Quotient of sum of exps

Percentage Accurate: 98.6% → 97.8%

Time: 5.7s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. a = 1.563576823241593e+264
  2. b = -4.5124542818184726e+203

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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999999999999:\\ \;\;\;\;\frac{e^{a}}{1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.9999999999999)
   (/ (exp a) (+ 1.0 (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 1.0)))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.9999999999999) {
		tmp = exp(a) / (1.0 + fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.9999999999999)
		tmp = Float64(exp(a) / Float64(1.0 + fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0)));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.9999999999999], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.999999999999899969

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) + 1\right)} + 1} \]

      2. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\left(\color{blue}{\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) \cdot a} + 1\right) + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right), a, 1\right)} + 1} \]

      4. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right) + 1}, a, 1\right) + 1} \]

      5. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot a\right) \cdot a} + 1, a, 1\right) + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot a, a, 1\right)}, a, 1\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot a + \frac{1}{2}}, a, 1\right), a, 1\right) + 1} \]

      8. lower-fma.f64 100.0

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, a, 0.5\right)}, a, 1\right), a, 1\right) + 1} \]

   8. Applied rewrites 100.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right)} + 1} \]

   if 0.999999999999899969 < (exp.f64 a)

   1. Initial program 98.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.9999999999999:\\ \;\;\;\;\frac{e^{a}}{1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Final simplification 98.8%

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

Alternative 3: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999999995:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.999999999995)
   (/ (exp a) (+ 1.0 1.0))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.999999999995) {
		tmp = exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.999999999995d0) then
        tmp = exp(a) / (1.0d0 + 1.0d0)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.999999999995) {
		tmp = Math.exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.999999999995:
		tmp = math.exp(a) / (1.0 + 1.0)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.999999999995)
		tmp = Float64(exp(a) / Float64(1.0 + 1.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.999999999995)
		tmp = exp(a) / (1.0 + 1.0);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.999999999995], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.999999999995

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.6%

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

   if 0.999999999995 < (exp.f64 a)

   1. Initial program 98.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.999999999995:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.8e+94)
   (/ 1.0 (+ 1.0 (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 1.0)))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.8e+94) {
		tmp = 1.0 / (1.0 + fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.8e+94)
		tmp = Float64(1.0 / Float64(1.0 + fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0)));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.8e+94], N[(1.0 / N[(1.0 + N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.79999999999999998e94

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) + 1\right)} + 1} \]

      2. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\left(\color{blue}{\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) \cdot a} + 1\right) + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right), a, 1\right)} + 1} \]

      4. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right) + 1}, a, 1\right) + 1} \]

      5. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot a\right) \cdot a} + 1, a, 1\right) + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot a, a, 1\right)}, a, 1\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot a + \frac{1}{2}}, a, 1\right), a, 1\right) + 1} \]

      8. lower-fma.f64 100.0

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, a, 0.5\right)}, a, 1\right), a, 1\right) + 1} \]

   8. Applied rewrites 100.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right)} + 1} \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, a, \frac{1}{2}\right), a, 1\right), a, 1\right) + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 95.6%

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

   if -2.79999999999999998e94 < a

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 94.7

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

   5. Applied rewrites 94.7%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.8% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.3e+101)
   (/ 1.0 (+ 1.0 (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 1.0)))
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.3e+101) {
		tmp = 1.0 / (1.0 + fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0));
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.3e+101)
		tmp = Float64(1.0 / Float64(1.0 + fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0)));
	else
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.3e+101], N[(1.0 / N[(1.0 + N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.3000000000000001e101

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) + 1\right)} + 1} \]

      2. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\left(\color{blue}{\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) \cdot a} + 1\right) + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right), a, 1\right)} + 1} \]

      4. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right) + 1}, a, 1\right) + 1} \]

      5. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot a\right) \cdot a} + 1, a, 1\right) + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot a, a, 1\right)}, a, 1\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot a + \frac{1}{2}}, a, 1\right), a, 1\right) + 1} \]

