Quotient of sum of exps

Percentage Accurate: 98.9% → 98.9%

Time: 5.9s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. a = 8.437960932887736e+41
  2. b = -1.0204541362874851e+204

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

Alternative 2: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -116000000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -116000000.0) (/ (exp a) 2.0) (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -116000000.0) {
		tmp = exp(a) / 2.0;
	} 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 (a <= (-116000000.0d0)) then
        tmp = exp(a) / 2.0d0
    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 (a <= -116000000.0) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -116000000.0:
		tmp = math.exp(a) / 2.0
	else:
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -116000000.0)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -116000000.0)
		tmp = exp(a) / 2.0;
	else
		tmp = (exp(b) + 1.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -116000000.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.16e8

   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%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if -1.16e8 < a

   1. Initial program 97.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 97.4

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

   5. Applied rewrites 97.4%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -116000000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right), a, -1\right), a, 1\right) + 1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.8e+102)
   (pow (+ (fma (fma (fma -0.16666666666666666 a 0.5) a -1.0) a 1.0) 1.0) -1.0)
   (pow (* (* (fma 0.16666666666666666 b 0.5) b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.8e+102) {
		tmp = pow((fma(fma(fma(-0.16666666666666666, a, 0.5), a, -1.0), a, 1.0) + 1.0), -1.0);
	} else {
		tmp = pow(((fma(0.16666666666666666, b, 0.5) * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.8e+102)
		tmp = Float64(fma(fma(fma(-0.16666666666666666, a, 0.5), a, -1.0), a, 1.0) + 1.0) ^ -1.0;
	else
		tmp = Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.8e+102], N[Power[N[(N[(N[(N[(-0.16666666666666666 * a + 0.5), $MachinePrecision] * a + -1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.7999999999999997e102

   1. Initial program 98.1%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 98.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lft-mult-inverse N/A

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

      6. lower-+.f64 N/A

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

      7. rec-exp N/A

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

      8. neg-mul-1 N/A

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

      9. lower-exp.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-neg.f64 73.8

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

   7. Applied rewrites 73.8%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\left(1 + a \cdot \left(a \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot a\right) - 1\right)\right) + 1} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.6%

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

   if 7.7999999999999997e102 < 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

      \[\leadsto \frac{1}{{b}^{3} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{2} \cdot \frac{1}{b}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right), a, -1\right), a, 1\right) + 1\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right), a, -1\right), a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.8e+102)
   (pow (fma (fma (fma -0.16666666666666666 a 0.5) a -1.0) a 2.0) -1.0)
   (pow (* (* (fma 0.16666666666666666 b 0.5) b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.8e+102) {
		tmp = pow(fma(fma(fma(-0.16666666666666666, a, 0.5), a, -1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(((fma(0.16666666666666666, b, 0.5) * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.8e+102)
		tmp = fma(fma(fma(-0.16666666666666666, a, 0.5), a, -1.0), a, 2.0) ^ -1.0;
	else
		tmp = Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.8e+102], N[Power[N[(N[(N[(-0.16666666666666666 * a + 0.5), $MachinePrecision] * a + -1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.7999999999999997e102

   1. Initial program 98.1%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 98.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lft-mult-inverse N/A

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

      6. lower-+.f64 N/A

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

      7. rec-exp N/A

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

      8. neg-mul-1 N/A

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

      9. lower-exp.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-neg.f64 73.8

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

   7. Applied rewrites 73.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 65.6%

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

   if 7.7999999999999997e102 < 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

      \[\leadsto \frac{1}{{b}^{3} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{2} \cdot \frac{1}{b}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right), a, -1\right), a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 7.8e+102)
   (/ (exp a) 2.0)
   (pow (* (* (fma 0.16666666666666666 b 0.5) b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 7.8e+102) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = pow(((fma(0.16666666666666666, b, 0.5) * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 7.8e+102)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 7.8e+102], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.7999999999999997e102

