Quotient of sum of exps

Percentage Accurate: 98.8% → 98.8%

Time: 6.4s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. a = 7.901385698112343e+19
  2. b = -1.1558738523012004e-80

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: 98.8% 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: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \frac{--1}{e^{b} + e^{a}} \cdot e^{a} \]
6. Add Preprocessing

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.005:\\ \;\;\;\;\frac{-1}{1 + e^{a}} \cdot \left(-e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.005)
   (* (/ -1.0 (+ 1.0 (exp a))) (- (exp a)))
   (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.005) {
		tmp = (-1.0 / (1.0 + exp(a))) * -exp(a);
	} else {
		tmp = pow((exp(b) + 1.0), -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) <= 0.005d0) then
        tmp = ((-1.0d0) / (1.0d0 + exp(a))) * -exp(a)
    else
        tmp = (exp(b) + 1.0d0) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.005) {
		tmp = (-1.0 / (1.0 + Math.exp(a))) * -Math.exp(a);
	} else {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.005:
		tmp = (-1.0 / (1.0 + math.exp(a))) * -math.exp(a)
	else:
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.005)
		tmp = Float64(Float64(-1.0 / Float64(1.0 + exp(a))) * Float64(-exp(a)));
	else
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.005)
		tmp = (-1.0 / (1.0 + exp(a))) * -exp(a);
	else
		tmp = (exp(b) + 1.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.005], N[(N[(-1.0 / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Exp[a], $MachinePrecision])), $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0050000000000000001

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 0.0050000000000000001 < (exp.f64 a)

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.005:\\ \;\;\;\;\frac{-1}{1 + e^{a}} \cdot \left(-e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.005:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.005)
   (/ (exp a) (+ (exp a) 1.0))
   (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.005) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = pow((exp(b) + 1.0), -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) <= 0.005d0) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = (exp(b) + 1.0d0) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.005) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.005:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.005)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.005)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = (exp(b) + 1.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.005], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0050000000000000001

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0050000000000000001 < (exp.f64 a)

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.005:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% 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 100.0%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 5: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, -0.5\right) \cdot \left(-e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.005)
   (* (fma 0.25 a -0.5) (- (exp a)))
   (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.005) {
		tmp = fma(0.25, a, -0.5) * -exp(a);
	} else {
		tmp = pow((exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.005)
		tmp = Float64(fma(0.25, a, -0.5) * Float64(-exp(a)));
	else
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.005], N[(N[(0.25 * a + -0.5), $MachinePrecision] * (-N[Exp[a], $MachinePrecision])), $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0050000000000000001

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.7%

       \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{a}, -0.5\right) \cdot \left(-e^{a}\right) \]

   if 0.0050000000000000001 < (exp.f64 a)

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, -0.5\right) \cdot \left(-e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+22}:\\ \;\;\;\;\left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e+22)
   (* (* 0.020833333333333332 (* b b)) b)
   (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e+22) {
		tmp = (0.020833333333333332 * (b * b)) * b;
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e+22)
		tmp = Float64(Float64(0.020833333333333332 * Float64(b * b)) * b);
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e+22], N[(N[(0.020833333333333332 * N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.4e22

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 33.8

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

   5. Applied rewrites 33.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.020833333333333332, -0.25\right), \color{blue}{b}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{48} \cdot {b}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.8%

       \[\leadsto {b}^{3} \cdot 0.020833333333333332 \]
   12. Step-by-step derivation

   13. Applied rewrites 53.8%

       \[\leadsto \left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b \]

   if -2.4e22 < a

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 67.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), \color{blue}{b}, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+22}:\\ \;\;\;\;\left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, -0.5\right) \cdot \left(-e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot b\right) \cdot b, b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9e+102)
   (* (fma 0.25 a -0.5) (- (exp a)))
   (pow (fma (* (* 0.16666666666666666 b) b) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9e+102) {
		tmp = fma(0.25, a, -0.5) * -exp(a);
	} else {
		tmp = pow(fma(((0.16666666666666666 * b) * b), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9e+102)
		tmp = Float64(fma(0.25, a, -0.5) * Float64(-exp(a)));
	else
		tmp = fma(Float64(Float64(0.16666666666666666 * b) * b), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9e+102], N[(N[(0.25 * a + -0.5), $MachinePrecision] * (-N[Exp[a], $MachinePrecision])), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b), $MachinePrecision] * b), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.00000000000000042e102

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 72.9

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

   7. Applied rewrites 72.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 72.5%

       \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{a}, -0.5\right) \cdot \left(-e^{a}\right) \]

   if 9.00000000000000042e102 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), \color{blue}{b}, 2\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1}{6} \cdot {b}^{2}, b, 2\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot b\right) \cdot b, b, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, -0.5\right) \cdot \left(-e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot b\right) \cdot b, b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+102}:\\ \;\;\;\;-0.5 \cdot \left(-e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot b\right) \cdot b, b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 9e+102)
   (* -0.5 (- (exp a)))
   (pow (fma (* (* 0.16666666666666666 b) b) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 9e+102) {
		tmp = -0.5 * -exp(a);
	} else {
		tmp = pow(fma(((0.16666666666666666 * b) * b), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 9e+102)
		tmp = Float64(-0.5 * Float64(-exp(a)));
	else
		tmp = fma(Float64(Float64(0.16666666666666666 * b) * b), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 9e+102], N[(-0.5 * (-N[Exp[a], $MachinePrecision])), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b), $MachinePrecision] * b), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.00000000000000042e102