      8. lower-fma.f64 67.9

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, a, 0.5\right)}, a, 1\right), a, 1\right) + 1} \]

   8. Applied rewrites 67.9%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right)} + 1} \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, a, \frac{1}{2}\right), a, 1\right), a, 1\right) + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 61.6%

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

   if 2.3000000000000001e101 < 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 94.6%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.5e+69)
   (/ 1.0 (+ (fma (fma 0.5 a 1.0) a 1.0) 1.0))
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.5e+69) {
		tmp = 1.0 / (fma(fma(0.5, a, 1.0), a, 1.0) + 1.0);
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.5e+69)
		tmp = Float64(1.0 / Float64(fma(fma(0.5, a, 1.0), a, 1.0) + 1.0));
	else
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.5e+69], N[(1.0 / N[(N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.49999999999999939e69

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) + 1\right)} + 1} \]

      2. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\left(\color{blue}{\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) \cdot a} + 1\right) + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right), a, 1\right)} + 1} \]

      4. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right) + 1}, a, 1\right) + 1} \]

      5. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot a\right) \cdot a} + 1, a, 1\right) + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot a, a, 1\right)}, a, 1\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot a + \frac{1}{2}}, a, 1\right), a, 1\right) + 1} \]

      8. lower-fma.f64 69.4

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, a, 0.5\right)}, a, 1\right), a, 1\right) + 1} \]

   8. Applied rewrites 69.4%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right)} + 1} \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, a, \frac{1}{2}\right), a, 1\right), a, 1\right) + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.7%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{2} \cdot a, a, 1\right)} + 1} \]

       4. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot a + 1}, a, 1\right) + 1} \]

       5. lower-fma.f64 61.5

          \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, a, 1\right)}, a, 1\right) + 1} \]

   14. Applied rewrites 61.5%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)} + 1} \]

   if 7.49999999999999939e69 < 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 72.1%

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

Alternative 7: 63.1% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.6 \cdot 10^{+145}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.6e+145)
   (/ 1.0 (+ (fma (fma 0.5 a 1.0) a 1.0) 1.0))
   (/ 1.0 (* (* b b) 0.5))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.6e+145) {
		tmp = 1.0 / (fma(fma(0.5, a, 1.0), a, 1.0) + 1.0);
	} else {
		tmp = 1.0 / ((b * b) * 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.6e+145)
		tmp = Float64(1.0 / Float64(fma(fma(0.5, a, 1.0), a, 1.0) + 1.0));
	else
		tmp = Float64(1.0 / Float64(Float64(b * b) * 0.5));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.6e+145], N[(1.0 / N[(N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.60000000000000025e145

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) + 1\right)} + 1} \]

      2. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\left(\color{blue}{\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) \cdot a} + 1\right) + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right), a, 1\right)} + 1} \]

      4. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right) + 1}, a, 1\right) + 1} \]

      5. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot a\right) \cdot a} + 1, a, 1\right) + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot a, a, 1\right)}, a, 1\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot a + \frac{1}{2}}, a, 1\right), a, 1\right) + 1} \]

      8. lower-fma.f64 66.1

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, a, 0.5\right)}, a, 1\right), a, 1\right) + 1} \]

   8. Applied rewrites 66.1%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right)} + 1} \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, a, \frac{1}{2}\right), a, 1\right), a, 1\right) + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 59.3%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(1 + \frac{1}{2} \cdot a, a, 1\right)} + 1} \]

       4. +-commutative N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{2} \cdot a + 1}, a, 1\right) + 1} \]

       5. lower-fma.f64 56.5

          \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5, a, 1\right)}, a, 1\right) + 1} \]

   14. Applied rewrites 56.5%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)} + 1} \]

   if 7.60000000000000025e145 < 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

