Quotient of sum of exps

Percentage Accurate: 99.1% → 97.8%

Time: 5.9s

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: 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: 97.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{+18}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4e+18) (exp a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4e+18) {
		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 <= (-4d+18)) 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 <= -4e+18) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4e+18:
		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 <= -4e+18)
		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 <= -4e+18)
		tmp = exp(a);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4e+18], 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 < -4e18

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

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

      3. pow-to-exp 100.0%

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

      4. pow3 100.0%

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

      5. log-pow 100.0%

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

      6. log-div 100.0%

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

      7. add-log-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in a around inf 100.0%

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

   if -4e18 < a

   1. Initial program 98.4%

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

   2. Taylor expanded in a around 0 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{+18}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 2: 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 98.8%

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

   1. add-exp-log 98.8%

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

   2. div-exp 98.9%

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

3. Applied egg-rr 98.9%

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

4. Final simplification 98.9%

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

Alternative 3: 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]

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: 78.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + a \cdot 0.25\\ \mathbf{if}\;b \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq -1.9 \cdot 10^{-143}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-188}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-164}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-146}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-22}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* a 0.25))))
   (if (<= b -2.15e-7)
     (exp a)
     (if (<= b -1.9e-143)
       t_0
       (if (<= b -1.6e-188)
         (exp a)
         (if (<= b 9e-164)
           0.5
           (if (<= b 3.1e-146) (exp a) (if (<= b 2.3e-22) t_0 0.0))))))))

C

double code(double a, double b) {
	double t_0 = 0.5 + (a * 0.25);
	double tmp;
	if (b <= -2.15e-7) {
		tmp = exp(a);
	} else if (b <= -1.9e-143) {
		tmp = t_0;
	} else if (b <= -1.6e-188) {
		tmp = exp(a);
	} else if (b <= 9e-164) {
		tmp = 0.5;
	} else if (b <= 3.1e-146) {
		tmp = exp(a);
	} else if (b <= 2.3e-22) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (a * 0.25d0)
    if (b <= (-2.15d-7)) then
        tmp = exp(a)
    else if (b <= (-1.9d-143)) then
        tmp = t_0
    else if (b <= (-1.6d-188)) then
        tmp = exp(a)
    else if (b <= 9d-164) then
        tmp = 0.5d0
    else if (b <= 3.1d-146) then
        tmp = exp(a)
    else if (b <= 2.3d-22) then
        tmp = t_0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 0.5 + (a * 0.25);
	double tmp;
	if (b <= -2.15e-7) {
		tmp = Math.exp(a);
	} else if (b <= -1.9e-143) {
		tmp = t_0;
	} else if (b <= -1.6e-188) {
		tmp = Math.exp(a);
	} else if (b <= 9e-164) {
		tmp = 0.5;
	} else if (b <= 3.1e-146) {
		tmp = Math.exp(a);
	} else if (b <= 2.3e-22) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 0.5 + (a * 0.25)
	tmp = 0
	if b <= -2.15e-7:
		tmp = math.exp(a)
	elif b <= -1.9e-143:
		tmp = t_0
	elif b <= -1.6e-188:
		tmp = math.exp(a)
	elif b <= 9e-164:
		tmp = 0.5
	elif b <= 3.1e-146:
		tmp = math.exp(a)
	elif b <= 2.3e-22:
		tmp = t_0
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	t_0 = Float64(0.5 + Float64(a * 0.25))
	tmp = 0.0
	if (b <= -2.15e-7)
		tmp = exp(a);
	elseif (b <= -1.9e-143)
		tmp = t_0;
	elseif (b <= -1.6e-188)
		tmp = exp(a);
	elseif (b <= 9e-164)
		tmp = 0.5;
	elseif (b <= 3.1e-146)
		tmp = exp(a);
	elseif (b <= 2.3e-22)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 0.5 + (a * 0.25);
	tmp = 0.0;
	if (b <= -2.15e-7)
		tmp = exp(a);
	elseif (b <= -1.9e-143)
		tmp = t_0;
	elseif (b <= -1.6e-188)
		tmp = exp(a);
	elseif (b <= 9e-164)
		tmp = 0.5;
	elseif (b <= 3.1e-146)
		tmp = exp(a);
	elseif (b <= 2.3e-22)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.15e-7], N[Exp[a], $MachinePrecision], If[LessEqual[b, -1.9e-143], t$95$0, If[LessEqual[b, -1.6e-188], N[Exp[a], $MachinePrecision], If[LessEqual[b, 9e-164], 0.5, If[LessEqual[b, 3.1e-146], N[Exp[a], $MachinePrecision], If[LessEqual[b, 2.3e-22], t$95$0, 0.0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.1500000000000001e-7 or -1.89999999999999991e-143 < b < -1.60000000000000011e-188 or 8.9999999999999995e-164 < b < 3.0999999999999998e-146

