Quotient of sum of exps

Percentage Accurate: 98.9% → 98.5%

Time: 6.2s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. a = 8.063026274235704e+76
  2. b = -1.64902330551805e-45

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: 98.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = exp(a) / (1.0d0 + 1.0d0)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 99.3%

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

Alternative 2: 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 98.8%

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

3. Final simplification 98.8%

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

Alternative 3: 94.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right) + 1}\\ \mathbf{elif}\;a \leq -620000000:\\ \;\;\;\;{b}^{5} \cdot -0.0020833333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.2e+103)
   (/ 1.0 (+ (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 1.0) 1.0))
   (if (<= a -620000000.0)
     (* (pow b 5.0) -0.0020833333333333333)
     (/ 1.0 (+ 1.0 (exp b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.2e+103) {
		tmp = 1.0 / (fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0) + 1.0);
	} else if (a <= -620000000.0) {
		tmp = pow(b, 5.0) * -0.0020833333333333333;
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7.2e+103)
		tmp = Float64(1.0 / Float64(fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0) + 1.0));
	elseif (a <= -620000000.0)
		tmp = Float64((b ^ 5.0) * -0.0020833333333333333);
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7.2e+103], N[(1.0 / N[(N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -620000000.0], N[(N[Power[b, 5.0], $MachinePrecision] * -0.0020833333333333333), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.20000000000000033e103

   1. Initial program 97.4%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 100.0%

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

   if -7.20000000000000033e103 < a < -6.2e8

   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 15.8

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

   5. Applied rewrites 15.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0020833333333333333, b \cdot b, 0.020833333333333332\right), b \cdot b, -0.25\right), \color{blue}{b}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{-1}{480} \cdot {b}^{\color{blue}{5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 74.7%

       \[\leadsto {b}^{5} \cdot -0.0020833333333333333 \]

   if -6.2e8 < a

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right) + 1}\\ \mathbf{elif}\;a \leq -620000000:\\ \;\;\;\;{b}^{5} \cdot -0.0020833333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right) + 1\\ \mathbf{if}\;b \leq 2.85 \cdot 10^{+29}:\\ \;\;\;\;\frac{1 + a}{t\_0}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot 0.5}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 1.0) 1.0)))
   (if (<= b 2.85e+29)
     (/ (+ 1.0 a) t_0)
     (if (<= b 1.02e+103)
       (/ (* (* a a) 0.5) t_0)
       (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0) + 1.0;
	double tmp;
	if (b <= 2.85e+29) {
		tmp = (1.0 + a) / t_0;
	} else if (b <= 1.02e+103) {
		tmp = ((a * a) * 0.5) / t_0;
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0) + 1.0)
	tmp = 0.0
	if (b <= 2.85e+29)
		tmp = Float64(Float64(1.0 + a) / t_0);
	elseif (b <= 1.02e+103)
		tmp = Float64(Float64(Float64(a * a) * 0.5) / t_0);
	else
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[b, 2.85e+29], N[(N[(1.0 + a), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[b, 1.02e+103], N[(N[(N[(a * a), $MachinePrecision] * 0.5), $MachinePrecision] / t$95$0), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.85e29

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 80.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 80.3

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

   8. Applied rewrites 80.3%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 71.3

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

   11. Applied rewrites 71.3%

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

   if 2.85e29 < b < 1.01999999999999991e103

   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 10.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 10.0

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

   8. Applied rewrites 10.0%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 3.2

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

   11. Applied rewrites 3.2%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 66.2%

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

   if 1.01999999999999991e103 < b

   1. Initial program 97.5%

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

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

Alternative 5: 68.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + a}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e+34)
   (/
    (+ 1.0 a)
    (+ (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 1.0) 1.0))
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.1e+34) {
		tmp = (1.0 + a) / (fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0) + 1.0);
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.1e+34)
		tmp = Float64(Float64(1.0 + a) / Float64(fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0) + 1.0));
	else
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.1e+34], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.10000000000000017e34

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 80.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 80.1

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

   8. Applied rewrites 80.1%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 70.6

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

   11. Applied rewrites 70.6%

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

   if 2.10000000000000017e34 < 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

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

   8. Applied rewrites 78.5%

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

Alternative 6: 68.4% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.1e+34)
   (/ 1.0 (+ (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 1.0) 1.0))
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.1e+34) {
		tmp = 1.0 / (fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0) + 1.0);
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.1e+34)
		tmp = Float64(1.0 / Float64(fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0) + 1.0));
	else
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.1e+34], N[(1.0 / N[(N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.10000000000000017e34

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 80.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 80.1

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

   8. Applied rewrites 80.1%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 70.6%

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

   if 2.10000000000000017e34 < 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

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

   8. Applied rewrites 78.5%

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

Alternative 7: 56.4% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -2

   1. Initial program 97.7%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 18.4%

      \[\leadsto 0.5 \]

   if -2 < b

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 77.7

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

   5. Applied rewrites 77.7%

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

   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 69.6%

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

Alternative 8: 52.1% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -2

   1. Initial program 97.7%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 18.4%

      \[\leadsto 0.5 \]

   if -2 < b

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 77.7

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

   5. Applied rewrites 77.7%

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

   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 64.0%

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

Alternative 9: 38.4% 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 98.8%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 81.1

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

5. Applied rewrites 81.1%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 45.0%

   \[\leadsto 0.5 \]
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 2024283 
(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))))