Quotient of sum of exps

Percentage Accurate: 98.9% → 98.1%

Time: 6.7s

Alternatives: 10

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: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999999996:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999999999996)
   (/ (exp a) (+ (exp a) 1.0))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999999999996) {
		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) <= 0.999999999996d0) 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) <= 0.999999999996) {
		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) <= 0.999999999996:
		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) <= 0.999999999996)
		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) <= 0.999999999996)
		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], 0.999999999996], 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) < 0.999999999995999977

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 100.0%

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

   if 0.999999999995999977 < (exp.f64 a)

   1. Initial program 99.4%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999999996:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 2: 99.1% 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 99.6%

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

   1. add-exp-log 99.6%

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

   2. div-exp 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

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 99.6%

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

2. Final simplification 99.6%

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

Alternative 4: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (exp a) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		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 (exp(a) <= 0.0d0) 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 (Math.exp(a) <= 0.0) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

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

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], 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 (exp.f64 a) < 0.0

   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 0.0 < (exp.f64 a)

   1. Initial program 99.4%

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

   2. Taylor expanded in a around 0 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

Alternative 5: 73.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-37}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot \left(1 - \left(b \cdot b\right) \cdot 0.25\right)}{1 + b \cdot -0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4e-37)
   (exp a)
   (/ 1.0 (+ 2.0 (/ (* b (- 1.0 (* (* b b) 0.25))) (+ 1.0 (* b -0.5)))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4e-37) {
		tmp = exp(a);
	} else {
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (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 (a <= (-4d-37)) then
        tmp = exp(a)
    else
        tmp = 1.0d0 / (2.0d0 + ((b * (1.0d0 - ((b * b) * 0.25d0))) / (1.0d0 + (b * (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4e-37) {
		tmp = Math.exp(a);
	} else {
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4e-37:
		tmp = math.exp(a)
	else:
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4e-37)
		tmp = exp(a);
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(Float64(b * Float64(1.0 - Float64(Float64(b * b) * 0.25))) / Float64(1.0 + Float64(b * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4e-37)
		tmp = exp(a);
	else
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4e-37], N[Exp[a], $MachinePrecision], N[(1.0 / N[(2.0 + N[(N[(b * N[(1.0 - N[(N[(b * b), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(b * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.00000000000000027e-37

   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 96.4%

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

   if -4.00000000000000027e-37 < a

   1. Initial program 99.4%

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

   2. Taylor expanded in a around 0 99.5%

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

   3. Taylor expanded in b around 0 67.1%

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

      1. unpow2 67.1%

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

   5. Simplified 67.1%

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

      1. *-un-lft-identity 67.1%

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

      2. associate-*r* 67.1%

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

      3. distribute-rgt-out 67.1%

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

      4. *-commutative 67.1%

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

   7. Applied egg-rr 67.1%

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

      1. *-commutative 67.1%

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

      2. flip-+ 67.1%

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

      3. associate-*l/ 69.7%

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

      4. metadata-eval 69.7%

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

      5. swap-sqr 69.7%

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

      6. metadata-eval 69.7%

         \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot \color{blue}{0.25}\right) \cdot b}{1 - b \cdot 0.5}} \]

      7. *-commutative 69.7%

         \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 - \color{blue}{0.5 \cdot b}}} \]

      8. cancel-sign-sub-inv 69.7%

         \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{\color{blue}{1 + \left(-0.5\right) \cdot b}}} \]

      9. metadata-eval 69.7%

         \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 + \color{blue}{-0.5} \cdot b}} \]

   9. Applied egg-rr 69.7%

      \[\leadsto \frac{1}{2 + \color{blue}{\frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 + -0.5 \cdot b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4 \cdot 10^{-37}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot \left(1 - \left(b \cdot b\right) \cdot 0.25\right)}{1 + b \cdot -0.5}}\\ \end{array} \]

Alternative 6: 60.0% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot 0.5\right)\\ \mathbf{if}\;b \leq -2.2:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 10^{+103}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot b - t_0 \cdot t_0}{b - t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot \left(1 - \left(b \cdot b\right) \cdot 0.25\right)}{1 + b \cdot -0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (* b 0.5))))
   (if (<= b -2.2)
     (+ 0.5 (* a 0.25))
     (if (<= b 1e+103)
       (/ 1.0 (+ 2.0 (/ (- (* b b) (* t_0 t_0)) (- b t_0))))
       (/
        1.0
        (+ 2.0 (/ (* b (- 1.0 (* (* b b) 0.25))) (+ 1.0 (* b -0.5)))))))))

