Quotient of sum of exps

Percentage Accurate: 99.0% → 98.7%

Time: 6.7s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. a = 2.467304974674741e+71
  2. b = -1.4259870679252605e+296

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: 99.0% 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: 98.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -720:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -720.0) (/ (exp a) 2.0) (pow (+ (exp b) 1.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (a <= -720.0) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = pow((exp(b) + 1.0), -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 <= (-720.0d0)) then
        tmp = exp(a) / 2.0d0
    else
        tmp = (exp(b) + 1.0d0) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -720.0) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = Math.pow((Math.exp(b) + 1.0), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -720.0:
		tmp = math.exp(a) / 2.0
	else:
		tmp = math.pow((math.exp(b) + 1.0), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -720.0)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(exp(b) + 1.0) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -720.0)
		tmp = exp(a) / 2.0;
	else
		tmp = (exp(b) + 1.0) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -720.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -720

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{e^{a}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \frac{e^{a}}{2} \]

   if -720 < a

   1. Initial program 98.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -720:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{b} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

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

C

double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

Alternative 3: 71.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.4 \cdot 10^{+89}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right) \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.4e+89)
   (pow (fma (- (* (fma -0.16666666666666666 a 0.5) a) 1.0) a 2.0) -1.0)
   (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.4e+89) {
		tmp = pow(fma(((fma(-0.16666666666666666, a, 0.5) * a) - 1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8.4e+89)
		tmp = fma(Float64(Float64(fma(-0.16666666666666666, a, 0.5) * a) - 1.0), a, 2.0) ^ -1.0;
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8.4e+89], N[Power[N[(N[(N[(N[(-0.16666666666666666 * a + 0.5), $MachinePrecision] * a), $MachinePrecision] - 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.39999999999999945e89

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh a \cdot \cosh a - \sinh a \cdot \sinh a}{\cosh a - \sinh a}}}{e^{a} + e^{b}} \]

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

         \[\leadsto \frac{\frac{\color{blue}{\cosh b \cdot \cosh b - \sinh b \cdot \sinh b}}{\cosh a - \sinh a}}{e^{a} + e^{b}} \]

      7. sinh---cosh-rev N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. sinh-cosh N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 99.0

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in b around 0

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

      1. distribute-rgt-in N/A

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

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

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

      3. *-lft-identity N/A

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

      4. distribute-lft-neg-out N/A

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

      5. exp-neg N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 72.8

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

   7. Applied rewrites 72.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 63.0%

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

   if 8.39999999999999945e89 < b

   1. Initial program 98.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 92.7%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.4 \cdot 10^{+89}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right) \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.4e+89)
   (/ (exp a) 2.0)
   (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.4e+89) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8.4e+89)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8.4e+89], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.39999999999999945e89

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{e^{a}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.3%

      \[\leadsto \frac{e^{a}}{2} \]

   if 8.39999999999999945e89 < b

   1. Initial program 98.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 92.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+42}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.8e+42)
   (pow (fma (- (* 0.5 a) 1.0) a 2.0) -1.0)
   (pow (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.8e+42) {
		tmp = pow(fma(((0.5 * a) - 1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.8e+42)
		tmp = fma(Float64(Float64(0.5 * a) - 1.0), a, 2.0) ^ -1.0;
	else
		tmp = fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.8e+42], N[Power[N[(N[(N[(0.5 * a), $MachinePrecision] - 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8e42

   1. Initial program 99.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh a \cdot \cosh a - \sinh a \cdot \sinh a}{\cosh a - \sinh a}}}{e^{a} + e^{b}} \]

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

         \[\leadsto \frac{\frac{\color{blue}{\cosh b \cdot \cosh b - \sinh b \cdot \sinh b}}{\cosh a - \sinh a}}{e^{a} + e^{b}} \]

      7. sinh---cosh-rev N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. sinh-cosh N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 99.4

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in b around 0

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

      1. distribute-rgt-in N/A

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

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

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

      3. *-lft-identity N/A

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

      4. distribute-lft-neg-out N/A

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

      5. exp-neg N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 74.7

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 61.5%

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

   if 1.8e42 < b

   1. Initial program 96.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 77.1%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+42}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{+143}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.4e+143)
   (pow (fma (- (* 0.5 a) 1.0) a 2.0) -1.0)
   (pow (fma (fma 0.5 b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.4e+143) {
		tmp = pow(fma(((0.5 * a) - 1.0), a, 2.0), -1.0);
	} else {
		tmp = pow(fma(fma(0.5, b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.4e+143)
		tmp = fma(Float64(Float64(0.5 * a) - 1.0), a, 2.0) ^ -1.0;
	else
		tmp = fma(fma(0.5, b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.4e+143], N[Power[N[(N[(N[(0.5 * a), $MachinePrecision] - 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.3999999999999998e143

   1. Initial program 98.6%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh a \cdot \cosh a - \sinh a \cdot \sinh a}{\cosh a - \sinh a}}}{e^{a} + e^{b}} \]

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

         \[\leadsto \frac{\frac{\color{blue}{\cosh b \cdot \cosh b - \sinh b \cdot \sinh b}}{\cosh a - \sinh a}}{e^{a} + e^{b}} \]

      7. sinh---cosh-rev N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. sinh-cosh N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 98.6

