Quotient of sum of exps

Percentage Accurate: 99.1% → 100.0%

Time: 5.1s

Alternatives: 7

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.1% 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 99.2%

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

   1. add-cube-cbrt 98.7%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}} \cdot \sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right) \cdot \sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}} \]

   2. pow3 98.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right)}^{3}} \]

   3. pow-to-exp 98.6%

      \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right) \cdot 3}} \]

   4. pow1/3 98.6%

      \[\leadsto e^{\log \color{blue}{\left({\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{0.3333333333333333}\right)} \cdot 3} \]

   5. log-pow 99.2%

      \[\leadsto e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)} \cdot 3} \]

   6. log-div 99.2%

      \[\leadsto e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log \left(e^{a}\right) - \log \left(e^{a} + e^{b}\right)\right)}\right) \cdot 3} \]

   7. add-log-exp 99.6%

      \[\leadsto e^{\left(0.3333333333333333 \cdot \left(\color{blue}{a} - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 3} \]

3. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{e^{\left(0.3333333333333333 \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 3}} \]

4. Taylor expanded in a around inf 99.6%

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

   1. sub-neg 99.6%

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

   2. +-commutative 99.6%

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

   3. neg-sub0 99.6%

      \[\leadsto e^{\color{blue}{\left(0 - \log \left(e^{a} + e^{b}\right)\right)} + a} \]

   4. associate--r- 99.6%

      \[\leadsto e^{\color{blue}{0 - \left(\log \left(e^{a} + e^{b}\right) - a\right)}} \]

   5. exp-diff 99.6%

      \[\leadsto \color{blue}{\frac{e^{0}}{e^{\log \left(e^{a} + e^{b}\right) - a}}} \]

   6. 1-exp 99.6%

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

   7. sub-neg 99.6%

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

   8. +-commutative 99.6%

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

   9. exp-sum 99.2%

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

   10. rem-exp-log 99.2%

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

   11. distribute-rgt-in 74.6%

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

   12. rec-exp 74.6%

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

   13. rgt-mult-inverse 99.6%

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

   14. prod-exp 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 98.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (exp a) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		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) <= 0.0d0) 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) <= 0.0) {
		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) <= 0.0:
		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) <= 0.0)
		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) <= 0.0)
		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], 0.0], 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) < 0.0

   1. Initial program 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}} \cdot \sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right) \cdot \sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}} \]

      2. pow3 100.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right)}^{3}} \]

      3. pow-to-exp 100.0%

         \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right) \cdot 3}} \]

      4. pow1/3 100.0%

         \[\leadsto e^{\log \color{blue}{\left({\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{0.3333333333333333}\right)} \cdot 3} \]

      5. log-pow 100.0%

         \[\leadsto e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)} \cdot 3} \]

      6. log-div 100.0%

         \[\leadsto e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log \left(e^{a}\right) - \log \left(e^{a} + e^{b}\right)\right)}\right) \cdot 3} \]

      7. add-log-exp 100.0%

         \[\leadsto e^{\left(0.3333333333333333 \cdot \left(\color{blue}{a} - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 3} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{\left(0.3333333333333333 \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 3}} \]

   4. Taylor expanded in a around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 98.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 3: 65.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99995:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.99995) (exp a) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.99995) {
		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) <= 0.99995d0) 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) <= 0.99995) {
		tmp = Math.exp(a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.99995:
		tmp = math.exp(a)
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.99995)
		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) <= 0.99995)
		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], 0.99995], 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) < 0.999950000000000006

   1. Initial program 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}} \cdot \sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right) \cdot \sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}} \]

      2. pow3 100.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right)}^{3}} \]

      3. pow-to-exp 99.9%

         \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right) \cdot 3}} \]

      4. pow1/3 100.0%

         \[\leadsto e^{\log \color{blue}{\left({\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{0.3333333333333333}\right)} \cdot 3} \]

      5. log-pow 100.0%

         \[\leadsto e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)} \cdot 3} \]

      6. log-div 100.0%

         \[\leadsto e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log \left(e^{a}\right) - \log \left(e^{a} + e^{b}\right)\right)}\right) \cdot 3} \]

      7. add-log-exp 100.0%

         \[\leadsto e^{\left(0.3333333333333333 \cdot \left(\color{blue}{a} - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 3} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{\left(0.3333333333333333 \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 3}} \]

   4. Taylor expanded in a around inf 93.3%

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

   if 0.999950000000000006 < (exp.f64 a)

   1. Initial program 98.9%

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

   2. Taylor expanded in b around 0 51.7%

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

   3. Taylor expanded in a around 0 51.7%

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

      1. *-commutative 51.7%

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

   5. Simplified 51.7%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99995:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 4: 79.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.98:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.98) 1.0 (/ (exp a) 2.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.98) {
		tmp = 1.0;
	} 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 <= (-0.98d0)) then
        tmp = 1.0d0
    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 <= -0.98) {
		tmp = 1.0;
	} else {
		tmp = Math.exp(a) / 2.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.98:
		tmp = 1.0
	else:
		tmp = math.exp(a) / 2.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.98)
		tmp = 1.0;
	else
		tmp = Float64(exp(a) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.98)
		tmp = 1.0;
	else
		tmp = exp(a) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.98], 1.0, N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.97999999999999998