   1. Initial program 98.1%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 72.8

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 71.2%

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

   if 7.7999999999999997e102 < 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

      \[\leadsto \frac{1}{{b}^{3} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{2} \cdot \frac{1}{b}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.7e+102)
   (pow (fma (fma 0.5 a -1.0) a 2.0) -1.0)
   (pow (* (* (fma 0.16666666666666666 b 0.5) b) b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.7e+102) {
		tmp = pow(fma(fma(0.5, a, -1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(((fma(0.16666666666666666, b, 0.5) * b) * b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.7e+102)
		tmp = fma(fma(0.5, a, -1.0), a, 2.0) ^ -1.0;
	else
		tmp = Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.7e+102], N[Power[N[(N[(0.5 * a + -1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.70000000000000023e102

   1. Initial program 98.1%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 98.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lft-mult-inverse N/A

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

      6. lower-+.f64 N/A

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

      7. rec-exp N/A

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

      8. neg-mul-1 N/A

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

      9. lower-exp.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-neg.f64 73.8

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

   7. Applied rewrites 73.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 61.6%

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

   if 3.70000000000000023e102 < 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

      \[\leadsto \frac{1}{{b}^{3} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{2} \cdot \frac{1}{b}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.7 \cdot 10^{+102}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{+143}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot b, b, b\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 6.2e+143)
   (pow (fma (fma 0.5 a -1.0) a 2.0) -1.0)
   (pow (fma (* 0.5 b) b b) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 6.2e+143) {
		tmp = pow(fma(fma(0.5, a, -1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(fma((0.5 * b), b, b), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 6.2e+143)
		tmp = fma(fma(0.5, a, -1.0), a, 2.0) ^ -1.0;
	else
		tmp = fma(Float64(0.5 * b), b, b) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 6.2e+143], N[Power[N[(N[(0.5 * a + -1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.5 * b), $MachinePrecision] * b + b), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.1999999999999998e143

   1. Initial program 98.1%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 98.1

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lft-mult-inverse N/A

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

      6. lower-+.f64 N/A

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

      7. rec-exp N/A

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

      8. neg-mul-1 N/A

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

      9. lower-exp.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-neg.f64 72.1

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

   7. Applied rewrites 72.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 60.3%

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

   if 6.1999999999999998e143 < 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 90.8%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 90.8%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{+143}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot b, b, b\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{-31}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.5e-31)
   (pow (- 2.0 a) -1.0)
   (pow (fma (fma 0.5 b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.5e-31) {
		tmp = pow((2.0 - a), -1.0);
	} 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 (b <= 2.5e-31)
		tmp = Float64(2.0 - a) ^ -1.0;
	else
		tmp = fma(fma(0.5, b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.5e-31], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $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 b < 2.5e-31

   1. Initial program 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 98.3

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lft-mult-inverse N/A

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

      6. lower-+.f64 N/A

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

      7. rec-exp N/A

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

      8. neg-mul-1 N/A

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

      9. lower-exp.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-neg.f64 77.5

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

   7. Applied rewrites 77.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 52.3%

       \[\leadsto \frac{1}{2 - \color{blue}{a}} \]

   if 2.5e-31 < b

   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 98.7

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

   5. Applied rewrites 98.7%

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

   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 58.1%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{-31}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.65 \cdot 10^{+78}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot b\right) \cdot 0.5\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.65e+78) (pow (- 2.0 a) -1.0) (pow (* (* b b) 0.5) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.65e+78) {
		tmp = pow((2.0 - a), -1.0);
	} else {
		tmp = pow(((b * b) * 0.5), -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 (b <= 3.65d+78) then
        tmp = (2.0d0 - a) ** (-1.0d0)
    else
        tmp = ((b * b) * 0.5d0) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.65e+78) {
		tmp = Math.pow((2.0 - a), -1.0);
	} else {
		tmp = Math.pow(((b * b) * 0.5), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.65e+78:
		tmp = math.pow((2.0 - a), -1.0)
	else:
		tmp = math.pow(((b * b) * 0.5), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.65e+78)
		tmp = Float64(2.0 - a) ^ -1.0;
	else
		tmp = Float64(Float64(b * b) * 0.5) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.65e+78)
		tmp = (2.0 - a) ^ -1.0;
	else
		tmp = ((b * b) * 0.5) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.65e+78], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.65e78