C

double code(double a, double b) {
	double t_0 = b * (b * 0.5);
	double tmp;
	if (b <= -2.2) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1e+103) {
		tmp = 1.0 / (2.0 + (((b * b) - (t_0 * t_0)) / (b - t_0)));
	} else {
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (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) :: t_0
    real(8) :: tmp
    t_0 = b * (b * 0.5d0)
    if (b <= (-2.2d0)) then
        tmp = 0.5d0 + (a * 0.25d0)
    else if (b <= 1d+103) then
        tmp = 1.0d0 / (2.0d0 + (((b * b) - (t_0 * t_0)) / (b - t_0)))
    else
        tmp = 1.0d0 / (2.0d0 + ((b * (1.0d0 - ((b * b) * 0.25d0))) / (1.0d0 + (b * (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = b * (b * 0.5);
	double tmp;
	if (b <= -2.2) {
		tmp = 0.5 + (a * 0.25);
	} else if (b <= 1e+103) {
		tmp = 1.0 / (2.0 + (((b * b) - (t_0 * t_0)) / (b - t_0)));
	} else {
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * (b * 0.5)
	tmp = 0
	if b <= -2.2:
		tmp = 0.5 + (a * 0.25)
	elif b <= 1e+103:
		tmp = 1.0 / (2.0 + (((b * b) - (t_0 * t_0)) / (b - t_0)))
	else:
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(b * 0.5))
	tmp = 0.0
	if (b <= -2.2)
		tmp = Float64(0.5 + Float64(a * 0.25));
	elseif (b <= 1e+103)
		tmp = Float64(1.0 / Float64(2.0 + Float64(Float64(Float64(b * b) - Float64(t_0 * t_0)) / Float64(b - t_0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(Float64(b * Float64(1.0 - Float64(Float64(b * b) * 0.25))) / Float64(1.0 + Float64(b * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = b * (b * 0.5);
	tmp = 0.0;
	if (b <= -2.2)
		tmp = 0.5 + (a * 0.25);
	elseif (b <= 1e+103)
		tmp = 1.0 / (2.0 + (((b * b) - (t_0 * t_0)) / (b - t_0)));
	else
		tmp = 1.0 / (2.0 + ((b * (1.0 - ((b * b) * 0.25))) / (1.0 + (b * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.2], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+103], N[(1.0 / N[(2.0 + N[(N[(N[(b * b), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(N[(b * N[(1.0 - N[(N[(b * b), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(b * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 18.8%

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

   3. Taylor expanded in a around 0 18.8%

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

      1. *-commutative 18.8%

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

   5. Simplified 18.8%

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

   if -2.2000000000000002 < b < 1e103

   1. Initial program 99.4%

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

   2. Taylor expanded in a around 0 72.9%

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

   3. Taylor expanded in b around 0 58.2%

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

      1. unpow2 58.2%

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

   5. Simplified 58.2%

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

      1. *-un-lft-identity 58.2%

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

      2. associate-*r* 58.2%

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

      3. distribute-rgt-out 58.2%

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

      4. *-commutative 58.2%

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

   7. Applied egg-rr 58.2%

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

      1. distribute-lft-in 58.2%

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

      2. *-rgt-identity 58.2%

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

      3. flip-+ 62.8%

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

   9. Applied egg-rr 62.8%

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

   if 1e103 < 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 83.8%

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

      1. unpow2 83.8%

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

   5. Simplified 83.8%

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

      1. *-un-lft-identity 83.8%

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

      2. associate-*r* 83.8%

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

      3. distribute-rgt-out 83.8%

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

      4. *-commutative 83.8%

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

   7. Applied egg-rr 83.8%

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

      1. *-commutative 83.8%

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

      2. flip-+ 83.8%

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

      3. associate-*l/ 100.0%

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

      4. metadata-eval 100.0%

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

      5. swap-sqr 100.0%

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

      6. metadata-eval 100.0%

         \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot \color{blue}{0.25}\right) \cdot b}{1 - b \cdot 0.5}} \]

