Quotient of sum of exps

Percentage Accurate: 98.9% → 100.0%

Time: 5.0s

Alternatives: 8

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, 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. add-exp-log 98.8%

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

   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. Step-by-step derivation

   1. exp-diff 98.8%

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

   2. add-exp-log 98.8%

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

   3. clear-num 98.8%

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

   4. frac-2neg 98.8%

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

   5. metadata-eval 98.8%

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

5. Applied egg-rr 98.8%

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

   1. rem-exp-log 98.8%

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

   2. exp-diff 99.2%

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

   3. sub-neg 99.2%

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

   4. +-commutative 99.2%

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

   5. prod-exp 98.8%

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

   6. rem-exp-log 98.8%

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

   7. distribute-lft-in 71.5%

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

   8. rec-exp 71.5%

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

   9. lft-mult-inverse 99.2%

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

   10. prod-exp 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 0 \lor \neg \left(e^{b} \leq 1.0000000005\right):\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= (exp b) 0.0) (not (<= (exp b) 1.0000000005)))
   (/ 1.0 (+ 1.0 (exp b)))
   (/ 1.0 (+ 1.0 (exp (- a))))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(b) <= 0.0) || !(exp(b) <= 1.0000000005)) {
		tmp = 1.0 / (1.0 + exp(b));
	} 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) <= 1.0000000005d0))) then
        tmp = 1.0d0 / (1.0d0 + exp(b))
    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) <= 1.0000000005)) {
		tmp = 1.0 / (1.0 + Math.exp(b));
	} 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) <= 1.0000000005):
		tmp = 1.0 / (1.0 + math.exp(b))
	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) <= 1.0000000005))
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	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) <= 1.0000000005)))
		tmp = 1.0 / (1.0 + exp(b));
	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], 1.0000000005]], $MachinePrecision]], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $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 1.0000000005 < (exp.f64 b)

   1. Initial program 98.2%

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

   2. Taylor expanded in a around 0 100.0%

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

   if 0.0 < (exp.f64 b) < 1.0000000005

   1. Initial program 99.3%

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

      1. add-exp-log 99.3%

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

      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. Step-by-step derivation

      1. exp-diff 99.3%

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

      2. add-exp-log 99.3%

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

      3. clear-num 99.3%

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

      4. frac-2neg 99.3%

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

      5. metadata-eval 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. rem-exp-log 99.3%

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

      2. exp-diff 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. prod-exp 99.3%

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

      6. rem-exp-log 99.3%

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

      7. distribute-lft-in 65.5%

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

      8. rec-exp 65.5%

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

      9. lft-mult-inverse 100.0%

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

      10. prod-exp 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 0 \lor \neg \left(e^{b} \leq 1.0000000005\right):\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \end{array} \]

Alternative 3: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-8) (exp a) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-8) {
		tmp = 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 (exp(a) <= 2d-8) then
        tmp = 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 (Math.exp(a) <= 2e-8) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 2e-8:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-8)
		tmp = exp(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 (exp(a) <= 2e-8)
		tmp = exp(a);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 2e-8], N[Exp[a], $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2e-8

   1. Initial program 98.6%

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

      1. add-exp-log 98.6%

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

      2. div-exp 98.6%

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

   3. Applied egg-rr 98.6%

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

   4. Taylor expanded in a around inf 98.1%

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

   if 2e-8 < (exp.f64 a)

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 96.7%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 4: 65.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-5) (exp a) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-5) {
		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) <= 5d-5) 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) <= 5e-5) {
		tmp = Math.exp(a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-5:
		tmp = math.exp(a)
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-5)
		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) <= 5e-5)
		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], 5e-5], 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) < 5.00000000000000024e-5

   1. Initial program 98.6%

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

      1. add-exp-log 98.6%

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

      2. div-exp 98.6%

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

   3. Applied egg-rr 98.6%

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

   4. Taylor expanded in a around inf 96.8%

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

   if 5.00000000000000024e-5 < (exp.f64 a)

   1. Initial program 98.9%

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

   2. Taylor expanded in b around 0 56.3%

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

   3. Taylor expanded in a around 0 55.2%

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

      1. *-commutative 55.2%

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

   5. Simplified 55.2%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 5: 79.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -15.5:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -15.5) (exp a) (/ (exp a) 2.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -15.5) {
		tmp = exp(a);
	} else {
		tmp = exp(a) / 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 <= (-15.5d0)) then
        tmp = exp(a)
    else
        tmp = exp(a) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -15.5) {
		tmp = Math.exp(a);
	} else {
		tmp = Math.exp(a) / 2.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -15.5:
		tmp = math.exp(a)
	else:
		tmp = math.exp(a) / 2.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -15.5)
		tmp = exp(a);
	else
		tmp = Float64(exp(a) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -15.5)
		tmp = exp(a);
	else
		tmp = exp(a) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -15.5], N[Exp[a], $MachinePrecision], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -15.5

   1. Initial program 97.3%

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

      1. add-exp-log 97.3%

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

      2. div-exp 97.4%

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

   3. Applied egg-rr 97.4%

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

   4. Taylor expanded in a around inf 97.0%

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

   if -15.5 < b

   1. Initial program 99.0%

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

   2. Taylor expanded in b around 0 76.4%

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

   3. Taylor expanded in a around 0 73.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -15.5:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \end{array} \]

Alternative 6: 39.8% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

2. Taylor expanded in b around 0 68.0%

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

3. Taylor expanded in a around 0 40.1%

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

   1. *-commutative 40.1%

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

5. Simplified 40.1%

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

6. Final simplification 40.1%

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

Alternative 7: 40.5% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. add-exp-log 98.8%

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

   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. Step-by-step derivation

   1. exp-diff 98.8%

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

   2. add-exp-log 98.8%

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

   3. clear-num 98.8%

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

   4. frac-2neg 98.8%

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

   5. metadata-eval 98.8%

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

5. Applied egg-rr 98.8%

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

   1. rem-exp-log 98.8%

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

   2. exp-diff 99.2%

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

   3. sub-neg 99.2%

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

   4. +-commutative 99.2%

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

   5. prod-exp 98.8%

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

   6. rem-exp-log 98.8%

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

   7. distribute-lft-in 71.5%

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

   8. rec-exp 71.5%

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

   9. lft-mult-inverse 99.2%

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

   10. prod-exp 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in b around 0 68.4%

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

9. Taylor expanded in a around 0 40.5%

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

    1. mul-1-neg 40.5%

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

    2. unsub-neg 40.5%

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

11. Simplified 40.5%

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

12. Final simplification 40.5%

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

Alternative 8: 39.7% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.8%

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

2. Taylor expanded in a around 0 78.7%

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

3. Taylor expanded in b around 0 39.3%

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

4. Final simplification 39.3%

   \[\leadsto 0.5 \]

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 2023185 
(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))))