Quotient of sum of exps

Percentage Accurate: 98.9% → 100.0%

Time: 6.1s

Alternatives: 9

Speedup: 2.9×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{b - a}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. add-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. Step-by-step derivation

   1. exp-diff 99.2%

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

   2. add-exp-log 99.2%

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

   3. +-commutative 99.2%

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

   4. clear-num 99.2%

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

   5. +-commutative 99.2%

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

   6. frac-2neg 99.2%

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

   7. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. rem-exp-log 99.2%

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

   2. exp-diff 99.2%

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

   3. sub-neg 99.2%

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

   4. +-commutative 99.2%

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

   5. prod-exp 99.2%

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

   6. rem-exp-log 99.2%

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

   7. distribute-lft-in 68.7%

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

   8. rec-exp 68.7%

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

   9. lft-mult-inverse 99.2%

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

   10. prod-exp 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in a around inf 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 72.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999996:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.99999996) (exp a) (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.99999996) {
		tmp = exp(a);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (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 (exp(a) <= 0.99999996d0) then
        tmp = exp(a)
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.99999996) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.99999996:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.99999996)
		tmp = exp(a);
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.99999996)
		tmp = exp(a);
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.99999996], N[Exp[a], $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.99999996000000002

   1. Initial program 97.6%

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

      1. add-exp-log 97.6%

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

      2. div-exp 97.7%

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

   3. Applied egg-rr 97.7%

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

   4. Taylor expanded in a around inf 95.6%

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

   if 0.99999996000000002 < (exp.f64 a)

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 98.9%

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

   3. Taylor expanded in b around 0 65.1%

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

      1. unpow2 65.1%

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

   5. Simplified 65.1%

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

      1. associate-*r* 65.1%

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

      2. distribute-rgt1-in 65.1%

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

   7. Applied egg-rr 65.1%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.99999996:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]

Alternative 3: 87.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+144}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.1)
   (exp a)
   (if (<= b 1.05e+144) (/ (exp a) 2.0) (/ 1.0 (+ 2.0 (* b (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = exp(a);
	} else if (b <= 1.05e+144) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 1.0 / (2.0 + (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 <= (-1.1d0)) then
        tmp = exp(a)
    else if (b <= 1.05d+144) then
        tmp = exp(a) / 2.0d0
    else
        tmp = 1.0d0 / (2.0d0 + (b * (b * 0.5d0)))
    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 if (b <= 1.05e+144) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 1.0 / (2.0 + (b * (b * 0.5)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.1:
		tmp = math.exp(a)
	elif b <= 1.05e+144:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 1.0 / (2.0 + (b * (b * 0.5)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.1)
		tmp = exp(a);
	elseif (b <= 1.05e+144)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(b * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.1)
		tmp = exp(a);
	elseif (b <= 1.05e+144)
		tmp = exp(a) / 2.0;
	else
		tmp = 1.0 / (2.0 + (b * (b * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.1], N[Exp[a], $MachinePrecision], If[LessEqual[b, 1.05e+144], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1000000000000001

   1. Initial program 95.1%

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

      1. add-exp-log 95.1%

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

      2. div-exp 95.2%

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

   3. Applied egg-rr 95.2%

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

   4. Taylor expanded in a around inf 92.4%

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

   if -1.1000000000000001 < b < 1.04999999999999998e144

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 98.3%

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

   3. Taylor expanded in b around 0 86.1%

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

   if 1.04999999999999998e144 < 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 93.2%

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

      1. unpow2 93.2%

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

   5. Simplified 93.2%

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

   6. Taylor expanded in b around inf 93.2%

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

      1. unpow2 93.2%

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

      2. *-commutative 93.2%

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

      3. associate-*r* 93.2%

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

   8. Simplified 93.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+144}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \]

Alternative 4: 98.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -28000000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -28000000.0) (exp a) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -28000000.0) {
		tmp = exp(a);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-28000000.0d0)) then
        tmp = exp(a)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -28000000.0) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -28000000.0:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -28000000.0)
		tmp = exp(a);
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -28000000.0)
		tmp = exp(a);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -28000000.0], N[Exp[a], $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.8e7

   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 98.8%

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

   3. Applied egg-rr 98.8%

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

   4. Taylor expanded in a around inf 100.0%

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

   if -2.8e7 < a

   1. Initial program 99.4%

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

   2. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -28000000:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

