Quotient of sum of exps

Percentage Accurate: 99.0% → 98.9%

Time: 8.2s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 0 \lor \neg \left(e^{b} \leq 1.0000002\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= (exp b) 0.0) (not (<= (exp b) 1.0000002)))
   (/ 1.0 (+ (exp b) 1.0))
   (/ 1.0 (+ 1.0 (exp (- a))))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(b) <= 0.0) || !(exp(b) <= 1.0000002)) {
		tmp = 1.0 / (exp(b) + 1.0);
	} else {
		tmp = 1.0 / (1.0 + exp(-a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((exp(b) <= 0.0d0) .or. (.not. (exp(b) <= 1.0000002d0))) then
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    else
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.exp(b) <= 0.0) || !(Math.exp(b) <= 1.0000002)) {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.exp(b) <= 0.0) or not (math.exp(b) <= 1.0000002):
		tmp = 1.0 / (math.exp(b) + 1.0)
	else:
		tmp = 1.0 / (1.0 + math.exp(-a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((exp(b) <= 0.0) || !(exp(b) <= 1.0000002))
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((exp(b) <= 0.0) || ~((exp(b) <= 1.0000002)))
		tmp = 1.0 / (exp(b) + 1.0);
	else
		tmp = 1.0 / (1.0 + exp(-a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[N[Exp[b], $MachinePrecision], 0.0], N[Not[LessEqual[N[Exp[b], $MachinePrecision], 1.0000002]], $MachinePrecision]], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 0.0 or 1.00000019999999989 < (exp.f64 b)

   1. Initial program 99.1%

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

   2. Taylor expanded in a around 0 100.0%

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

   if 0.0 < (exp.f64 b) < 1.00000019999999989

   1. Initial program 99.2%

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

      1. add-exp-log 99.2%

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

      2. div-exp 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in b around 0 99.3%

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

      1. log1p-def 99.3%

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

   6. Simplified 99.3%

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

      1. exp-diff 99.2%

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

      2. log1p-udef 99.2%

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

      3. add-exp-log 99.2%

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

      4. frac-2neg 99.2%

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

      5. distribute-neg-in 99.2%

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

      6. metadata-eval 99.2%

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

   8. Applied egg-rr 99.2%

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

      1. neg-mul-1 99.2%

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

      2. associate-/l* 99.2%

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

      3. unsub-neg 99.2%

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

      4. div-sub 69.1%

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

      5. sub-neg 69.1%

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

      6. *-inverses 100.0%

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

      7. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in a around inf 100.0%

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

       1. rec-exp 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0 \lor \neg \left(e^{b} \leq 1.0000002\right):\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 (exp (- a (log (+ (exp a) (exp b))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. add-exp-log 99.2%

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

   2. div-exp 99.2%

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

3. Applied egg-rr 99.2%

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

4. Final simplification 99.2%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

2. Final simplification 99.2%

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

Alternative 4: 66.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (exp a) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(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 (exp(a) <= 0.0d0) then
        tmp = exp(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 (Math.exp(a) <= 0.0) {
		tmp = Math.exp(a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = math.exp(a)
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[Exp[a], $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 98.5%

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

      1. add-exp-log 98.5%

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

      2. div-exp 98.5%

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

   3. Applied egg-rr 98.5%

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

   4. Taylor expanded in a around inf 98.5%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in b around 0 55.8%

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

   5. Taylor expanded in a around 0 54.9%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 5: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -38000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -38000.0) (exp a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -38000.0) {
		tmp = exp(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 <= (-38000.0d0)) then
        tmp = exp(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 <= -38000.0) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -38000.0:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -38000.0)
		tmp = exp(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 <= -38000.0)
		tmp = exp(a);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -38000.0], N[Exp[a], $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -38000

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. div-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around inf 100.0%

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

   if -38000 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 97.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -38000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 6: 39.1% 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 99.2%

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

   1. clear-num 99.2%

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

   2. associate-/r/ 99.2%

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

3. Applied egg-rr 99.2%

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

4. Taylor expanded in b around 0 66.6%

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

5. Taylor expanded in a around 0 41.5%

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

6. Final simplification 41.5%

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

Alternative 7: 39.8% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. add-exp-log 99.2%

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

   2. div-exp 99.2%

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

3. Applied egg-rr 99.2%

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

4. Taylor expanded in b around 0 66.7%

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

   1. log1p-def 66.7%

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

6. Simplified 66.7%

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

   1. exp-diff 66.6%

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

   2. log1p-udef 66.6%

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

   3. add-exp-log 66.6%

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

   4. frac-2neg 66.6%

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

   5. distribute-neg-in 66.6%

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

   6. metadata-eval 66.6%

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

8. Applied egg-rr 66.6%

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

   1. neg-mul-1 66.6%

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

   2. associate-/l* 66.6%

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

   3. unsub-neg 66.6%

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

   4. div-sub 41.6%

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

   5. sub-neg 41.6%

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

   6. *-inverses 67.0%

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

   7. metadata-eval 67.0%

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

10. Simplified 67.0%

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

11. Taylor expanded in a around 0 41.8%

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

12. Final simplification 41.8%

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

Alternative 8: 38.9% 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 99.2%

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

2. Taylor expanded in a around 0 82.5%

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

3. Taylor expanded in b around 0 40.6%

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

4. Final simplification 40.6%

   \[\leadsto 0.5 \]

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 2023338 
(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))))