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

   8. Applied rewrites 84.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 91.6%

       \[\leadsto \frac{1}{\left(b \cdot b\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 52.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.9e+48) (/ 1.0 (+ (+ 1.0 a) 1.0)) (/ 1.0 (* (* b b) 0.5))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.9e+48) {
		tmp = 1.0 / ((1.0 + a) + 1.0);
	} else {
		tmp = 1.0 / ((b * b) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 3.9d+48) then
        tmp = 1.0d0 / ((1.0d0 + a) + 1.0d0)
    else
        tmp = 1.0d0 / ((b * b) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.9e+48) {
		tmp = 1.0 / ((1.0 + a) + 1.0);
	} else {
		tmp = 1.0 / ((b * b) * 0.5);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.9e+48:
		tmp = 1.0 / ((1.0 + a) + 1.0)
	else:
		tmp = 1.0 / ((b * b) * 0.5)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.9e+48)
		tmp = Float64(1.0 / Float64(Float64(1.0 + a) + 1.0));
	else
		tmp = Float64(1.0 / Float64(Float64(b * b) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.9e+48)
		tmp = 1.0 / ((1.0 + a) + 1.0);
	else
		tmp = 1.0 / ((b * b) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.9e+48], N[(1.0 / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.9000000000000001e48

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) + 1\right)} + 1} \]

      2. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\left(\color{blue}{\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) \cdot a} + 1\right) + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right), a, 1\right)} + 1} \]

      4. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right) + 1}, a, 1\right) + 1} \]

      5. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot a\right) \cdot a} + 1, a, 1\right) + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot a, a, 1\right)}, a, 1\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot a + \frac{1}{2}}, a, 1\right), a, 1\right) + 1} \]

      8. lower-fma.f64 69.4

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, a, 0.5\right)}, a, 1\right), a, 1\right) + 1} \]

   8. Applied rewrites 69.4%

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right)} + 1} \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, a, \frac{1}{2}\right), a, 1\right), a, 1\right) + 1} \]
   10. Step-by-step derivation

   11. Applied rewrites 63.7%

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

   12. Taylor expanded in a around 0

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

       1. lower-+.f64 48.7

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

   14. Applied rewrites 48.7%

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

   if 3.9000000000000001e48 < 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

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

   8. Applied rewrites 44.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 47.7%

       \[\leadsto \frac{1}{\left(b \cdot b\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 39.5% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + a\right) + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 1.0 / ((1.0 + a) + 1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 63.1%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{e^{a}}{\color{blue}{\left(a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) + 1\right)} + 1} \]

   2. *-commutative N/A

      \[\leadsto \frac{e^{a}}{\left(\color{blue}{\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) \cdot a} + 1\right) + 1} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right), a, 1\right)} + 1} \]

   4. +-commutative N/A

      \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right) + 1}, a, 1\right) + 1} \]

   5. *-commutative N/A

      \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot a\right) \cdot a} + 1, a, 1\right) + 1} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot a, a, 1\right)}, a, 1\right) + 1} \]

   7. +-commutative N/A

      \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot a + \frac{1}{2}}, a, 1\right), a, 1\right) + 1} \]

   8. lower-fma.f64 62.8

      \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, a, 0.5\right)}, a, 1\right), a, 1\right) + 1} \]

8. Applied rewrites 62.8%

   \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right)} + 1} \]

9. Taylor expanded in a around 0

   \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, a, \frac{1}{2}\right), a, 1\right), a, 1\right) + 1} \]
10. Step-by-step derivation

11. Applied rewrites 55.5%

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

12. Taylor expanded in a around 0

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

    1. lower-+.f64 41.1

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

14. Applied rewrites 41.1%

    \[\leadsto \frac{1}{\color{blue}{\left(1 + a\right)} + 1} \]
15. Add Preprocessing

Alternative 10: 39.1% 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 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 83.8

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

5. Applied rewrites 83.8%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 40.6%

   \[\leadsto 0.5 \]
9. 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 2024255 
(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))))