Alternative 5: 53.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-91}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.65e-91)
   (+ 0.5 (* a 0.25))
   (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.65e-91) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (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 <= 1.65d-91) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.65e-91) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.65e-91:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.65e-91)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.65e-91)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.65e-91], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.65000000000000006e-91

   1. Initial program 98.7%

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

   2. Taylor expanded in b around 0 79.0%

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

   3. Taylor expanded in a around 0 51.7%

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

      1. *-commutative 51.7%

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

   5. Simplified 51.7%

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

   if 1.65000000000000006e-91 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 92.9%

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

   3. Taylor expanded in b around 0 53.8%

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

      1. unpow2 53.8%

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

   5. Simplified 53.8%

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

      1. associate-*r* 53.8%

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

      2. distribute-rgt1-in 53.8%

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

   7. Applied egg-rr 53.8%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-91}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]

Alternative 6: 52.7% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{-92}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9.8e-92) (+ 0.5 (* a 0.25)) (/ 1.0 (+ 2.0 (* b (* b 0.5))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9.8e-92) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0 / (2.0 + (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 <= 9.8d-92) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 1.0d0 / (2.0d0 + (b * (b * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 9.8e-92) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 1.0 / (2.0 + (b * (b * 0.5)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 9.8e-92:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 1.0 / (2.0 + (b * (b * 0.5)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9.8e-92)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(b * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 9.8e-92)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 1.0 / (2.0 + (b * (b * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9.8e-92], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.8e-92

   1. Initial program 98.7%

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

   2. Taylor expanded in b around 0 79.0%

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

   3. Taylor expanded in a around 0 51.7%

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

      1. *-commutative 51.7%

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

   5. Simplified 51.7%

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

   if 9.8e-92 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 92.9%

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

   3. Taylor expanded in b around 0 53.8%

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

      1. unpow2 53.8%

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

   5. Simplified 53.8%

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

   6. Taylor expanded in b around inf 53.3%

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

      1. unpow2 53.3%

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

      2. *-commutative 53.3%

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

      3. associate-*r* 53.3%

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

   8. Simplified 53.3%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{-92}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(b \cdot 0.5\right)}\\ \end{array} \]

Alternative 7: 52.8% accurate, 43.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.6) (+ 0.5 (* a 0.25)) (/ 2.0 (* b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.6) {
		tmp = 0.5 + (a * 0.25);
	} 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 <= 1.6d0) then
        tmp = 0.5d0 + (a * 0.25d0)
    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 <= 1.6) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 2.0 / (b * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.6:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 2.0 / (b * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.6)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = Float64(2.0 / Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.6)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 2.0 / (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.6], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001

   1. Initial program 98.9%

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

   2. Taylor expanded in b around 0 80.9%

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

   3. Taylor expanded in a around 0 52.8%

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

      1. *-commutative 52.8%

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

   5. Simplified 52.8%

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

   if 1.6000000000000001 < b

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0 99.8%

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

   3. Taylor expanded in b around 0 51.0%

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

      1. unpow2 51.0%

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

   5. Simplified 51.0%

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

   6. Taylor expanded in b around inf 51.0%

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

      1. unpow2 51.0%

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

   8. Simplified 51.0%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{b \cdot b}\\ \end{array} \]

Alternative 8: 39.3% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + a \cdot 0.25 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (+ 0.5 (* a 0.25)))

C

double code(double a, double b) {
	return 0.5 + (a * 0.25);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0 + (a * 0.25d0)
end function

Java

public static double code(double a, double b) {
	return 0.5 + (a * 0.25);
}

Python

def code(a, b):
	return 0.5 + (a * 0.25)

Julia

function code(a, b)
	return Float64(0.5 + Float64(a * 0.25))
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5 + (a * 0.25);
end

Wolfram

code[a_, b_] := N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in b around 0 68.1%

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

3. Taylor expanded in a around 0 37.8%

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

   1. *-commutative 37.8%

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

5. Simplified 37.8%

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

6. Final simplification 37.8%

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

Alternative 9: 39.2% 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 79.5%

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

3. Taylor expanded in b around 0 37.3%

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

4. Final simplification 37.3%

   \[\leadsto 0.5 \]

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 2023213 
(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))))