   1. Initial program 97.4%

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

      1. add-cbrt-cube 97.3%

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

      2. pow1/3 97.4%

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

      3. pow-to-exp 97.3%

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

      4. pow3 97.3%

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

      5. log-pow 97.3%

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

      6. log-div 97.3%

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

      7. add-log-exp 97.4%

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

   3. Applied egg-rr 97.4%

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

   4. Taylor expanded in a around inf 91.4%

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

   if -2.1500000000000001e-7 < b < -1.89999999999999991e-143 or 3.0999999999999998e-146 < b < 2.2999999999999998e-22

   1. Initial program 98.1%

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

   2. Taylor expanded in b around 0 98.1%

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

   3. Taylor expanded in a around 0 80.4%

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

      1. *-commutative 80.4%

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

   5. Simplified 80.4%

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

   if -1.60000000000000011e-188 < b < 8.9999999999999995e-164

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 71.8%

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

   3. Taylor expanded in b around 0 71.8%

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

   if 2.2999999999999998e-22 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 93.1%

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

   3. Taylor expanded in b around 0 6.9%

      \[\leadsto \frac{1}{\color{blue}{2 + b}} \]
   4. Step-by-step derivation

      1. +-commutative 6.9%

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

   5. Simplified 6.9%

      \[\leadsto \frac{1}{\color{blue}{b + 2}} \]
   6. Step-by-step derivation

      1. add-log-exp 86.5%

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

   7. Applied egg-rr 86.5%

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

   8. Taylor expanded in b around inf 97.2%

      \[\leadsto \log \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq -1.9 \cdot 10^{-143}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-188}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-164}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-146}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-22}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 66.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{-32}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-180}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-125}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{-95}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;e^{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9.2e-32)
   (exp a)
   (if (<= a 1.7e-180)
     (/ 1.0 (+ (+ b 2.0) (* 0.5 (* b b))))
     (if (<= a 5.8e-125) (exp a) (if (<= a 4.3e-95) 0.5 (exp a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9.2e-32) {
		tmp = exp(a);
	} else if (a <= 1.7e-180) {
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)));
	} else if (a <= 5.8e-125) {
		tmp = exp(a);
	} else if (a <= 4.3e-95) {
		tmp = 0.5;
	} else {
		tmp = exp(a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-9.2d-32)) then
        tmp = exp(a)
    else if (a <= 1.7d-180) then
        tmp = 1.0d0 / ((b + 2.0d0) + (0.5d0 * (b * b)))
    else if (a <= 5.8d-125) then
        tmp = exp(a)
    else if (a <= 4.3d-95) then
        tmp = 0.5d0
    else
        tmp = exp(a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -9.2e-32) {
		tmp = Math.exp(a);
	} else if (a <= 1.7e-180) {
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)));
	} else if (a <= 5.8e-125) {
		tmp = Math.exp(a);
	} else if (a <= 4.3e-95) {
		tmp = 0.5;
	} else {
		tmp = Math.exp(a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9.2e-32:
		tmp = math.exp(a)
	elif a <= 1.7e-180:
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)))
	elif a <= 5.8e-125:
		tmp = math.exp(a)
	elif a <= 4.3e-95:
		tmp = 0.5
	else:
		tmp = math.exp(a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9.2e-32)
		tmp = exp(a);
	elseif (a <= 1.7e-180)
		tmp = Float64(1.0 / Float64(Float64(b + 2.0) + Float64(0.5 * Float64(b * b))));
	elseif (a <= 5.8e-125)
		tmp = exp(a);
	elseif (a <= 4.3e-95)
		tmp = 0.5;
	else
		tmp = exp(a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9.2e-32)
		tmp = exp(a);
	elseif (a <= 1.7e-180)
		tmp = 1.0 / ((b + 2.0) + (0.5 * (b * b)));
	elseif (a <= 5.8e-125)
		tmp = exp(a);
	elseif (a <= 4.3e-95)
		tmp = 0.5;
	else
		tmp = exp(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9.2e-32], N[Exp[a], $MachinePrecision], If[LessEqual[a, 1.7e-180], N[(1.0 / N[(N[(b + 2.0), $MachinePrecision] + N[(0.5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.8e-125], N[Exp[a], $MachinePrecision], If[LessEqual[a, 4.3e-95], 0.5, N[Exp[a], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.2000000000000002e-32 or 1.69999999999999991e-180 < a < 5.8000000000000004e-125 or 4.29999999999999997e-95 < a

