Quotient of sum of exps

Percentage Accurate: 99.2% → 99.3%

Time: 5.7s

Alternatives: 11

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: 99.2% 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: 99.3% 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 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.99) (/ (exp a) (+ (exp a) 1.0)) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.99) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.99d0) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.99) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.99:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.99)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.99)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.99], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.98999999999999999

   1. Initial program 98.6%

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

   2. Taylor expanded in b around 0 98.7%

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

   if 0.98999999999999999 < (exp.f64 a)

   1. Initial program 99.4%

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

   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 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 3: 99.2% 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: 98.4% 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. Taylor expanded in a around 0 97.4%

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

   1. associate-+r+ 97.4%

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

   2. +-commutative 97.4%

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

4. Simplified 97.4%

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

5. Final simplification 97.4%

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

Alternative 5: 88.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.06:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{+155}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {b}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.06)
   (exp a)
   (if (<= b 1.42e+155) (/ (exp a) 2.0) (* 2.0 (pow b -2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.06) {
		tmp = exp(a);
	} else if (b <= 1.42e+155) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 2.0 * pow(b, -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 <= (-1.06d0)) then
        tmp = exp(a)
    else if (b <= 1.42d+155) then
        tmp = exp(a) / 2.0d0
    else
        tmp = 2.0d0 * (b ** (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.06) {
		tmp = Math.exp(a);
	} else if (b <= 1.42e+155) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 2.0 * Math.pow(b, -2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.06:
		tmp = math.exp(a)
	elif b <= 1.42e+155:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 2.0 * math.pow(b, -2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.06)
		tmp = exp(a);
	elseif (b <= 1.42e+155)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(2.0 * (b ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.06)
		tmp = exp(a);
	elseif (b <= 1.42e+155)
		tmp = exp(a) / 2.0;
	else
		tmp = 2.0 * (b ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.06], N[Exp[a], $MachinePrecision], If[LessEqual[b, 1.42e+155], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(2.0 * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.0600000000000001

   1. Initial program 95.5%

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

      1. add-exp-log 95.5%

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

      2. div-exp 95.6%

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

   3. Applied egg-rr 95.6%

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

   4. Taylor expanded in a around inf 90.6%

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

   if -1.0600000000000001 < b < 1.41999999999999994e155

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 97.7%

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

   3. Taylor expanded in b around 0 84.4%

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

   if 1.41999999999999994e155 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 100.0%

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

   4. Taylor expanded in b around inf 100.0%

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

      1. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{{b}^{2}}{2}}} \]

      2. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{{b}^{2}} \cdot 2} \]

      3. pow-flip 98.2%

         \[\leadsto \color{blue}{{b}^{\left(-2\right)}} \cdot 2 \]

      4. metadata-eval 98.2%

         \[\leadsto {b}^{\color{blue}{-2}} \cdot 2 \]

   6. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{{b}^{-2} \cdot 2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.06:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{+155}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {b}^{-2}\\ \end{array} \]

Alternative 6: 88.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.06:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{b}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.06)
   (exp a)
   (if (<= b 1.35e+154) (/ (exp a) 2.0) (/ 2.0 (pow b 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.06) {
		tmp = exp(a);
	} else if (b <= 1.35e+154) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 2.0 / pow(b, 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 <= (-1.06d0)) then
        tmp = exp(a)
    else if (b <= 1.35d+154) then
        tmp = exp(a) / 2.0d0
    else
        tmp = 2.0d0 / (b ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.06) {
		tmp = Math.exp(a);
	} else if (b <= 1.35e+154) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 2.0 / Math.pow(b, 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.06:
		tmp = math.exp(a)
	elif b <= 1.35e+154:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 2.0 / math.pow(b, 2.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.06)
		tmp = exp(a);
	elseif (b <= 1.35e+154)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(2.0 / (b ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.06)
		tmp = exp(a);
	elseif (b <= 1.35e+154)
		tmp = exp(a) / 2.0;
	else
		tmp = 2.0 / (b ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.06], N[Exp[a], $MachinePrecision], If[LessEqual[b, 1.35e+154], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(2.0 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.0600000000000001

   1. Initial program 95.5%

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

      1. add-exp-log 95.5%

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

      2. div-exp 95.6%

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

   3. Applied egg-rr 95.6%

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

   4. Taylor expanded in a around inf 90.6%

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

   if -1.0600000000000001 < b < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 97.7%

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

   3. Taylor expanded in b around 0 84.4%

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

   if 1.35000000000000003e154 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 100.0%

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

   4. Taylor expanded in b around inf 100.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.06:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{b}^{2}}\\ \end{array} \]

Alternative 7: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -118000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -118000.0) (exp a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -118000.0) {
		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 <= (-118000.0d0)) 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 <= -118000.0) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -118000.0:
		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 <= -118000.0)
		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 <= -118000.0)
		tmp = exp(a);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -118000.0], 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 < -118000

   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 -118000 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 97.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -118000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 8: 80.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -1.1) (exp a) (/ (exp a) 2.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		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 <= (-1.1d0)) 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 <= -1.1) {
		tmp = Math.exp(a);
	} else {
		tmp = Math.exp(a) / 2.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.1:
		tmp = math.exp(a)
	else:
		tmp = math.exp(a) / 2.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.1)
		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 <= -1.1)
		tmp = exp(a);
	else
		tmp = exp(a) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.1], N[Exp[a], $MachinePrecision], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001

   1. Initial program 95.5%

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

      1. add-exp-log 95.5%

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

      2. div-exp 95.6%

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

   3. Applied egg-rr 95.6%

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

   4. Taylor expanded in a around inf 90.6%

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

   if -1.1000000000000001 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 98.1%

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

   3. Taylor expanded in b around 0 75.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \end{array} \]

Alternative 9: 66.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-20}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.5e-20) (exp a) (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.5e-20) {
		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 (a <= (-6.5d-20)) 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 (a <= -6.5e-20) {
		tmp = Math.exp(a);
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.5e-20:
		tmp = math.exp(a)
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.5e-20)
		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 (a <= -6.5e-20)
		tmp = exp(a);
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.5e-20], N[Exp[a], $MachinePrecision], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.50000000000000032e-20

   1. Initial program 98.7%

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

      1. add-exp-log 98.7%

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

      2. div-exp 98.7%

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in a around inf 96.4%

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

   if -6.50000000000000032e-20 < a

   1. Initial program 99.4%

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

   2. Taylor expanded in a around 0 97.9%

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

      1. associate-+r+ 97.9%

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

      2. +-commutative 97.9%

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

   4. Simplified 97.9%

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

   5. Taylor expanded in b around 0 54.2%

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

      1. +-commutative 54.2%

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

   7. Simplified 54.2%

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

   8. Taylor expanded in a around 0 54.5%

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

      1. *-commutative 54.5%

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

   10. Simplified 54.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-20}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]

Alternative 10: 40.8% 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 99.2%

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

2. Taylor expanded in a around 0 97.4%

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

   1. associate-+r+ 97.4%

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

   2. +-commutative 97.4%

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

4. Simplified 97.4%

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

5. Taylor expanded in b around 0 65.9%

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

   1. +-commutative 65.9%

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

7. Simplified 65.9%

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

8. Taylor expanded in a around 0 39.4%

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

   1. *-commutative 39.4%

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

10. Simplified 39.4%

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

11. Final simplification 39.4%

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

Alternative 11: 40.6% 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 77.9%

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

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 2023334 
(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))))