Quotient of sum of exps

Percentage Accurate: 98.9% → 99.4%

Time: 7.2s

Alternatives: 14

Speedup: 1.4×

• could not determine a ground truth

  1. a = 1.556072048295322e+168
  2. b = -1.5609089357755705e-162

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right)\\ \mathbf{if}\;a \leq -330:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, a, 1\right)}{\mathsf{fma}\left(t\_0, a, 1 + e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma 0.16666666666666666 a 0.5) a 1.0)))
   (if (<= a -330.0)
     (/ (exp a) (+ (exp a) 1.0))
     (/ (fma t_0 a 1.0) (fma t_0 a (+ 1.0 (exp b)))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(0.16666666666666666, a, 0.5), a, 1.0);
	double tmp;
	if (a <= -330.0) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = fma(t_0, a, 1.0) / fma(t_0, a, (1.0 + exp(b)));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(0.16666666666666666, a, 0.5), a, 1.0)
	tmp = 0.0
	if (a <= -330.0)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(fma(t_0, a, 1.0) / fma(t_0, a, Float64(1.0 + exp(b))));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision]}, If[LessEqual[a, -330.0], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * a + 1.0), $MachinePrecision] / N[(t$95$0 * a + N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -330

   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 -330 < a

   1. Initial program 98.4%

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 96.6%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 99.6%

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

Alternative 2: 58.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), a, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.0)
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))
   (fma (fma -0.020833333333333332 (* a a) 0.25) a 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.0) {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	} else {
		tmp = fma(fma(-0.020833333333333332, (a * a), 0.25), a, 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.0)
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	else
		tmp = fma(fma(-0.020833333333333332, Float64(a * a), 0.25), a, 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(a * a), $MachinePrecision] + 0.25), $MachinePrecision] * a + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.0

   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

   5. Applied rewrites 57.4%

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

   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 41.6%

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

   if 0.0 < (/.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 b around 0

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \color{blue}{a \cdot \left(\left(\frac{1}{2} \cdot \left(1 + -1 \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right) + a \cdot \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} + -1 \cdot \left(\frac{1}{4} \cdot b - \left(\frac{1}{8} \cdot b + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right)\right)\right) + a \cdot \left(\frac{1}{2} \cdot \left(\frac{1}{6} + -1 \cdot \left(\frac{1}{12} \cdot b - \left(\frac{1}{24} \cdot b + \left(\frac{1}{4} \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot b - \left(\frac{1}{8} \cdot b + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right)\right)\right)\right)\right)\right) - \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \left(\frac{1}{4} \cdot \left(\frac{1}{2} \cdot \left(1 + -1 \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(\frac{1}{2} + -1 \cdot \left(\frac{1}{4} \cdot b - \left(\frac{1}{8} \cdot b + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right)\right)\right) - \left(\frac{1}{8} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(1 + -1 \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right)\right)\right)\right)\right) - \left(\frac{1}{8} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(1 + -1 \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)} \]

   8. Applied rewrites 66.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, b, 0.16666666666666666\right) \cdot 0.5 - \mathsf{fma}\left(0.125 - \mathsf{fma}\left(-0.0625, b, 0.125\right), 0.5, 0.10416666666666667 - 0.020833333333333332 \cdot b\right), a, 0.125 - \mathsf{fma}\left(-0.0625, b, 0.125\right)\right), a, 0.25\right), \color{blue}{a}, \mathsf{fma}\left(-0.25, b, 0.5\right)\right) \]

   9. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{48}, a, \frac{1}{8} - \mathsf{fma}\left(\frac{-1}{16}, b, \frac{1}{8}\right)\right), a, \frac{1}{4}\right), a, \mathsf{fma}\left(\frac{-1}{4}, b, \frac{1}{2}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 66.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.020833333333333332, a, 0.125 - \mathsf{fma}\left(-0.0625, b, 0.125\right)\right), a, 0.25\right), a, \mathsf{fma}\left(-0.25, b, 0.5\right)\right) \]

