Quotient of sum of exps

Percentage Accurate: 98.9% → 98.9%

Time: 6.1s

Alternatives: 18

Speedup: 1.4×

• could not determine a ground truth

  1. a = 1.4940183222521086e+152
  2. b = -3.377904276899657e-37

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-129}:\\ \;\;\;\;\frac{e^{a}}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-129)
   (/ (exp a) (+ 1.0 (exp a)))
   (/ (+ 1.0 a) (+ (+ 1.0 a) (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-129) {
		tmp = exp(a) / (1.0 + exp(a));
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + 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) <= 5d-129) then
        tmp = exp(a) / (1.0d0 + exp(a))
    else
        tmp = (1.0d0 + a) / ((1.0d0 + a) + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-129) {
		tmp = Math.exp(a) / (1.0 + Math.exp(a));
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-129:
		tmp = math.exp(a) / (1.0 + math.exp(a))
	else:
		tmp = (1.0 + a) / ((1.0 + a) + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-129)
		tmp = Float64(exp(a) / Float64(1.0 + exp(a)));
	else
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-129)
		tmp = exp(a) / (1.0 + exp(a));
	else
		tmp = (1.0 + a) / ((1.0 + a) + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-129], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 5.00000000000000027e-129

   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%

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

   if 5.00000000000000027e-129 < (exp.f64 a)

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 99.6

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

   8. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-129}:\\ \;\;\;\;\frac{e^{a}}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.49999995:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.027777777777777776, b \cdot b, -0.25\right)}{-0.5}, b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.49999995)
   (/
    1.0
    (fma (fma (/ (fma 0.027777777777777776 (* b b) -0.25) -0.5) b 1.0) b 2.0))
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.49999995) {
		tmp = 1.0 / fma(fma((fma(0.027777777777777776, (b * b), -0.25) / -0.5), b, 1.0), b, 2.0);
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.49999995)
		tmp = Float64(1.0 / fma(fma(Float64(fma(0.027777777777777776, Float64(b * b), -0.25) / -0.5), b, 1.0), b, 2.0));
	else
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.49999995], N[(1.0 / N[(N[(N[(N[(0.027777777777777776 * N[(b * b), $MachinePrecision] + -0.25), $MachinePrecision] / -0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.499999950000000026

   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 62.4

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

   5. Applied rewrites 62.4%

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

   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 48.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), \color{blue}{b}, 2\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 48.2%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.027777777777777776, b \cdot b, -0.25\right)}{\mathsf{fma}\left(0.16666666666666666, b, -0.5\right)}, b, 1\right), b, 2\right)} \]

   11. Taylor expanded in b around 0

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{36}, b \cdot b, \frac{-1}{4}\right)}{\frac{-1}{2}}, b, 1\right), b, 2\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 49.0%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.027777777777777776, b \cdot b, -0.25\right)}{-0.5}, b, 1\right), b, 2\right)} \]

   if 0.499999950000000026 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 68.9%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.49999995:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.027777777777777776, b \cdot b, -0.25\right)}{-0.5}, b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.5000905931675143:\\ \;\;\;\;\frac{1 + a}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) + \left(1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.5000905931675143)
   (/
    (+ 1.0 a)
    (+ (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0) (+ 1.0 a)))
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.5000905931675143) {
		tmp = (1.0 + a) / (fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) + (1.0 + a));
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.5000905931675143)
		tmp = Float64(Float64(1.0 + a) / Float64(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) + Float64(1.0 + a)));
	else
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5000905931675143], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.500090593167514252

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 77.8

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

   8. Applied rewrites 77.8%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lft-mult-inverse N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       11. distribute-lft-in N/A

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

       12. +-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. distribute-lft-in N/A

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

       15. *-commutative N/A

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

       16. *-commutative N/A

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

       17. associate-*l* N/A

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

       18. lft-mult-inverse N/A

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

       19. metadata-eval N/A

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

       20. lower-fma.f64 69.4

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

   11. Applied rewrites 69.4%

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

   if 0.500090593167514252 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 98.8

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 19.1%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.5000905931675143:\\ \;\;\;\;\frac{1 + a}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) + \left(1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.49999995:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.49999995)
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.49999995) {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.49999995)
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	else
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.49999995], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.499999950000000026

   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 62.4

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

   5. Applied rewrites 62.4%

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

   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 48.2%

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

   if 0.499999950000000026 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 68.9%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.49999995:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.49999995:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.49999995)
   (/ 1.0 (fma (fma (* 0.16666666666666666 b) b 1.0) b 2.0))
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.49999995) {
		tmp = 1.0 / fma(fma((0.16666666666666666 * b), b, 1.0), b, 2.0);
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.49999995)
		tmp = Float64(1.0 / fma(fma(Float64(0.16666666666666666 * b), b, 1.0), b, 2.0));
	else
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.49999995], N[(1.0 / N[(N[(N[(0.16666666666666666 * b), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.499999950000000026

