Quotient of sum of exps

Percentage Accurate: 99.1% → 99.4%

Time: 4.3s

Alternatives: 13

Speedup: 0.5×

• could not determine a ground truth

  1. a = 4.710732173224056e+232
  2. b = -1.0856725459332207e+48

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: 99.1% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e^{a}}{e^{a} + e^{b}}\\ \mathbf{if}\;t\_0 \leq 0.72:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(e^{b} - -1\right)}^{-0.5}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ (exp a) (+ (exp a) (exp b)))))
   (if (<= t_0 0.72) t_0 (pow (pow (- (exp b) -1.0) -0.5) 2.0))))

C

double code(double a, double b) {
	double t_0 = exp(a) / (exp(a) + exp(b));
	double tmp;
	if (t_0 <= 0.72) {
		tmp = t_0;
	} else {
		tmp = pow(pow((exp(b) - -1.0), -0.5), 2.0);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = exp(a) / (exp(a) + exp(b))
    if (t_0 <= 0.72d0) then
        tmp = t_0
    else
        tmp = ((exp(b) - (-1.0d0)) ** (-0.5d0)) ** 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = Math.exp(a) / (Math.exp(a) + Math.exp(b));
	double tmp;
	if (t_0 <= 0.72) {
		tmp = t_0;
	} else {
		tmp = Math.pow(Math.pow((Math.exp(b) - -1.0), -0.5), 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = math.exp(a) / (math.exp(a) + math.exp(b))
	tmp = 0
	if t_0 <= 0.72:
		tmp = t_0
	else:
		tmp = math.pow(math.pow((math.exp(b) - -1.0), -0.5), 2.0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(exp(a) / Float64(exp(a) + exp(b)))
	tmp = 0.0
	if (t_0 <= 0.72)
		tmp = t_0;
	else
		tmp = (Float64(exp(b) - -1.0) ^ -0.5) ^ 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = exp(a) / (exp(a) + exp(b));
	tmp = 0.0;
	if (t_0 <= 0.72)
		tmp = t_0;
	else
		tmp = ((exp(b) - -1.0) ^ -0.5) ^ 2.0;
	end
	tmp_2 = 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]}, If[LessEqual[t$95$0, 0.72], t$95$0, N[Power[N[Power[N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision], -0.5], $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.71999999999999997

   1. Initial program 100.0%

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

   if 0.71999999999999997 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 84.0%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto {\left(e^{b} + 1\right)}^{-1} \]

      4. metadata-eval N/A

         \[\leadsto {\left(e^{b} + 1 \cdot 1\right)}^{-1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto {\left(e^{b} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)}^{-1} \]

      6. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1 \cdot 1\right)}^{-1} \]

      7. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      8. lower--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      9. lift-exp.f64 94.8

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   5. Applied rewrites 94.8%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      3. lift-exp.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto {\left({\left(e^{b} - -1\right)}^{\frac{-1}{2}}\right)}^{2} \]

      8. lower-pow.f64 N/A

         \[\leadsto {\left({\left(e^{b} - -1\right)}^{\frac{-1}{2}}\right)}^{2} \]

      9. lift-exp.f64 N/A

         \[\leadsto {\left({\left(e^{b} - -1\right)}^{\frac{-1}{2}}\right)}^{2} \]

      10. lift--.f64 94.8

          \[\leadsto {\left({\left(e^{b} - -1\right)}^{-0.5}\right)}^{2} \]

   7. Applied rewrites 94.8%

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

Alternative 2: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.2e-6) (/ (exp a) (+ (exp a) 1.0)) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.2e-6) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = 1.0 / (exp(b) - -1.0);
	}
	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 <= (-7.2d-6)) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = 1.0d0 / (exp(b) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -7.2e-6) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -7.2e-6:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7.2e-6)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7.2e-6)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7.2e-6], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.19999999999999967e-6