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 70.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), a, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 53.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5 \cdot b, b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), a, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.0)
   (/ 1.0 (fma (* 0.5 b) b 2.0))
   (fma (fma -0.020833333333333332 (* a a) 0.25) a 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.0) {
		tmp = 1.0 / fma((0.5 * b), b, 2.0);
	} else {
		tmp = fma(fma(-0.020833333333333332, (a * a), 0.25), a, 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.0)
		tmp = Float64(1.0 / fma(Float64(0.5 * b), b, 2.0));
	else
		tmp = fma(fma(-0.020833333333333332, Float64(a * a), 0.25), a, 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(N[(0.5 * b), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(a * a), $MachinePrecision] + 0.25), $MachinePrecision] * a + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.0

   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

   5. Applied rewrites 57.4%

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

   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 29.3%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 29.3%

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

   if 0.0 < (/.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 b around 0

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \color{blue}{a \cdot \left(\left(\frac{1}{2} \cdot \left(1 + -1 \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right) + a \cdot \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} + -1 \cdot \left(\frac{1}{4} \cdot b - \left(\frac{1}{8} \cdot b + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right)\right)\right) + a \cdot \left(\frac{1}{2} \cdot \left(\frac{1}{6} + -1 \cdot \left(\frac{1}{12} \cdot b - \left(\frac{1}{24} \cdot b + \left(\frac{1}{4} \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot b - \left(\frac{1}{8} \cdot b + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right)\right)\right)\right)\right)\right) - \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \left(\frac{1}{4} \cdot \left(\frac{1}{2} \cdot \left(1 + -1 \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(\frac{1}{2} + -1 \cdot \left(\frac{1}{4} \cdot b - \left(\frac{1}{8} \cdot b + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right)\right)\right) - \left(\frac{1}{8} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(1 + -1 \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right)\right)\right)\right)\right) - \left(\frac{1}{8} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(1 + -1 \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)} \]

   8. Applied rewrites 66.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, b, 0.16666666666666666\right) \cdot 0.5 - \mathsf{fma}\left(0.125 - \mathsf{fma}\left(-0.0625, b, 0.125\right), 0.5, 0.10416666666666667 - 0.020833333333333332 \cdot b\right), a, 0.125 - \mathsf{fma}\left(-0.0625, b, 0.125\right)\right), a, 0.25\right), \color{blue}{a}, \mathsf{fma}\left(-0.25, b, 0.5\right)\right) \]

   9. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{48}, a, \frac{1}{8} - \mathsf{fma}\left(\frac{-1}{16}, b, \frac{1}{8}\right)\right), a, \frac{1}{4}\right), a, \mathsf{fma}\left(\frac{-1}{4}, b, \frac{1}{2}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 66.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.020833333333333332, a, 0.125 - \mathsf{fma}\left(-0.0625, b, 0.125\right)\right), a, 0.25\right), a, \mathsf{fma}\left(-0.25, b, 0.5\right)\right) \]

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 70.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), a, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 53.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5 \cdot b, b, b\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), a, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.0)
   (/ 1.0 (fma (* 0.5 b) b b))
   (fma (fma -0.020833333333333332 (* a a) 0.25) a 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 0.0) {
		tmp = 1.0 / fma((0.5 * b), b, b);
	} else {
		tmp = fma(fma(-0.020833333333333332, (a * a), 0.25), a, 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 0.0)
		tmp = Float64(1.0 / fma(Float64(0.5 * b), b, b));
	else
		tmp = fma(fma(-0.020833333333333332, Float64(a * a), 0.25), a, 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(N[(0.5 * b), $MachinePrecision] * b + b), $MachinePrecision]), $MachinePrecision], N[(N[(-0.020833333333333332 * N[(a * a), $MachinePrecision] + 0.25), $MachinePrecision] * a + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.0

   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

   5. Applied rewrites 57.4%

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

   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 29.3%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 28.9%