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 72.9

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

   7. Applied rewrites 72.9%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{-1}{2} \cdot \left(-e^{a}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.2%

       \[\leadsto -0.5 \cdot \left(-e^{a}\right) \]

   if 9.00000000000000042e102 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), \color{blue}{b}, 2\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1}{6} \cdot {b}^{2}, b, 2\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot b\right) \cdot b, b, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+102}:\\ \;\;\;\;-0.5 \cdot \left(-e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot b\right) \cdot b, b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+22}:\\ \;\;\;\;\left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot b\right) \cdot b, b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e+22)
   (* (* 0.020833333333333332 (* b b)) b)
   (pow (fma (* (* 0.16666666666666666 b) b) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e+22) {
		tmp = (0.020833333333333332 * (b * b)) * b;
	} else {
		tmp = pow(fma(((0.16666666666666666 * b) * b), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e+22)
		tmp = Float64(Float64(0.020833333333333332 * Float64(b * b)) * b);
	else
		tmp = fma(Float64(Float64(0.16666666666666666 * b) * b), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e+22], N[(N[(0.020833333333333332 * N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b), $MachinePrecision] * b), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.4e22

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 33.8

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

   5. Applied rewrites 33.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.020833333333333332, -0.25\right), \color{blue}{b}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{48} \cdot {b}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.8%

       \[\leadsto {b}^{3} \cdot 0.020833333333333332 \]
   12. Step-by-step derivation

   13. Applied rewrites 53.8%

       \[\leadsto \left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b \]

   if -2.4e22 < a

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 67.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), \color{blue}{b}, 2\right)} \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1}{6} \cdot {b}^{2}, b, 2\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 65.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\left(0.16666666666666666 \cdot b\right) \cdot b, b, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+22}:\\ \;\;\;\;\left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot b\right) \cdot b, b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+22}:\\ \;\;\;\;\left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e+22)
   (* (* 0.020833333333333332 (* b b)) b)
   (pow (fma (fma 0.5 b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e+22) {
		tmp = (0.020833333333333332 * (b * b)) * b;
	} else {
		tmp = pow(fma(fma(0.5, b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e+22)
		tmp = Float64(Float64(0.020833333333333332 * Float64(b * b)) * b);
	else
		tmp = fma(fma(0.5, b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e+22], N[(N[(0.020833333333333332 * N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[Power[N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.4e22

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 33.8

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

   5. Applied rewrites 33.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.020833333333333332, -0.25\right), \color{blue}{b}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{48} \cdot {b}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.8%

       \[\leadsto {b}^{3} \cdot 0.020833333333333332 \]
   12. Step-by-step derivation

   13. Applied rewrites 53.8%

       \[\leadsto \left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b \]

   if -2.4e22 < a

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 61.4%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+22}:\\ \;\;\;\;\left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.6% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+21}:\\ \;\;\;\;\left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.8e+21) (* (* 0.020833333333333332 (* b b)) b) (fma 0.25 a 0.5)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.8e+21) {
		tmp = (0.020833333333333332 * (b * b)) * b;
	} else {
		tmp = fma(0.25, a, 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7.8e+21)
		tmp = Float64(Float64(0.020833333333333332 * Float64(b * b)) * b);
	else
		tmp = fma(0.25, a, 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7.8e+21], N[(N[(0.020833333333333332 * N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(0.25 * a + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.8e21

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 33.8

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

   5. Applied rewrites 33.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.020833333333333332, -0.25\right), \color{blue}{b}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{48} \cdot {b}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.8%

       \[\leadsto {b}^{3} \cdot 0.020833333333333332 \]
   12. Step-by-step derivation

   13. Applied rewrites 53.8%

       \[\leadsto \left(0.020833333333333332 \cdot \left(b \cdot b\right)\right) \cdot b \]

   if -7.8e21 < a

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-frac-neg N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 49.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, b, 0.5\right) - \mathsf{fma}\left(-0.125, b, 0.25\right), \color{blue}{a}, \mathsf{fma}\left(-0.25, b, 0.5\right)\right) \]

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 52.0%

       \[\leadsto \mathsf{fma}\left(0.25, a, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 39.2% accurate, 315.0× 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 100.0%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 83.9

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

5. Applied rewrites 83.9%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 38.1%

   \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{b}, 0.5\right) \]

9. Taylor expanded in b around 0

   \[\leadsto \frac{1}{2} \]
10. Step-by-step derivation

11. Applied rewrites 40.3%

    \[\leadsto 0.5 \]
12. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× 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 2024303 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ 1 (exp (- b a)))))

  (/ (exp a) (+ (exp a) (exp b))))