   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 62.4

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

   5. Applied rewrites 62.4%

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

   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 48.2%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 47.8%

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

   if 0.499999950000000026 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 68.9%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.49999995:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.49999995:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.49999995)
   (/ 1.0 (fma (fma b 0.5 1.0) b 2.0))
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.49999995) {
		tmp = 1.0 / fma(fma(b, 0.5, 1.0), b, 2.0);
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.49999995)
		tmp = Float64(1.0 / fma(fma(b, 0.5, 1.0), b, 2.0));
	else
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.49999995], N[(1.0 / N[(N[(b * 0.5 + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.499999950000000026

   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 62.4

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

   5. Applied rewrites 62.4%

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

   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 34.3%

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

   if 0.499999950000000026 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 68.9%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.49999995:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% 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 99.6%

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

3. Final simplification 99.6%

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

Alternative 8: 98.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-129}:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-129)
   (/ (exp a) (+ 1.0 1.0))
   (/ (+ 1.0 a) (+ (+ 1.0 a) (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-129) {
		tmp = exp(a) / (1.0 + 1.0);
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + 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) <= 5d-129) then
        tmp = exp(a) / (1.0d0 + 1.0d0)
    else
        tmp = (1.0d0 + a) / ((1.0d0 + a) + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-129) {
		tmp = Math.exp(a) / (1.0 + 1.0);
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-129:
		tmp = math.exp(a) / (1.0 + 1.0)
	else:
		tmp = (1.0 + a) / ((1.0 + a) + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-129)
		tmp = Float64(exp(a) / Float64(1.0 + 1.0));
	else
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-129)
		tmp = exp(a) / (1.0 + 1.0);
	else
		tmp = (1.0 + a) / ((1.0 + a) + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-129], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 5.00000000000000027e-129

   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 98.8%

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

   if 5.00000000000000027e-129 < (exp.f64 a)

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 99.6

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

   8. Applied rewrites 99.6%

      \[\leadsto \frac{1 + a}{\color{blue}{\left(1 + a\right)} + e^{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-129}:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-129) (/ (exp a) (+ 1.0 1.0)) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-129) {
		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) <= 5d-129) 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) <= 5e-129) {
		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) <= 5e-129:
		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) <= 5e-129)
		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) <= 5e-129)
		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], 5e-129], 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) < 5.00000000000000027e-129

   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 98.8%

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

   if 5.00000000000000027e-129 < (exp.f64 a)

   1. Initial program 99.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-129}:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 2:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(0.16666666666666666 \cdot b\right) \cdot \left(b \cdot b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 2.0)
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))
   (/ 1.0 (* (* 0.16666666666666666 b) (* b b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(b) <= 2.0) {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	} else {
		tmp = 1.0 / ((0.16666666666666666 * b) * (b * b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(b) <= 2.0d0) then
        tmp = (1.0d0 + a) / ((1.0d0 + a) + 1.0d0)
    else
        tmp = 1.0d0 / ((0.16666666666666666d0 * b) * (b * b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(b) <= 2.0) {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	} else {
		tmp = 1.0 / ((0.16666666666666666 * b) * (b * b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(b) <= 2.0:
		tmp = (1.0 + a) / ((1.0 + a) + 1.0)
	else:
		tmp = 1.0 / ((0.16666666666666666 * b) * (b * b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(b) <= 2.0)
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	else
		tmp = Float64(1.0 / Float64(Float64(0.16666666666666666 * b) * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(b) <= 2.0)
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	else
		tmp = 1.0 / ((0.16666666666666666 * b) * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 2

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 74.8

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 75.3

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

   8. Applied rewrites 75.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 52.2%

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

   if 2 < (exp.f64 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 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

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

   8. Applied rewrites 75.2%

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

   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 75.2%

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

   12. Taylor expanded in b around inf

       \[\leadsto \frac{1}{\left(\frac{1}{6} \cdot b\right) \cdot \left(b \cdot b\right)} \]
   13. Step-by-step derivation

   14. Applied rewrites 75.2%

       \[\leadsto \frac{1}{\left(0.16666666666666666 \cdot b\right) \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 83.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+41}:\\ \;\;\;\;{b}^{3} \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5e+41)
   (* (pow b 3.0) 0.020833333333333332)
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5e+41) {
		tmp = pow(b, 3.0) * 0.020833333333333332;
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5d+41)) then
        tmp = (b ** 3.0d0) * 0.020833333333333332d0
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5e+41) {
		tmp = Math.pow(b, 3.0) * 0.020833333333333332;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5e+41:
		tmp = math.pow(b, 3.0) * 0.020833333333333332
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5e+41)
		tmp = Float64((b ^ 3.0) * 0.020833333333333332);
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5e+41)
		tmp = (b ^ 3.0) * 0.020833333333333332;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5e+41], N[(N[Power[b, 3.0], $MachinePrecision] * 0.020833333333333332), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.00000000000000022e41