   1. Initial program 98.7%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 -7.19999999999999967e-6 < a

   1. Initial program 96.0%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto {\left(e^{b} + 1\right)}^{-1} \]

      4. metadata-eval N/A

         \[\leadsto {\left(e^{b} + 1 \cdot 1\right)}^{-1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto {\left(e^{b} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)}^{-1} \]

      6. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1 \cdot 1\right)}^{-1} \]

      7. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      8. lower--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      9. lift-exp.f64 97.8

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   5. Applied rewrites 97.8%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      3. lift-exp.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 97.8

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

   7. Applied rewrites 97.8%

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

Alternative 3: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0009:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -0.0009) (/ (exp a) (+ 1.0 1.0)) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -0.0009) {
		tmp = exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (exp(b) - -1.0);
	}
	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 <= (-0.0009d0)) then
        tmp = exp(a) / (1.0d0 + 1.0d0)
    else
        tmp = 1.0d0 / (exp(b) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -0.0009) {
		tmp = Math.exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -0.0009:
		tmp = math.exp(a) / (1.0 + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -0.0009)
		tmp = Float64(exp(a) / Float64(1.0 + 1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -0.0009)
		tmp = exp(a) / (1.0 + 1.0);
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -0.0009], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.9999999999999998e-4

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. sinh-+-cosh-rev 97.3

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

   8. Applied rewrites 97.3%

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

   if -8.9999999999999998e-4 < a

   1. Initial program 96.0%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto {\left(e^{b} + 1\right)}^{-1} \]

      4. metadata-eval N/A

         \[\leadsto {\left(e^{b} + 1 \cdot 1\right)}^{-1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto {\left(e^{b} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)}^{-1} \]

      6. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1 \cdot 1\right)}^{-1} \]

      7. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      8. lower--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      9. lift-exp.f64 97.8

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   5. Applied rewrites 97.8%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      3. lift-exp.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 97.8

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

   7. Applied rewrites 97.8%

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

Alternative 4: 85.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -130000000:\\ \;\;\;\;{b}^{5} \cdot -0.0020833333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -130000000.0)
   (* (pow b 5.0) -0.0020833333333333333)
   (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -130000000.0) {
		tmp = pow(b, 5.0) * -0.0020833333333333333;
	} else {
		tmp = 1.0 / (exp(b) - -1.0);
	}
	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 <= (-130000000.0d0)) then
        tmp = (b ** 5.0d0) * (-0.0020833333333333333d0)
    else
        tmp = 1.0d0 / (exp(b) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -130000000.0) {
		tmp = Math.pow(b, 5.0) * -0.0020833333333333333;
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -130000000.0:
		tmp = math.pow(b, 5.0) * -0.0020833333333333333
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -130000000.0)
		tmp = Float64((b ^ 5.0) * -0.0020833333333333333);
	else
		tmp = Float64(1.0 / Float64(exp(b) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -130000000.0)
		tmp = (b ^ 5.0) * -0.0020833333333333333;
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -130000000.0], N[(N[Power[b, 5.0], $MachinePrecision] * -0.0020833333333333333), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3e8

   1. Initial program 98.7%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto {\left(e^{b} + 1\right)}^{-1} \]

      4. metadata-eval N/A

         \[\leadsto {\left(e^{b} + 1 \cdot 1\right)}^{-1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto {\left(e^{b} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)}^{-1} \]

      6. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1 \cdot 1\right)}^{-1} \]

      7. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      8. lower--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      9. lift-exp.f64 26.7

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   5. Applied rewrites 26.7%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto b \cdot \left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}\right) \cdot b + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}, b, \frac{1}{2}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}, b, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{480} \cdot {b}^{2} + \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, {b}^{2}, \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, b \cdot b, \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, b \cdot b, \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, b \cdot b, \frac{1}{48}\right) \cdot \left(b \cdot b\right) - \frac{1}{4}, b, \frac{1}{2}\right) \]

      12. lower-*.f64 2.8

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

   8. Applied rewrites 2.8%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

          \[\leadsto {b}^{5} \cdot \frac{-1}{480} \]