      7. *-commutative 100.0%

         \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 - \color{blue}{0.5 \cdot b}}} \]

      8. cancel-sign-sub-inv 100.0%

         \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{\color{blue}{1 + \left(-0.5\right) \cdot b}}} \]

      9. metadata-eval 100.0%

         \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 + \color{blue}{-0.5} \cdot b}} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{2 + \color{blue}{\frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 + -0.5 \cdot b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{elif}\;b \leq 10^{+103}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot b - \left(b \cdot \left(b \cdot 0.5\right)\right) \cdot \left(b \cdot \left(b \cdot 0.5\right)\right)}{b - b \cdot \left(b \cdot 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot \left(1 - \left(b \cdot b\right) \cdot 0.25\right)}{1 + b \cdot -0.5}}\\ \end{array} \]

Alternative 7: 57.7% accurate, 14.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -2

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 18.8%

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

   3. Taylor expanded in a around 0 18.8%

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

      1. *-commutative 18.8%

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

   5. Simplified 18.8%

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

   if -2 < b

   1. Initial program 99.5%

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

   2. Taylor expanded in a around 0 78.8%

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

   3. Taylor expanded in b around 0 63.8%

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

      1. unpow2 63.8%

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

   5. Simplified 63.8%

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

      1. *-un-lft-identity 63.8%

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

      2. associate-*r* 63.8%

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

      3. distribute-rgt-out 63.8%

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

      4. *-commutative 63.8%

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

   7. Applied egg-rr 63.8%

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

      1. *-commutative 63.8%

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

      2. flip-+ 63.8%

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

      3. associate-*l/ 67.3%

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

      4. metadata-eval 67.3%

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

      5. swap-sqr 67.3%

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

      6. metadata-eval 67.3%

         \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot \color{blue}{0.25}\right) \cdot b}{1 - b \cdot 0.5}} \]

      7. *-commutative 67.3%

         \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 - \color{blue}{0.5 \cdot b}}} \]

      8. cancel-sign-sub-inv 67.3%

         \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{\color{blue}{1 + \left(-0.5\right) \cdot b}}} \]

      9. metadata-eval 67.3%

         \[\leadsto \frac{1}{2 + \frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 + \color{blue}{-0.5} \cdot b}} \]

   9. Applied egg-rr 67.3%

      \[\leadsto \frac{1}{2 + \color{blue}{\frac{\left(1 - \left(b \cdot b\right) \cdot 0.25\right) \cdot b}{1 + -0.5 \cdot b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

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

Alternative 8: 53.4% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -2.1) (+ 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 <= -2.1) {
		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 <= (-2.1d0)) 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 <= -2.1) {
		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 <= -2.1:
		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 <= -2.1)
		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 <= -2.1)
		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, -2.1], 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 < -2.10000000000000009

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 18.8%

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

   3. Taylor expanded in a around 0 18.8%

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

      1. *-commutative 18.8%

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

   5. Simplified 18.8%

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

   if -2.10000000000000009 < b

   1. Initial program 99.5%

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

   2. Taylor expanded in a around 0 78.8%

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

   3. Taylor expanded in b around 0 63.8%

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

      1. unpow2 63.8%

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

   5. Simplified 63.8%

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

      1. *-un-lft-identity 63.8%

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

      2. associate-*r* 63.8%

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

      3. distribute-rgt-out 63.8%

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

      4. *-commutative 63.8%

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

   7. Applied egg-rr 63.8%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot 0.5\right)}\\ \end{array} \]

Alternative 9: 53.0% 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 99.4%

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

   2. Taylor expanded in a around 0 76.0%

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

   3. Taylor expanded in b around 0 54.5%

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

   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 56.6%

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

      1. unpow2 56.6%

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

   5. Simplified 56.6%

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

   6. Taylor expanded in b around inf 56.6%

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

      1. unpow2 56.6%

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

   8. Simplified 56.6%

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

4. Final simplification 55.1%

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

Alternative 10: 39.0% 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 99.6%

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

2. Taylor expanded in a around 0 82.6%

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

3. Taylor expanded in b around 0 40.5%

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

4. Final simplification 40.5%

   \[\leadsto 0.5 \]

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 2023215 
(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))))