Quotient of sum of exps

Percentage Accurate: 98.9% → 97.8%

Time: 4.3s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 97.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.05e+18) (/ (exp a) a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.05e+18) {
		tmp = exp(a) / 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 <= (-2.05d+18)) then
        tmp = exp(a) / 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 <= -2.05e+18) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -2.05e+18:
		tmp = math.exp(a) / a
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.05e+18)
		tmp = Float64(exp(a) / 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 <= -2.05e+18)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.05e+18], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.05e18

   1. Initial program 98.6%

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

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around inf 100.0%

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

   if -2.05e18 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 2: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. add-log-exp 98.8%

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

3. Applied egg-rr 98.8%

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

4. Final simplification 98.8%

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

Alternative 3: 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 98.8%

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

2. Final simplification 98.8%

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

Alternative 4: 71.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -540000000000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -540000000000.0)
   (/ (exp a) a)
   (/ 1.0 (+ (+ b 2.0) (* 0.5 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -540000000000.0) {
		tmp = exp(a) / a;
	} else {
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-540000000000.0d0)) then
        tmp = exp(a) / a
    else
        tmp = 1.0d0 / ((b + 2.0d0) + (0.5d0 * (b * b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -540000000000.0) {
		tmp = Math.exp(a) / a;
	} else {
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -540000000000.0:
		tmp = math.exp(a) / a
	else:
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -540000000000.0)
		tmp = Float64(exp(a) / a);
	else
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) + Float64(0.5 * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -540000000000.0)
		tmp = exp(a) / a;
	else
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -540000000000.0], N[(N[Exp[a], $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.4e11

   1. Initial program 98.6%

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

   2. Taylor expanded in b around 0 100.0%

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

   3. Taylor expanded in a around 0 100.0%

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

      1. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around inf 100.0%

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

   if -5.4e11 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 99.5%

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

   3. Taylor expanded in b around 0 64.4%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + 0.5 \cdot {b}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-+r+ 64.4%

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

      2. +-commutative 64.4%

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

      3. unpow2 64.4%

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

   5. Simplified 64.4%

      \[\leadsto \frac{1}{\color{blue}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -540000000000:\\ \;\;\;\;\frac{e^{a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 5: 53.1% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -3100.0) 0.5 (/ 1.0 (+ (+ b 2.0) (* 0.5 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -3100.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * 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 <= (-3100.0d0)) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / ((b + 2.0d0) + (0.5d0 * (b * b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -3100.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -3100.0:
		tmp = 0.5
	else:
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -3100.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) + Float64(0.5 * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -3100.0)
		tmp = 0.5;
	else
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -3100.0], 0.5, N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3100

   1. Initial program 97.7%

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

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 18.8%

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

   if -3100 < b

   1. Initial program 99.0%

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

   2. Taylor expanded in a around 0 74.5%

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

   3. Taylor expanded in b around 0 60.7%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + 0.5 \cdot {b}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-+r+ 60.7%

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

      2. +-commutative 60.7%

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

      3. unpow2 60.7%

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

   5. Simplified 60.7%

      \[\leadsto \frac{1}{\color{blue}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 6: 52.7% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.25) 0.5 (/ 1.0 (+ b (* 0.5 (* b b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.25) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / (b + (0.5 * (b * 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 <= 1.25d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0 / (b + (0.5d0 * (b * b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.25) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / (b + (0.5 * (b * b)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.25:
		tmp = 0.5
	else:
		tmp = 1.0 / (b + (0.5 * (b * b)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.25)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / Float64(b + Float64(0.5 * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.25)
		tmp = 0.5;
	else
		tmp = 1.0 / (b + (0.5 * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.25], 0.5, N[(1.0 / N[(b + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.25

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 70.6%

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

   3. Taylor expanded in b around 0 49.8%

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

   if 1.25 < b

   1. Initial program 98.6%

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

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 60.7%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + 0.5 \cdot {b}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-+r+ 60.7%

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

      2. +-commutative 60.7%

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

      3. unpow2 60.7%

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

   5. Simplified 60.7%

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

   6. Taylor expanded in b around inf 60.7%

      \[\leadsto \frac{1}{\color{blue}{b + 0.5 \cdot {b}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 60.7%

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

   8. Simplified 60.7%

      \[\leadsto \frac{1}{\color{blue}{b + 0.5 \cdot \left(b \cdot b\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 0.5 \cdot \left(b \cdot b\right)}\\ \end{array} \]

Alternative 7: 52.7% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 2.0) 0.5 (/ 2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 2.0 / (b * 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 <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = 2.0d0 / (b * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.0:
		tmp = 0.5
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.0)
		tmp = 0.5;
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.0], 0.5, N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0 70.6%

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

   3. Taylor expanded in b around 0 49.8%

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

   if 2 < b

   1. Initial program 98.6%

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

   2. Taylor expanded in a around 0 100.0%

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

   3. Taylor expanded in b around 0 60.7%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(b + 0.5 \cdot {b}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-+r+ 60.7%

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

      2. +-commutative 60.7%

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

      3. unpow2 60.7%

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

   5. Simplified 60.7%

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

   6. Taylor expanded in b around inf 60.7%

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

      1. unpow2 60.7%

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

   8. Simplified 60.7%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 8: 39.3% 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 78.9%

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

3. Taylor expanded in b around 0 36.7%

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

4. Final simplification 36.7%

   \[\leadsto 0.5 \]

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 2023224 
(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))))