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

   if 0.0 < (/.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 b around 0

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \color{blue}{a \cdot \left(\left(\frac{1}{2} \cdot \left(1 + -1 \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right) + a \cdot \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} + -1 \cdot \left(\frac{1}{4} \cdot b - \left(\frac{1}{8} \cdot b + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right)\right)\right) + a \cdot \left(\frac{1}{2} \cdot \left(\frac{1}{6} + -1 \cdot \left(\frac{1}{12} \cdot b - \left(\frac{1}{24} \cdot b + \left(\frac{1}{4} \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot b - \left(\frac{1}{8} \cdot b + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right)\right)\right)\right)\right)\right) - \left(\frac{1}{24} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \left(\frac{1}{4} \cdot \left(\frac{1}{2} \cdot \left(1 + -1 \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(\frac{1}{2} + -1 \cdot \left(\frac{1}{4} \cdot b - \left(\frac{1}{8} \cdot b + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right)\right)\right) - \left(\frac{1}{8} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(1 + -1 \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right)\right)\right)\right)\right) - \left(\frac{1}{8} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(1 + -1 \cdot \left(\frac{1}{2} \cdot b - \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)\right)\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)} \]

   8. Applied rewrites 66.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, b, 0.16666666666666666\right) \cdot 0.5 - \mathsf{fma}\left(0.125 - \mathsf{fma}\left(-0.0625, b, 0.125\right), 0.5, 0.10416666666666667 - 0.020833333333333332 \cdot b\right), a, 0.125 - \mathsf{fma}\left(-0.0625, b, 0.125\right)\right), a, 0.25\right), \color{blue}{a}, \mathsf{fma}\left(-0.25, b, 0.5\right)\right) \]

   9. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{48}, a, \frac{1}{8} - \mathsf{fma}\left(\frac{-1}{16}, b, \frac{1}{8}\right)\right), a, \frac{1}{4}\right), a, \mathsf{fma}\left(\frac{-1}{4}, b, \frac{1}{2}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 66.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.020833333333333332, a, 0.125 - \mathsf{fma}\left(-0.0625, b, 0.125\right)\right), a, 0.25\right), a, \mathsf{fma}\left(-0.25, b, 0.5\right)\right) \]

   12. Taylor expanded in b around 0

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

   14. Applied rewrites 70.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.020833333333333332, a \cdot a, 0.25\right), a, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

Alternative 6: 93.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right)\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{+74}:\\ \;\;\;\;\frac{a - -1}{\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right) \cdot a\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, a, 1\right)}{\mathsf{fma}\left(t\_0, a, 1 + e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma 0.16666666666666666 a 0.5) a 1.0)))
   (if (<= a -2.4e+74)
     (/ (- a -1.0) (* (* (fma 0.16666666666666666 a 0.5) a) a))
     (/ (fma t_0 a 1.0) (fma t_0 a (+ 1.0 (exp b)))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(0.16666666666666666, a, 0.5), a, 1.0);
	double tmp;
	if (a <= -2.4e+74) {
		tmp = (a - -1.0) / ((fma(0.16666666666666666, a, 0.5) * a) * a);
	} else {
		tmp = fma(t_0, a, 1.0) / fma(t_0, a, (1.0 + exp(b)));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(0.16666666666666666, a, 0.5), a, 1.0)
	tmp = 0.0
	if (a <= -2.4e+74)
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(fma(0.16666666666666666, a, 0.5) * a) * a));
	else
		tmp = Float64(fma(t_0, a, 1.0) / fma(t_0, a, Float64(1.0 + exp(b))));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision]}, If[LessEqual[a, -2.4e+74], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * a + 1.0), $MachinePrecision] / N[(t$95$0 * a + N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.40000000000000008e74

   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

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 66.7%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 87.4%

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

   if -2.40000000000000008e74 < a

   1. Initial program 98.5%

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 92.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 95.3%

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

Alternative 7: 93.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+74}:\\ \;\;\;\;\frac{a - -1}{\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right) \cdot a\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1 + e^{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e+74)
   (/ (- a -1.0) (* (* (fma 0.16666666666666666 a 0.5) a) a))
   (/ (fma (fma 0.5 a 1.0) a 1.0) (fma (fma 0.5 a 1.0) a (+ 1.0 (exp b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e+74) {
		tmp = (a - -1.0) / ((fma(0.16666666666666666, a, 0.5) * a) * a);
	} else {
		tmp = fma(fma(0.5, a, 1.0), a, 1.0) / fma(fma(0.5, a, 1.0), a, (1.0 + exp(b)));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e+74)
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(fma(0.16666666666666666, a, 0.5) * a) * a));
	else
		tmp = Float64(fma(fma(0.5, a, 1.0), a, 1.0) / fma(fma(0.5, a, 1.0), a, Float64(1.0 + exp(b))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e+74], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] / N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.40000000000000008e74