       2. lower-*.f64 N/A

          \[\leadsto {b}^{5} \cdot \frac{-1}{480} \]

       3. lower-pow.f64 56.2

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

   11. Applied rewrites 56.2%

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

   if -1.3e8 < a

   1. Initial program 96.0%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto {\left(e^{b} + 1\right)}^{-1} \]

      4. metadata-eval N/A

         \[\leadsto {\left(e^{b} + 1 \cdot 1\right)}^{-1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto {\left(e^{b} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)}^{-1} \]

      6. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1 \cdot 1\right)}^{-1} \]

      7. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      8. lower--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      9. lift-exp.f64 96.5

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   5. Applied rewrites 96.5%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      3. lift-exp.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 96.5

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

   7. Applied rewrites 96.5%

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

Alternative 5: 61.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 430:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+102}:\\ \;\;\;\;{a}^{3} \cdot -0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 430.0)
   (fma 0.25 a 0.5)
   (if (<= b 4.8e+102)
     (* (pow a 3.0) -0.020833333333333332)
     (/ 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 <= 430.0) {
		tmp = fma(0.25, a, 0.5);
	} else if (b <= 4.8e+102) {
		tmp = pow(a, 3.0) * -0.020833333333333332;
	} 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 <= 430.0)
		tmp = fma(0.25, a, 0.5);
	elseif (b <= 4.8e+102)
		tmp = Float64((a ^ 3.0) * -0.020833333333333332);
	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, 430.0], N[(0.25 * a + 0.5), $MachinePrecision], If[LessEqual[b, 4.8e+102], N[(N[Power[a, 3.0], $MachinePrecision] * -0.020833333333333332), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 430

   1. Initial program 96.9%

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

   5. Applied rewrites 76.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{e^{a} - -1}\right)} \]

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot a + -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(-1 \cdot b\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\left(\mathsf{neg}\left(2\right)\right) \cdot \log 2}\right) + \frac{1}{2} \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{-2 \cdot \log 2}\right) + \frac{1}{2} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\log 2 \cdot -2}\right) + \frac{1}{2} \]

      10. pow-to-exp N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot {2}^{-2}\right) + \frac{1}{2} \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

      13. lift-neg.f64 45.9

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

   8. Applied rewrites 45.9%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \frac{1}{4} \cdot \color{blue}{a} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 48.2

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

   11. Applied rewrites 48.2%

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

   if 430 < b < 4.79999999999999989e102

   1. Initial program 100.0%

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

   5. Applied rewrites 28.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{e^{a} - -1}\right)} \]

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot a + -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(-1 \cdot b\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\left(\mathsf{neg}\left(2\right)\right) \cdot \log 2}\right) + \frac{1}{2} \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{-2 \cdot \log 2}\right) + \frac{1}{2} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\log 2 \cdot -2}\right) + \frac{1}{2} \]

      10. pow-to-exp N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot {2}^{-2}\right) + \frac{1}{2} \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

      13. lift-neg.f64 2.6

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

   8. Applied rewrites 2.6%

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

   9. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + a \cdot \left(\frac{1}{4} + a \cdot \left(\frac{-1}{48} \cdot a + \frac{1}{4} \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + a \cdot \left(\frac{1}{4} + a \cdot \left(\frac{-1}{48} \cdot a + \frac{1}{4} \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)\right)\right) + \frac{1}{2} \]

       2. lower-+.f64 N/A

          \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + a \cdot \left(\frac{1}{4} + a \cdot \left(\frac{-1}{48} \cdot a + \frac{1}{4} \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)\right)\right) + \frac{1}{2} \]

   11. Applied rewrites 2.5%

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

   12. Taylor expanded in a around inf

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

       1. *-commutative N/A

          \[\leadsto {a}^{3} \cdot \frac{-1}{48} \]

       2. lower-*.f64 N/A

          \[\leadsto {a}^{3} \cdot \frac{-1}{48} \]

