Quotient of sum of exps

Percentage Accurate: 98.9% → 99.0%

Time: 8.7s

Alternatives: 9

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: 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: 99.0% 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. Add Preprocessing
3. 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%

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

4. Applied egg-rr 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 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]

Derivation

1. Initial program 99.2%

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

3. Final simplification 99.2%

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

Alternative 3: 65.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999995:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.99999995) (/ (exp a) b) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.99999995) {
		tmp = exp(a) / b;
	} 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) <= 0.99999995d0) then
        tmp = exp(a) / b
    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) <= 0.99999995) {
		tmp = Math.exp(a) / b;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.99999995:
		tmp = math.exp(a) / b
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.99999995)
		tmp = Float64(exp(a) / b);
	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) <= 0.99999995)
		tmp = exp(a) / b;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.99999995], N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.999999949999999971

   1. Initial program 98.6%

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

   3. Taylor expanded in a around 0 98.6%

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

      1. associate-+r+ 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in b around 0 95.9%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + \left(a + b\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 95.9%

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(a + b\right) + 2}} \]

      2. +-commutative 95.9%

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(b + a\right)} + 2} \]

      3. associate-+l+ 95.9%

         \[\leadsto \frac{e^{a}}{\color{blue}{b + \left(a + 2\right)}} \]

   8. Simplified 95.9%

      \[\leadsto \frac{e^{a}}{\color{blue}{b + \left(a + 2\right)}} \]

   9. Taylor expanded in b around inf 95.9%

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

   if 0.999999949999999971 < (exp.f64 a)

   1. Initial program 99.4%

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

   3. Taylor expanded in b around 0 49.0%

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

   4. Taylor expanded in a around 0 48.4%

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

      1. *-commutative 48.4%

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

   6. Simplified 48.4%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999995:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{b} + \left(a + 1\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in a around 0 98.4%

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

   1. associate-+r+ 98.4%

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

5. Simplified 98.4%

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

6. Final simplification 98.4%

   \[\leadsto \frac{e^{a}}{e^{b} + \left(a + 1\right)} \]
7. Add Preprocessing

Alternative 5: 65.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-13) (exp a) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-13) {
		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-13) 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-13) {
		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-13:
		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-13)
		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-13)
		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-13], 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) < 4.9999999999999999e-13

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in a around inf 97.2%

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

   if 4.9999999999999999e-13 < (exp.f64 a)

   1. Initial program 99.4%

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

   3. Taylor expanded in b around 0 48.8%

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

   4. Taylor expanded in a around 0 48.1%

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

      1. *-commutative 48.1%

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

   6. Simplified 48.1%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-13}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.155:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-253}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-233}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-32}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 3.15 \cdot 10^{+63}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+276}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -0.155)
   (exp a)
   (if (<= b 4.3e-253)
     (+ 0.5 (* b -0.25))
     (if (<= b 4.8e-233)
       (exp a)
       (if (<= b 7.5e-32)
         (+ 0.5 (* a 0.25))
         (if (<= b 3.15e+63)
           (exp a)
           (if (<= b 1.15e+276)
             (* -0.020833333333333332 (pow a 3.0))
             (/ (exp a) b))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.155) {
		tmp = exp(a);
	} else if (b <= 4.3e-253) {
		tmp = 0.5 + (b * -0.25);
	} else if (b <= 4.8e-233) {
		tmp = exp(a);
	} else if (b <= 7.5e-32) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 3.15e+63) {
		tmp = exp(a);
	} else if (b <= 1.15e+276) {
		tmp = -0.020833333333333332 * pow(a, 3.0);
	} else {
		tmp = exp(a) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.155d0)) then
        tmp = exp(a)
    else if (b <= 4.3d-253) then
        tmp = 0.5d0 + (b * (-0.25d0))
    else if (b <= 4.8d-233) then
        tmp = exp(a)
    else if (b <= 7.5d-32) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= 3.15d+63) then
        tmp = exp(a)
    else if (b <= 1.15d+276) then
        tmp = (-0.020833333333333332d0) * (a ** 3.0d0)
    else
        tmp = exp(a) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.155) {
		tmp = Math.exp(a);
	} else if (b <= 4.3e-253) {
		tmp = 0.5 + (b * -0.25);
	} else if (b <= 4.8e-233) {
		tmp = Math.exp(a);
	} else if (b <= 7.5e-32) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 3.15e+63) {
		tmp = Math.exp(a);
	} else if (b <= 1.15e+276) {
		tmp = -0.020833333333333332 * Math.pow(a, 3.0);
	} else {
		tmp = Math.exp(a) / b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.155:
		tmp = math.exp(a)
	elif b <= 4.3e-253:
		tmp = 0.5 + (b * -0.25)
	elif b <= 4.8e-233:
		tmp = math.exp(a)
	elif b <= 7.5e-32:
		tmp = 0.5 + (a * 0.25)
	elif b <= 3.15e+63:
		tmp = math.exp(a)
	elif b <= 1.15e+276:
		tmp = -0.020833333333333332 * math.pow(a, 3.0)
	else:
		tmp = math.exp(a) / b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.155)
		tmp = exp(a);
	elseif (b <= 4.3e-253)
		tmp = Float64(0.5 + Float64(b * -0.25));
	elseif (b <= 4.8e-233)
		tmp = exp(a);
	elseif (b <= 7.5e-32)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= 3.15e+63)
		tmp = exp(a);
	elseif (b <= 1.15e+276)
		tmp = Float64(-0.020833333333333332 * (a ^ 3.0));
	else
		tmp = Float64(exp(a) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.155)
		tmp = exp(a);
	elseif (b <= 4.3e-253)
		tmp = 0.5 + (b * -0.25);
	elseif (b <= 4.8e-233)
		tmp = exp(a);
	elseif (b <= 7.5e-32)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= 3.15e+63)
		tmp = exp(a);
	elseif (b <= 1.15e+276)
		tmp = -0.020833333333333332 * (a ^ 3.0);
	else
		tmp = exp(a) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.155], N[Exp[a], $MachinePrecision], If[LessEqual[b, 4.3e-253], N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.8e-233], N[Exp[a], $MachinePrecision], If[LessEqual[b, 7.5e-32], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.15e+63], N[Exp[a], $MachinePrecision], If[LessEqual[b, 1.15e+276], N[(-0.020833333333333332 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -0.154999999999999999 or 4.3000000000000002e-253 < b < 4.7999999999999998e-233 or 7.49999999999999953e-32 < b < 3.1499999999999999e63

