Quotient of sum of exps

Percentage Accurate: 98.9% → 99.1%

Time: 5.8s

Alternatives: 11

Speedup: 1.2×

• could not determine a ground truth

  1. a = 3.3781185945407653e+28
  2. b = -2.3273397181949038e+20

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: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)\\ \mathbf{if}\;a \leq -4.5:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_0 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 a 1.0) a 1.0)))
   (if (<= a -4.5) (/ (exp a) (+ (exp a) 1.0)) (/ t_0 (+ t_0 (exp b))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(0.5, a, 1.0), a, 1.0);
	double tmp;
	if (a <= -4.5) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = t_0 / (t_0 + exp(b));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(0.5, a, 1.0), a, 1.0)
	tmp = 0.0
	if (a <= -4.5)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(t_0 / Float64(t_0 + exp(b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision]}, If[LessEqual[a, -4.5], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(t$95$0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5

   1. Initial program 98.7%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 98.5%

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

   if -4.5 < a

   1. Initial program 99.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 99.4%

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

Alternative 2: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -4.5)
   (/ (exp a) (+ (exp a) 1.0))
   (/ (+ 1.0 a) (+ (+ 1.0 a) (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -4.5) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + exp(b));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (a <= (-4.5d0)) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = (1.0d0 + a) / ((1.0d0 + a) + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -4.5) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -4.5:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = (1.0 + a) / ((1.0 + a) + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -4.5)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -4.5)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = (1.0 + a) / ((1.0 + a) + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -4.5], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.5

   1. Initial program 98.7%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 98.5%

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

   if -4.5 < a

   1. Initial program 99.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 99.1%

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

Alternative 3: 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 98.9%

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

Alternative 4: 90.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9e+146)
   (/ 1.0 (* 0.5 (* a a)))
   (/ (+ 1.0 a) (+ (+ 1.0 a) (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9e+146) {
		tmp = 1.0 / (0.5 * (a * a));
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + exp(b));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (a <= (-9d+146)) then
        tmp = 1.0d0 / (0.5d0 * (a * a))
    else
        tmp = (1.0d0 + a) / ((1.0d0 + a) + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -9e+146) {
		tmp = 1.0 / (0.5 * (a * a));
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9e+146:
		tmp = 1.0 / (0.5 * (a * a))
	else:
		tmp = (1.0 + a) / ((1.0 + a) + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9e+146)
		tmp = Float64(1.0 / Float64(0.5 * Float64(a * a)));
	else
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9e+146)
		tmp = 1.0 / (0.5 * (a * a));
	else
		tmp = (1.0 + a) / ((1.0 + a) + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9e+146], N[(1.0 / N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.00000000000000051e146

   1. Initial program 99.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 34.8%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 34.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 3.9%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 97.2%

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

   14. Taylor expanded in a around inf

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

   16. Applied rewrites 95.6%

       \[\leadsto \frac{1}{0.5 \cdot \color{blue}{\left(a \cdot a\right)}} \]

   if -9.00000000000000051e146 < a

   1. Initial program 98.8%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 89.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 90.1%

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

Alternative 5: 90.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -9e+146) (/ 1.0 (* 0.5 (* a a))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9e+146) {
		tmp = 1.0 / (0.5 * (a * a));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (a <= (-9d+146)) then
        tmp = 1.0d0 / (0.5d0 * (a * a))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -9e+146) {
		tmp = 1.0 / (0.5 * (a * a));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9e+146:
		tmp = 1.0 / (0.5 * (a * a))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9e+146)
		tmp = Float64(1.0 / Float64(0.5 * Float64(a * a)));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9e+146)
		tmp = 1.0 / (0.5 * (a * a));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -9e+146], N[(1.0 / N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.00000000000000051e146

   1. Initial program 99.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 34.8%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 34.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 3.9%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 97.2%

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

   14. Taylor expanded in a around inf

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

   16. Applied rewrites 95.6%

       \[\leadsto \frac{1}{0.5 \cdot \color{blue}{\left(a \cdot a\right)}} \]

   if -9.00000000000000051e146 < a

   1. Initial program 98.8%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 89.6%

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

Alternative 6: 90.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \mathsf{fma}\left(0.5 \cdot a, a, e^{b}\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 1.0 (fma (* 0.5 a) a (exp b)))))

C

double code(double a, double b) {
	return 1.0 / (1.0 + fma((0.5 * a), a, exp(b)));
}

Julia

function code(a, b)
	return Float64(1.0 / Float64(1.0 + fma(Float64(0.5 * a), a, exp(b))))
end

Wolfram

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in a around 0

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

4. Applied rewrites 81.1%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 82.0%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 37.8%