       3. lower-pow.f64 58.3

          \[\leadsto {a}^{3} \cdot -0.020833333333333332 \]

   14. Applied rewrites 58.3%

       \[\leadsto {a}^{3} \cdot -0.020833333333333332 \]

   if 4.79999999999999989e102 < b

   1. Initial program 95.0%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto {\left(e^{b} + 1\right)}^{-1} \]

      4. metadata-eval N/A

         \[\leadsto {\left(e^{b} + 1 \cdot 1\right)}^{-1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto {\left(e^{b} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)}^{-1} \]

      6. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1 \cdot 1\right)}^{-1} \]

      7. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      8. lower--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      9. lift-exp.f64 100.0

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   5. Applied rewrites 100.0%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      3. lift-exp.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 97.9

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

   10. Applied rewrites 97.9%

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

Alternative 6: 62.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -700:\\ \;\;\;\;{b}^{5} \cdot -0.0020833333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -700.0)
   (* (pow b 5.0) -0.0020833333333333333)
   (/ 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 <= -700.0) {
		tmp = pow(b, 5.0) * -0.0020833333333333333;
	} 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 <= -700.0)
		tmp = Float64((b ^ 5.0) * -0.0020833333333333333);
	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, -700.0], N[(N[Power[b, 5.0], $MachinePrecision] * -0.0020833333333333333), $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 < -700

   1. Initial program 98.7%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto {\left(e^{b} + 1\right)}^{-1} \]

      4. metadata-eval N/A

         \[\leadsto {\left(e^{b} + 1 \cdot 1\right)}^{-1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto {\left(e^{b} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)}^{-1} \]

      6. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1 \cdot 1\right)}^{-1} \]

      7. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      8. lower--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      9. lift-exp.f64 26.7

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   5. Applied rewrites 26.7%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto b \cdot \left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}\right) \cdot b + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}, b, \frac{1}{2}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}, b, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{480} \cdot {b}^{2} + \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, {b}^{2}, \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, b \cdot b, \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, b \cdot b, \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, b \cdot b, \frac{1}{48}\right) \cdot \left(b \cdot b\right) - \frac{1}{4}, b, \frac{1}{2}\right) \]

      12. lower-*.f64 2.8

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

   8. Applied rewrites 2.8%

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

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

          \[\leadsto {b}^{5} \cdot \frac{-1}{480} \]

       2. lower-*.f64 N/A

          \[\leadsto {b}^{5} \cdot \frac{-1}{480} \]

       3. lower-pow.f64 56.2

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

   11. Applied rewrites 56.2%

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

   if -700 < a

   1. Initial program 96.0%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto {\left(e^{b} + 1\right)}^{-1} \]

      4. metadata-eval N/A

         \[\leadsto {\left(e^{b} + 1 \cdot 1\right)}^{-1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto {\left(e^{b} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)}^{-1} \]

      6. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1 \cdot 1\right)}^{-1} \]

      7. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      8. lower--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      9. lift-exp.f64 96.5

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   5. Applied rewrites 96.5%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      3. lift-exp.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 96.5

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 62.6

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

   10. Applied rewrites 62.6%

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

Alternative 7: 58.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.5e-49)
   (fma 0.25 a 0.5)
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.5e-49) {
		tmp = fma(0.25, a, 0.5);
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.5e-49)
		tmp = fma(0.25, a, 0.5);
	else
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.5e-49], N[(0.25 * a + 0.5), $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 < 1.5e-49

   1. Initial program 96.7%

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{e^{a} - -1}\right)} \]

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot a + -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(-1 \cdot b\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\left(\mathsf{neg}\left(2\right)\right) \cdot \log 2}\right) + \frac{1}{2} \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{-2 \cdot \log 2}\right) + \frac{1}{2} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\log 2 \cdot -2}\right) + \frac{1}{2} \]

      10. pow-to-exp N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot {2}^{-2}\right) + \frac{1}{2} \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

      13. lift-neg.f64 46.4

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

   8. Applied rewrites 46.4%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \frac{1}{4} \cdot \color{blue}{a} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 49.1

