{0.5} \]
7. Add Preprocessing

Developer target: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024103 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :alt
  (/ 1.0 (+ 1.0 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))