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

11. Taylor expanded in a around 0

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

13. Applied rewrites 90.0%

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

14. Taylor expanded in a around inf

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

16. Applied rewrites 90.2%

    \[\leadsto \frac{1}{1 + \mathsf{fma}\left(0.5 \cdot a, a, e^{b}\right)} \]
17. Add Preprocessing

Alternative 7: 68.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{a}}{e^{a} + e^{b}}\\ t_1 := \frac{1}{1 + 0.5 \cdot \left(a \cdot a\right)}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b, 0.5\right) + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (exp a) (+ (exp a) (exp b))))
        (t_1 (/ 1.0 (+ 1.0 (* 0.5 (* a a))))))
   (if (<= t_0 2e-12)
     t_1
     (if (<= t_0 0.6) (+ (fma -0.25 b 0.5) (* a 0.25)) t_1))))

C

double code(double a, double b) {
	double t_0 = exp(a) / (exp(a) + exp(b));
	double t_1 = 1.0 / (1.0 + (0.5 * (a * a)));
	double tmp;
	if (t_0 <= 2e-12) {
		tmp = t_1;
	} else if (t_0 <= 0.6) {
		tmp = fma(-0.25, b, 0.5) + (a * 0.25);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(exp(a) / Float64(exp(a) + exp(b)))
	t_1 = Float64(1.0 / Float64(1.0 + Float64(0.5 * Float64(a * a))))
	tmp = 0.0
	if (t_0 <= 2e-12)
		tmp = t_1;
	elseif (t_0 <= 0.6)
		tmp = Float64(fma(-0.25, b, 0.5) + Float64(a * 0.25));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-12], t$95$1, If[LessEqual[t$95$0, 0.6], N[(N[(-0.25 * b + 0.5), $MachinePrecision] + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 1.99999999999999996e-12 or 0.599999999999999978 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 98.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 72.9%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 4.0%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 85.4%

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

   14. Taylor expanded in a around inf

       \[\leadsto \frac{1}{1 + \frac{1}{2} \cdot \color{blue}{{a}^{2}}} \]

   16. Applied rewrites 51.5%

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

   if 1.99999999999999996e-12 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.599999999999999978

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 98.8%

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

Alternative 8: 53.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.82:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(a \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, 0.25, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.82) (/ 1.0 (* 0.5 (* a a))) (fma a 0.25 0.5)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.82) {
		tmp = 1.0 / (0.5 * (a * a));
	} else {
		tmp = fma(a, 0.25, 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.82)
		tmp = Float64(1.0 / Float64(0.5 * Float64(a * a)));
	else
		tmp = fma(a, 0.25, 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.82], N[(1.0 / N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * 0.25 + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.82000000000000006

   1. Initial program 98.7%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 34.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 36.0%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 3.9%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 68.4%

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

   14. Taylor expanded in a around inf

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

   16. Applied rewrites 52.8%

       \[\leadsto \frac{1}{0.5 \cdot \color{blue}{\left(a \cdot a\right)}} \]

   if -1.82000000000000006 < a

   1. Initial program 99.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\frac{1}{4} \cdot a} \]

   7. Applied rewrites 53.6%

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

Alternative 9: 40.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 + a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 2.0 a)))

C

double code(double a, double b) {
	return 1.0 / (2.0 + 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 / (2.0d0 + a)
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 + a);
}

Python

def code(a, b):
	return 1.0 / (2.0 + a)

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 + a))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 + a);
end

Wolfram

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in a around 0

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

4. Applied rewrites 81.1%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 82.0%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 37.8%

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

11. Taylor expanded in a around 0

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

13. Applied rewrites 81.5%

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

14. Taylor expanded in b around 0

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

16. Applied rewrites 40.4%

    \[\leadsto \frac{1}{2 + \color{blue}{a}} \]
17. Add Preprocessing

Alternative 10: 40.1% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in a around 0

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

4. Applied rewrites 82.3%

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

5. Taylor expanded in b around 0

   \[\leadsto \frac{1}{2} + \color{blue}{\frac{1}{4} \cdot a} \]

7. Applied rewrites 40.1%

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

Alternative 11: 39.9% accurate, 37.5× 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 98.9%

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

2. Taylor expanded in a around 0

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

4. Applied rewrites 82.3%

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

5. Taylor expanded in b around 0

   \[\leadsto \frac{1}{2} + \color{blue}{\frac{1}{4} \cdot a} \]

7. Applied rewrites 40.1%

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

8. Taylor expanded in a around 0

   \[\leadsto \frac{1}{2} \]

10. Applied rewrites 39.9%

    \[\leadsto 0.5 \]
11. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.9× 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 2025132 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :alt
  (! :herbie-platform c (/ 1 (+ 1 (exp (- b a)))))

  (/ (exp a) (+ (exp a) (exp b))))