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

   11. Applied rewrites 49.1%

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

   if 1.5e-49 < b

   1. Initial program 97.3%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto {\left(e^{b} + 1\right)}^{-1} \]

      4. metadata-eval N/A

         \[\leadsto {\left(e^{b} + 1 \cdot 1\right)}^{-1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto {\left(e^{b} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)}^{-1} \]

      6. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1 \cdot 1\right)}^{-1} \]

      7. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      8. lower--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      9. lift-exp.f64 90.8

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   5. Applied rewrites 90.8%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      3. lift-exp.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 90.8

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

   7. Applied rewrites 90.8%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 60.2

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

   10. Applied rewrites 60.2%

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

Alternative 8: 54.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.5e-49) (fma 0.25 a 0.5) (/ 1.0 (fma (fma 0.5 b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.5e-49) {
		tmp = fma(0.25, a, 0.5);
	} else {
		tmp = 1.0 / fma(fma(0.5, b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.5e-49)
		tmp = fma(0.25, a, 0.5);
	else
		tmp = Float64(1.0 / fma(fma(0.5, b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.5e-49], N[(0.25 * a + 0.5), $MachinePrecision], N[(1.0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5e-49

   1. Initial program 96.7%

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{e^{a} - -1}\right)} \]

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot a + -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(-1 \cdot b\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\left(\mathsf{neg}\left(2\right)\right) \cdot \log 2}\right) + \frac{1}{2} \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{-2 \cdot \log 2}\right) + \frac{1}{2} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\log 2 \cdot -2}\right) + \frac{1}{2} \]

      10. pow-to-exp N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot {2}^{-2}\right) + \frac{1}{2} \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

      13. lift-neg.f64 46.4

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

   8. Applied rewrites 46.4%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \frac{1}{4} \cdot \color{blue}{a} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 49.1

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

   11. Applied rewrites 49.1%

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

   if 1.5e-49 < b

   1. Initial program 97.3%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto {\left(e^{b} + 1\right)}^{-1} \]

      4. metadata-eval N/A

         \[\leadsto {\left(e^{b} + 1 \cdot 1\right)}^{-1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto {\left(e^{b} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)}^{-1} \]

      6. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1 \cdot 1\right)}^{-1} \]

      7. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      8. lower--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      9. lift-exp.f64 90.8

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   5. Applied rewrites 90.8%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      3. lift-exp.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 90.8

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

   7. Applied rewrites 90.8%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 48.5

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

   10. Applied rewrites 48.5%

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

Alternative 9: 53.7% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5 \cdot b, b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.5e-49) (fma 0.25 a 0.5) (/ 1.0 (fma (* 0.5 b) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.5e-49) {
		tmp = fma(0.25, a, 0.5);
	} else {
		tmp = 1.0 / fma((0.5 * b), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.5e-49)
		tmp = fma(0.25, a, 0.5);
	else
		tmp = Float64(1.0 / fma(Float64(0.5 * b), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.5e-49], N[(0.25 * a + 0.5), $MachinePrecision], N[(1.0 / N[(N[(0.5 * b), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5e-49

   1. Initial program 96.7%

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{e^{a} - -1}\right)} \]

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot a + -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(-1 \cdot b\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\left(\mathsf{neg}\left(2\right)\right) \cdot \log 2}\right) + \frac{1}{2} \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{-2 \cdot \log 2}\right) + \frac{1}{2} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\log 2 \cdot -2}\right) + \frac{1}{2} \]

      10. pow-to-exp N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot {2}^{-2}\right) + \frac{1}{2} \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

      13. lift-neg.f64 46.4

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

   8. Applied rewrites 46.4%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \frac{1}{4} \cdot \color{blue}{a} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 49.1

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

   11. Applied rewrites 49.1%

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

   if 1.5e-49 < b

   1. Initial program 97.3%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto {\left(e^{b} + 1\right)}^{-1} \]

      4. metadata-eval N/A

         \[\leadsto {\left(e^{b} + 1 \cdot 1\right)}^{-1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto {\left(e^{b} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)}^{-1} \]