   1. Initial program 97.5%

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

      1. add-cbrt-cube 97.3%

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

      2. pow1/3 97.5%

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

      3. pow-to-exp 97.3%

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

      4. pow3 97.3%

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

      5. log-pow 97.3%

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

      6. log-div 97.3%

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

      7. add-log-exp 97.4%

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

   3. Applied egg-rr 97.4%

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

   4. Taylor expanded in a around inf 82.4%

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

   if -9.2000000000000002e-32 < a < 1.69999999999999991e-180

   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 68.9%

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

      1. associate-+r+ 68.9%

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

      2. +-commutative 68.9%

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

      3. unpow2 68.9%

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

   5. Simplified 68.9%

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

   if 5.8000000000000004e-125 < a < 4.29999999999999997e-95

   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 81.9%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.2 \cdot 10^{-32}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-180}:\\ \;\;\;\;\frac{1}{\left(b + 2\right) + 0.5 \cdot \left(b \cdot b\right)}\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-125}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;a \leq 4.3 \cdot 10^{-95}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;e^{a}\\ \end{array} \]

Alternative 6: 53.9% accurate, 23.3× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.0) {
		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 <= (-2.0d0)) 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 <= -2.0) {
		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 <= -2.0:
		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 <= -2.0)
		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 <= -2.0)
		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, -2.0], 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 < -2

   1. Initial program 96.7%

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

   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 -2 < b

   1. Initial program 99.5%

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

   2. Taylor expanded in a around 0 77.6%

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

   3. Taylor expanded in b around 0 62.9%

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

      1. associate-+r+ 62.9%

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

      2. +-commutative 62.9%

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

      3. unpow2 62.9%

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

   5. Simplified 62.9%

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

4. Final simplification 52.6%

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

Alternative 7: 53.4% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

Python

def code(a, b):
	tmp = 0
	if b <= 1.25:
		tmp = 0.5
	else:
		tmp = 1.0 / (b + (b * (b * 0.5)))
	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(b * Float64(b * 0.5))));
	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 + (b * (b * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 1.25

   1. Initial program 98.4%

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

   2. Taylor expanded in a around 0 77.3%

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

   3. Taylor expanded in b around 0 51.6%

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

   if 1.25 < 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 55.1%

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

      1. associate-+r+ 55.1%

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

      2. +-commutative 55.1%

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

      3. unpow2 55.1%

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

   5. Simplified 55.1%

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

   6. Taylor expanded in b around inf 55.1%

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

      1. unpow2 55.1%

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

      2. *-commutative 55.1%

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

      3. associate-*r* 55.1%

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

   8. Simplified 55.1%

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

4. Final simplification 52.4%

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

Alternative 8: 53.4% 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 98.4%

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

   2. Taylor expanded in a around 0 77.3%

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

   3. Taylor expanded in b around 0 51.6%

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

   if 2 < 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 55.1%

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

      1. associate-+r+ 55.1%

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

      2. +-commutative 55.1%

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

      3. unpow2 55.1%

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

   5. Simplified 55.1%

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

   6. Taylor expanded in b around inf 55.1%

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

      1. unpow2 55.1%

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

   8. Simplified 55.1%

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

4. Final simplification 52.4%

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

Alternative 9: 39.6% 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 98.8%

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

2. Taylor expanded in a around 0 82.9%

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

3. Taylor expanded in b around 0 39.6%

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

4. Final simplification 39.6%

   \[\leadsto 0.5 \]

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 2023174 
(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))))