   1. Initial program 98.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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. lower-/.f64 N/A

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

      4. lower-/.f64 98.5

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in b around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lft-mult-inverse N/A

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

      6. lower-+.f64 N/A

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

      7. rec-exp N/A

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

      8. neg-mul-1 N/A

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

      9. lower-exp.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-neg.f64 75.0

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

   7. Applied rewrites 75.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 49.4%

       \[\leadsto \frac{1}{2 - \color{blue}{a}} \]

   if 3.65e78 < b

   1. Initial program 98.1%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 69.7%

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

   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 69.7%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.65 \cdot 10^{+78}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot b\right) \cdot 0.5\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ {\left(2 - a\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (pow (- 2.0 a) -1.0))

C

double code(double a, double b) {
	return pow((2.0 - a), -1.0);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (2.0d0 - a) ** (-1.0d0)
end function

Java

public static double code(double a, double b) {
	return Math.pow((2.0 - a), -1.0);
}

Python

def code(a, b):
	return math.pow((2.0 - a), -1.0)

Julia

function code(a, b)
	return Float64(2.0 - a) ^ -1.0
end

MATLAB

function tmp = code(a, b)
	tmp = (2.0 - a) ^ -1.0;
end

Wolfram

code[a_, b_] := N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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. lower-/.f64 N/A

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

   4. lower-/.f64 98.4

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in b around 0

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

   1. *-lft-identity N/A

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

   2. associate-*l/ N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lft-mult-inverse N/A

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

   6. lower-+.f64 N/A

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

   7. rec-exp N/A

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

   8. neg-mul-1 N/A

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

   9. lower-exp.f64 N/A

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

   10. neg-mul-1 N/A

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

   11. lower-neg.f64 66.5

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

7. Applied rewrites 66.5%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 40.2%

    \[\leadsto \frac{1}{2 - \color{blue}{a}} \]

11. Final simplification 40.2%

    \[\leadsto {\left(2 - a\right)}^{-1} \]
12. Add Preprocessing

Alternative 11: 39.8% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.25, a, 0.5\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma 0.25 a 0.5))

C

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

Julia

function code(a, b)
	return fma(0.25, a, 0.5)
end

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{a \cdot \left(\left(a \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{1 + e^{b}} + \frac{1}{{\left(1 + e^{b}\right)}^{3}}\right) - \frac{3}{2} \cdot \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right) + \frac{1}{1 + e^{b}}\right) - \frac{1}{{\left(1 + e^{b}\right)}^{2}}\right) + \frac{1}{1 + e^{b}}} \]

5. Applied rewrites 81.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), \frac{1}{e^{b} + 1}, \frac{a}{{\left(e^{b} + 1\right)}^{3}} + \mathsf{fma}\left(a, 1.5, 1\right) \cdot \frac{-1}{{\left(e^{b} + 1\right)}^{2}}\right), a, \frac{1}{e^{b} + 1}\right)} \]

6. Taylor expanded in b around 0

   \[\leadsto \frac{1}{2} + \color{blue}{a \cdot \left(\frac{1}{2} + \left(\frac{-1}{4} \cdot \left(1 + \frac{3}{2} \cdot a\right) + \left(\frac{1}{8} \cdot a + \frac{1}{4} \cdot a\right)\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 39.5%

   \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{a}, 0.5\right) \]
9. Add Preprocessing

Alternative 12: 39.7% 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 98.4%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 81.0

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

5. Applied rewrites 81.0%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 39.2%

   \[\leadsto 0.5 \]
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 2024308 
(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))))