      6. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1 \cdot 1\right)}^{-1} \]

      7. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      8. lower--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      9. lift-exp.f64 90.8

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   5. Applied rewrites 90.8%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      3. lift-exp.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 90.8

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

   7. Applied rewrites 90.8%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 48.5

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

   10. Applied rewrites 48.5%

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

   11. Taylor expanded in b around inf

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

       1. lower-*.f64 47.9

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

   13. Applied rewrites 47.9%

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

Alternative 10: 53.7% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b \cdot b, 0.5, b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.1) (fma 0.25 a 0.5) (/ 1.0 (fma (* b b) 0.5 b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.1) {
		tmp = fma(0.25, a, 0.5);
	} else {
		tmp = 1.0 / fma((b * b), 0.5, b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.1)
		tmp = fma(0.25, a, 0.5);
	else
		tmp = Float64(1.0 / fma(Float64(b * b), 0.5, b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.1], N[(0.25 * a + 0.5), $MachinePrecision], N[(1.0 / N[(N[(b * b), $MachinePrecision] * 0.5 + b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001

   1. Initial program 96.9%

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

   5. Applied rewrites 76.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{e^{a} - -1}\right)} \]

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot a + -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(-1 \cdot b\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\left(\mathsf{neg}\left(2\right)\right) \cdot \log 2}\right) + \frac{1}{2} \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{-2 \cdot \log 2}\right) + \frac{1}{2} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\log 2 \cdot -2}\right) + \frac{1}{2} \]

      10. pow-to-exp N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot {2}^{-2}\right) + \frac{1}{2} \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

      13. lift-neg.f64 45.9

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

   8. Applied rewrites 45.9%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \frac{1}{4} \cdot \color{blue}{a} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 48.2

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

   11. Applied rewrites 48.2%

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

   if 1.1000000000000001 < b

   1. Initial program 96.8%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto {\left(e^{b} + 1\right)}^{-1} \]

      4. metadata-eval N/A

         \[\leadsto {\left(e^{b} + 1 \cdot 1\right)}^{-1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto {\left(e^{b} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)}^{-1} \]

      6. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1 \cdot 1\right)}^{-1} \]

      7. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      8. lower--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      9. lift-exp.f64 100.0

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   5. Applied rewrites 100.0%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      3. lift-exp.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 50.3

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

   10. Applied rewrites 50.3%

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

   11. Taylor expanded in b around inf

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

       1. distribute-lft-in N/A

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

       2. inv-pow N/A

          \[\leadsto \frac{1}{{b}^{2} \cdot \frac{1}{2} + {b}^{2} \cdot {b}^{-1}} \]

       3. pow-prod-up N/A

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

       4. metadata-eval N/A

          \[\leadsto \frac{1}{{b}^{2} \cdot \frac{1}{2} + {b}^{1}} \]

       5. unpow1 N/A

          \[\leadsto \frac{1}{{b}^{2} \cdot \frac{1}{2} + b} \]

       6. lower-fma.f64 N/A

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

       7. pow2 N/A

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

       8. lower-*.f64 50.3

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

   13. Applied rewrites 50.3%

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

Alternative 11: 53.7% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.5) (fma 0.25 a 0.5) (/ 1.0 (* (* b b) 0.5))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.5) {
		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 (b <= 1.5)
		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[b, 1.5], 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 b < 1.5

   1. Initial program 96.9%

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

      12. lift-exp.f64 N/A

          \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

   5. Applied rewrites 76.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{e^{a} - -1}\right)} \]

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot a + -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(-1 \cdot b\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\left(\mathsf{neg}\left(2\right)\right) \cdot \log 2}\right) + \frac{1}{2} \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{-2 \cdot \log 2}\right) + \frac{1}{2} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\log 2 \cdot -2}\right) + \frac{1}{2} \]