   1. Initial program 97.7%

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

      1. add-cube-cbrt 97.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}} \cdot \sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right) \cdot \sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}} \]

      2. pow3 97.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right)}^{3}} \]

      3. pow-to-exp 97.7%

         \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right) \cdot 3}} \]

      4. pow1/3 97.7%

         \[\leadsto e^{\log \color{blue}{\left({\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{0.3333333333333333}\right)} \cdot 3} \]

      5. log-pow 97.7%

         \[\leadsto e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)} \cdot 3} \]

      6. log-div 97.7%

         \[\leadsto e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log \left(e^{a}\right) - \log \left(e^{a} + e^{b}\right)\right)}\right) \cdot 3} \]

      7. add-log-exp 97.8%

         \[\leadsto e^{\left(0.3333333333333333 \cdot \left(\color{blue}{a} - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 3} \]

   3. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{e^{\left(0.3333333333333333 \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 3}} \]

   4. Taylor expanded in a around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. log1p-def 100.0%

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

   6. Simplified 100.0%

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

   8. Applied egg-rr 96.6%

      \[\leadsto e^{-\color{blue}{0}} \]

   if -0.97999999999999998 < b

   1. Initial program 99.5%

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

   2. Taylor expanded in b around 0 72.5%

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

   3. Taylor expanded in a around 0 71.3%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.98:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \end{array} \]

Alternative 5: 39.9% 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. Taylor expanded in b around 0 63.0%

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

3. Taylor expanded in a around 0 38.5%

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

   1. *-commutative 38.5%

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

5. Simplified 38.5%

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

6. Final simplification 38.5%

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

Alternative 6: 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 99.2%

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

   1. add-cube-cbrt 98.7%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}} \cdot \sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right) \cdot \sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}} \]

   2. pow3 98.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right)}^{3}} \]

   3. pow-to-exp 98.6%

      \[\leadsto \color{blue}{e^{\log \left(\sqrt[3]{\frac{e^{a}}{e^{a} + e^{b}}}\right) \cdot 3}} \]

   4. pow1/3 98.6%

      \[\leadsto e^{\log \color{blue}{\left({\left(\frac{e^{a}}{e^{a} + e^{b}}\right)}^{0.3333333333333333}\right)} \cdot 3} \]

   5. log-pow 99.2%

      \[\leadsto e^{\color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{e^{a}}{e^{a} + e^{b}}\right)\right)} \cdot 3} \]

   6. log-div 99.2%

      \[\leadsto e^{\left(0.3333333333333333 \cdot \color{blue}{\left(\log \left(e^{a}\right) - \log \left(e^{a} + e^{b}\right)\right)}\right) \cdot 3} \]

   7. add-log-exp 99.6%

      \[\leadsto e^{\left(0.3333333333333333 \cdot \left(\color{blue}{a} - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 3} \]

3. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{e^{\left(0.3333333333333333 \cdot \left(a - \log \left(e^{a} + e^{b}\right)\right)\right) \cdot 3}} \]

4. Taylor expanded in a around inf 99.6%

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

   1. sub-neg 99.6%

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

   2. +-commutative 99.6%

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

   3. neg-sub0 99.6%

      \[\leadsto e^{\color{blue}{\left(0 - \log \left(e^{a} + e^{b}\right)\right)} + a} \]

   4. associate--r- 99.6%

      \[\leadsto e^{\color{blue}{0 - \left(\log \left(e^{a} + e^{b}\right) - a\right)}} \]

   5. exp-diff 99.6%

      \[\leadsto \color{blue}{\frac{e^{0}}{e^{\log \left(e^{a} + e^{b}\right) - a}}} \]

   6. 1-exp 99.6%

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

   7. sub-neg 99.6%

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

   8. +-commutative 99.6%

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

   9. exp-sum 99.2%

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

   10. rem-exp-log 99.2%

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

   11. distribute-rgt-in 74.6%

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

   12. rec-exp 74.6%

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

   13. rgt-mult-inverse 99.6%

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

   14. prod-exp 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in b around 0 63.4%

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

8. Taylor expanded in a around 0 39.1%

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

   1. mul-1-neg 39.1%

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

   2. unsub-neg 39.1%

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

10. Simplified 39.1%

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

11. Final simplification 39.1%

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

Alternative 7: 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 99.2%

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

2. Taylor expanded in a around 0 81.9%

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

3. Taylor expanded in b around 0 38.2%

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

4. Final simplification 38.2%

   \[\leadsto 0.5 \]

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 2023318 
(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))))