   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 31.3

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 2.7%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 50.5%

       \[\leadsto {b}^{3} \cdot 0.020833333333333332 \]

   if -5.00000000000000022e41 < a

   1. Initial program 99.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+41}:\\ \;\;\;\;{b}^{3} \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+41}:\\ \;\;\;\;{b}^{3} \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) + \left(1 + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.9e+41)
   (* (pow b 3.0) 0.020833333333333332)
   (/
    (+ 1.0 a)
    (+ (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0) (+ 1.0 a)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.9e+41) {
		tmp = pow(b, 3.0) * 0.020833333333333332;
	} else {
		tmp = (1.0 + a) / (fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) + (1.0 + a));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.9e+41)
		tmp = Float64((b ^ 3.0) * 0.020833333333333332);
	else
		tmp = Float64(Float64(1.0 + a) / Float64(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) + Float64(1.0 + a)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.9e+41], N[(N[Power[b, 3.0], $MachinePrecision] * 0.020833333333333332), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.9000000000000001e41

   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 31.3

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 2.7%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 50.5%

       \[\leadsto {b}^{3} \cdot 0.020833333333333332 \]

   if -1.9000000000000001e41 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 95.8

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

   8. Applied rewrites 95.8%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. metadata-eval N/A

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

       7. lft-mult-inverse N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       11. distribute-lft-in N/A

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

       12. +-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. distribute-lft-in N/A

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

       15. *-commutative N/A

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

       16. *-commutative N/A

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

       17. associate-*l* N/A

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

       18. lft-mult-inverse N/A

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

       19. metadata-eval N/A

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

       20. lower-fma.f64 64.5

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

   11. Applied rewrites 64.5%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+41}:\\ \;\;\;\;{b}^{3} \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) + \left(1 + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.3% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.62:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.62)
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))
   (/ 1.0 (* (* b b) (fma 0.16666666666666666 b 0.5)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.62) {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	} else {
		tmp = 1.0 / ((b * b) * fma(0.16666666666666666, b, 0.5));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.62)
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	else
		tmp = Float64(1.0 / Float64(Float64(b * b) * fma(0.16666666666666666, b, 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.62], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b * b), $MachinePrecision] * N[(0.16666666666666666 * b + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6200000000000001

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 74.8

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 75.3

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

   8. Applied rewrites 75.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 52.2%

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

   if 1.6200000000000001 < 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 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

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

   8. Applied rewrites 75.2%

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

   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 75.2%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.62:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.16666666666666666, b, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.2% accurate, 11.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2.0d0) then
        tmp = (1.0d0 + a) / ((1.0d0 + a) + 1.0d0)
    else
        tmp = 1.0d0 / (0.5d0 * (b * b))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-+.f64 74.8

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 75.3

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

   8. Applied rewrites 75.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 52.2%

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

   if 2 < 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 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

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

   8. Applied rewrites 75.2%

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

   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 75.2%

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

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 52.3%

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

Alternative 15: 40.0% accurate, 15.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (1.0 + a) / ((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 + a) / ((1.0d0 + a) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in a around 0

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

   1. lower-+.f64 81.6

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

5. Applied rewrites 81.6%

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

6. Taylor expanded in a around 0

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

   1. lower-+.f64 81.9

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

8. Applied rewrites 81.9%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 38.2%

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

Alternative 16: 39.6% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.375, a, -0.375 \cdot a + 0.25\right), a, 0.5\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (fma (fma 0.375 a (+ (* -0.375 a) 0.25)) a 0.5))

C

double code(double a, double b) {
	return fma(fma(0.375, a, ((-0.375 * a) + 0.25)), a, 0.5);
}

Julia

function code(a, b)
	return fma(fma(0.375, a, Float64(Float64(-0.375 * a) + 0.25)), a, 0.5)
end

Wolfram

code[a_, b_] := N[(N[(0.375 * a + N[(N[(-0.375 * a), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision] * a + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 81.7%

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

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 37.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.375, a, 0.25 + -0.375 \cdot a\right), \color{blue}{a}, 0.5\right) \]

9. Final simplification 37.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.375, a, -0.375 \cdot a + 0.25\right), a, 0.5\right) \]
10. Add Preprocessing

Alternative 17: 39.6% 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 99.6%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 81.7%

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

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 37.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.375, a, 0.25 + -0.375 \cdot a\right), \color{blue}{a}, 0.5\right) \]

9. Taylor expanded in a around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \frac{1}{2}\right) \]
10. Step-by-step derivation

11. Applied rewrites 37.9%

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

Alternative 18: 39.4% 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 99.6%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 81.4

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

5. Applied rewrites 81.4%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 37.6%

   \[\leadsto 0.5 \]
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 2024268 
(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))))