      10. pow-to-exp N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot {2}^{-2}\right) + \frac{1}{2} \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

      13. lift-neg.f64 45.9

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

   8. Applied rewrites 45.9%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \frac{1}{4} \cdot \color{blue}{a} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 48.2

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

   11. Applied rewrites 48.2%

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

   if 1.5 < b

   1. Initial program 96.8%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto {\left(e^{b} + 1\right)}^{-1} \]

      4. metadata-eval N/A

         \[\leadsto {\left(e^{b} + 1 \cdot 1\right)}^{-1} \]

      5. fp-cancel-sign-sub-inv N/A

         \[\leadsto {\left(e^{b} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)}^{-1} \]

      6. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1 \cdot 1\right)}^{-1} \]

      7. metadata-eval N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      8. lower--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      9. lift-exp.f64 100.0

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   5. Applied rewrites 100.0%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      3. lift-exp.f64 N/A

         \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 50.3

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

   10. Applied rewrites 50.3%

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

   11. Taylor expanded in b around inf

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

       1. *-commutative N/A

          \[\leadsto \frac{1}{{b}^{2} \cdot \frac{1}{2}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{{b}^{2} \cdot \frac{1}{2}} \]

       3. pow2 N/A

          \[\leadsto \frac{1}{\left(b \cdot b\right) \cdot \frac{1}{2}} \]

       4. lower-*.f64 50.3

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

   13. Applied rewrites 50.3%

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

Alternative 12: 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 96.9%

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

   1. mul-1-neg N/A

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

   2. associate-/l* N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. pow-to-exp N/A

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

   7. div-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-log1p.f64 N/A

       \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

   12. lift-exp.f64 N/A

       \[\leadsto \mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{1 + e^{a}}\right) \]

5. Applied rewrites 65.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-b, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, \frac{e^{a}}{e^{a} - -1}\right)} \]

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

   2. lower-+.f64 N/A

      \[\leadsto \left(-1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{4} \cdot a\right) + \frac{1}{2} \]

   3. +-commutative N/A

      \[\leadsto \left(\frac{1}{4} \cdot a + -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, -1 \cdot \left(b \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(-1 \cdot b\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) + \frac{1}{2} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\left(\mathsf{neg}\left(2\right)\right) \cdot \log 2}\right) + \frac{1}{2} \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{-2 \cdot \log 2}\right) + \frac{1}{2} \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot e^{\log 2 \cdot -2}\right) + \frac{1}{2} \]

   10. pow-to-exp N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot {2}^{-2}\right) + \frac{1}{2} \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \frac{1}{4}\right) + \frac{1}{2} \]

   13. lift-neg.f64 35.1

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

8. Applied rewrites 35.1%

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

9. Taylor expanded in b around 0

   \[\leadsto \frac{1}{2} + \frac{1}{4} \cdot \color{blue}{a} \]
10. Step-by-step derivation

    1. +-commutative N/A

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

    2. lower-fma.f64 37.0

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

11. Applied rewrites 37.0%

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

Alternative 13: 39.4% accurate, 315.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

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 96.9%

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

3. Taylor expanded in a around 0

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

   1. inv-pow N/A

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

   2. lower-pow.f64 N/A

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

   3. +-commutative N/A

      \[\leadsto {\left(e^{b} + 1\right)}^{-1} \]

   4. metadata-eval N/A

      \[\leadsto {\left(e^{b} + 1 \cdot 1\right)}^{-1} \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto {\left(e^{b} - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)}^{-1} \]

   6. metadata-eval N/A

      \[\leadsto {\left(e^{b} - -1 \cdot 1\right)}^{-1} \]

   7. metadata-eval N/A

      \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   8. lower--.f64 N/A

      \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

   9. lift-exp.f64 75.3

      \[\leadsto {\left(e^{b} - -1\right)}^{-1} \]

5. Applied rewrites 75.3%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 36.6%

   \[\leadsto 0.5 \]
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 2025064 
(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))))