   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

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 66.7%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 87.4%

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

   if -2.40000000000000008e74 < a

   1. Initial program 98.5%

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 92.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 94.8%

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

Alternative 8: 53.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 2

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 51.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 54.2%

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

   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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 49.7%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 49.7%

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

Alternative 9: 92.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+74}:\\ \;\;\;\;\frac{a - -1}{\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right) \cdot a\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - -1}{\left(e^{b} + a\right) - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e+74)
   (/ (- a -1.0) (* (* (fma 0.16666666666666666 a 0.5) a) a))
   (/ (- a -1.0) (- (+ (exp b) a) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e+74) {
		tmp = (a - -1.0) / ((fma(0.16666666666666666, a, 0.5) * a) * a);
	} else {
		tmp = (a - -1.0) / ((exp(b) + a) - -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e+74)
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(fma(0.16666666666666666, a, 0.5) * a) * a));
	else
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(exp(b) + a) - -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e+74], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(N[Exp[b], $MachinePrecision] + a), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.40000000000000008e74

   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

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 66.7%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 87.4%

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

   if -2.40000000000000008e74 < a

   1. Initial program 98.5%

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 94.2%

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

Alternative 10: 92.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{+74}:\\ \;\;\;\;\frac{a - -1}{\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right) \cdot a\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.4e+74)
   (/ (- a -1.0) (* (* (fma 0.16666666666666666 a 0.5) a) a))
   (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.4e+74) {
		tmp = (a - -1.0) / ((fma(0.16666666666666666, a, 0.5) * a) * a);
	} else {
		tmp = 1.0 / (exp(b) - -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -2.4e+74)
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(fma(0.16666666666666666, a, 0.5) * a) * a));
	else
		tmp = Float64(1.0 / Float64(exp(b) - -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -2.4e+74], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.40000000000000008e74

   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

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 66.7%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 87.4%

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

   if -2.40000000000000008e74 < a

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 93.1%

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

Alternative 11: 67.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.75 \cdot 10^{+49}:\\ \;\;\;\;\frac{a - -1}{\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right) \cdot a\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.75e+49)
   (/ (- a -1.0) (* (* (fma 0.16666666666666666 a 0.5) a) a))
   (/ 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 (a <= -1.75e+49) {
		tmp = (a - -1.0) / ((fma(0.16666666666666666, a, 0.5) * a) * a);
	} 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 (a <= -1.75e+49)
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(fma(0.16666666666666666, a, 0.5) * a) * a));
	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[a, -1.75e+49], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a), $MachinePrecision] * a), $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 a < -1.74999999999999987e49

   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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 64.0%

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

   9. Taylor expanded in a around inf

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

   11. Applied rewrites 81.6%

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

   if -1.74999999999999987e49 < a

   1. Initial program 98.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 94.4%

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

   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 64.7%

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

Alternative 12: 67.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{a - -1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.5e+102)
   (/ (- a -1.0) (fma (fma 0.5 a 1.0) a (+ 1.0 1.0)))
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.5e+102) {
		tmp = (a - -1.0) / fma(fma(0.5, a, 1.0), a, (1.0 + 1.0));
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8.5e+102)
		tmp = Float64(Float64(a - -1.0) / fma(fma(0.5, a, 1.0), a, Float64(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, 8.5e+102], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.4999999999999996e102

   1. Initial program 98.6%

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 85.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 60.7%

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

   if 8.4999999999999996e102 < 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

   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 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 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 98.8%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 65.4%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 39.9%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 42.2%

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

Alternative 14: 39.5% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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

5. Applied rewrites 80.1%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 41.6%

   \[\leadsto 0.5 \]
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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    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 2025021 
(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))))