   1. Initial program 98.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in a around inf 83.9%

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

   if -0.154999999999999999 < b < 4.3000000000000002e-253

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0 70.8%

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

   4. Taylor expanded in b around 0 69.7%

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

      1. *-commutative 69.7%

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

   6. Simplified 69.7%

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

   if 4.7999999999999998e-233 < b < 7.49999999999999953e-32

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 70.6%

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

      1. *-commutative 70.6%

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

   6. Simplified 70.6%

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

   if 3.1499999999999999e63 < b < 1.15000000000000002e276

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0 21.9%

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

   4. Taylor expanded in a around 0 2.9%

      \[\leadsto \color{blue}{0.5 + \left(-0.020833333333333332 \cdot {a}^{3} + 0.25 \cdot a\right)} \]

   5. Taylor expanded in a around inf 52.7%

      \[\leadsto \color{blue}{-0.020833333333333332 \cdot {a}^{3}} \]

   if 1.15000000000000002e276 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0 100.0%

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

      1. associate-+r+ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around 0 67.3%

      \[\leadsto \frac{e^{a}}{\color{blue}{2 + \left(a + b\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 67.3%

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(a + b\right) + 2}} \]

      2. +-commutative 67.3%

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(b + a\right)} + 2} \]

      3. associate-+l+ 67.3%

         \[\leadsto \frac{e^{a}}{\color{blue}{b + \left(a + 2\right)}} \]

   8. Simplified 67.3%

      \[\leadsto \frac{e^{a}}{\color{blue}{b + \left(a + 2\right)}} \]

   9. Taylor expanded in b around inf 67.3%

      \[\leadsto \color{blue}{\frac{e^{a}}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.155:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-253}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-233}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-32}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 3.15 \cdot 10^{+63}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+276}:\\ \;\;\;\;-0.020833333333333332 \cdot {a}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1600000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1600000.0) (exp a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1600000.0) {
		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 <= (-1600000.0d0)) 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 <= -1600000.0) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1600000.0:
		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 <= -1600000.0)
		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 <= -1600000.0)
		tmp = exp(a);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1600000.0], 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 < -1.6e6

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around inf 100.0%

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

   if -1.6e6 < a

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1600000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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. Add Preprocessing

3. Taylor expanded in b around 0 61.8%

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

4. Taylor expanded in a around 0 35.8%

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

   1. *-commutative 35.8%

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

6. Simplified 35.8%

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

7. Final simplification 35.8%

   \[\leadsto 0.5 + a \cdot 0.25 \]
8. Add Preprocessing

Alternative 9: 39.8% 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. Add Preprocessing

3. Taylor expanded in a around 0 82.9%

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

4. Taylor expanded in b around 0 35.6%

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

5. Final simplification 35.6%

   \[\leadsto 0.5 \]
6. 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 2024046 
(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))))