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in b around 0

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

      1. distribute-rgt-in N/A

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

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

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

      3. *-lft-identity N/A

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

      4. distribute-lft-neg-out N/A

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

      5. exp-neg N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 70.9

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

   7. Applied rewrites 70.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 56.4%

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

   if 2.3999999999999998e143 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 95.7%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{+143}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5 \cdot a - 1, a, 2\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.56 \cdot 10^{-15}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.56e-15)
   (pow (- 2.0 a) -1.0)
   (pow (fma (fma 0.5 b 1.0) b 2.0) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.56e-15) {
		tmp = pow((2.0 - a), -1.0);
	} else {
		tmp = pow(fma(fma(0.5, b, 1.0), b, 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.56e-15)
		tmp = Float64(2.0 - a) ^ -1.0;
	else
		tmp = fma(fma(0.5, b, 1.0), b, 2.0) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.56e-15], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.55999999999999991e-15

   1. Initial program 99.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh a \cdot \cosh a - \sinh a \cdot \sinh a}{\cosh a - \sinh a}}}{e^{a} + e^{b}} \]

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

         \[\leadsto \frac{\frac{\color{blue}{\cosh b \cdot \cosh b - \sinh b \cdot \sinh b}}{\cosh a - \sinh a}}{e^{a} + e^{b}} \]

      7. sinh---cosh-rev N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. sinh-cosh N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 99.4

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in b around 0

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

      1. distribute-rgt-in N/A

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

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

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

      3. *-lft-identity N/A

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

      4. distribute-lft-neg-out N/A

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

      5. exp-neg N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 75.8

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 50.0%

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

   if 1.55999999999999991e-15 < b

   1. Initial program 97.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 95.9

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

   5. Applied rewrites 95.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 58.1%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.56 \cdot 10^{-15}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+35}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot b\right) \cdot 0.5\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.15e+35) (pow (- 2.0 a) -1.0) (pow (* (* b b) 0.5) -1.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.15e+35) {
		tmp = pow((2.0 - a), -1.0);
	} else {
		tmp = pow(((b * b) * 0.5), -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 (b <= 1.15d+35) then
        tmp = (2.0d0 - a) ** (-1.0d0)
    else
        tmp = ((b * b) * 0.5d0) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.15e+35) {
		tmp = Math.pow((2.0 - a), -1.0);
	} else {
		tmp = Math.pow(((b * b) * 0.5), -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.15e+35:
		tmp = math.pow((2.0 - a), -1.0)
	else:
		tmp = math.pow(((b * b) * 0.5), -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.15e+35)
		tmp = Float64(2.0 - a) ^ -1.0;
	else
		tmp = Float64(Float64(b * b) * 0.5) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.15e+35)
		tmp = (2.0 - a) ^ -1.0;
	else
		tmp = ((b * b) * 0.5) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.15e+35], N[Power[N[(2.0 - a), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1499999999999999e35

   1. Initial program 99.5%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh a \cdot \cosh a - \sinh a \cdot \sinh a}{\cosh a - \sinh a}}}{e^{a} + e^{b}} \]

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

         \[\leadsto \frac{\frac{\color{blue}{\cosh b \cdot \cosh b - \sinh b \cdot \sinh b}}{\cosh a - \sinh a}}{e^{a} + e^{b}} \]

      7. sinh---cosh-rev N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. sinh-cosh N/A

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

      11. lower-*.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 99.4

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in b around 0

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

      1. distribute-rgt-in N/A

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

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

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

      3. *-lft-identity N/A

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

      4. distribute-lft-neg-out N/A

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

      5. exp-neg N/A

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

      6. rgt-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 74.7

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 48.3%

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

   if 1.1499999999999999e35 < b

   1. Initial program 96.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 64.6%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 64.6%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+35}:\\ \;\;\;\;{\left(2 - a\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(b \cdot b\right) \cdot 0.5\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ {\left(2 - a\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (pow (- 2.0 a) -1.0))

C

double code(double a, double b) {
	return pow((2.0 - a), -1.0);
}

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 = (2.0d0 - a) ** (-1.0d0)
end function

Java

public static double code(double a, double b) {
	return Math.pow((2.0 - a), -1.0);
}

Python

def code(a, b):
	return math.pow((2.0 - a), -1.0)

Julia

function code(a, b)
	return Float64(2.0 - a) ^ -1.0
end

MATLAB

function tmp = code(a, b)
	tmp = (2.0 - a) ^ -1.0;
end

Wolfram

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

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\cosh a \cdot \cosh a - \sinh a \cdot \sinh a}{\cosh a - \sinh a}}}{e^{a} + e^{b}} \]

   5. sinh-cosh N/A

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

   6. sinh-cosh N/A

      \[\leadsto \frac{\frac{\color{blue}{\cosh b \cdot \cosh b - \sinh b \cdot \sinh b}}{\cosh a - \sinh a}}{e^{a} + e^{b}} \]

   7. sinh---cosh-rev N/A

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

   8. associate-/l/ N/A

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

   9. lower-/.f64 N/A

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

   10. sinh-cosh N/A

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

   11. lower-*.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-neg.f64 98.8

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in b around 0

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

   1. distribute-rgt-in N/A

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

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

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

   3. *-lft-identity N/A

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

   4. distribute-lft-neg-out N/A

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

   5. exp-neg N/A

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

   6. rgt-mult-inverse N/A

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

   7. metadata-eval N/A

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

   8. lower--.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 65.2

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

7. Applied rewrites 65.2%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 37.7%

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

11. Final simplification 37.7%

    \[\leadsto {\left(2 - a\right)}^{-1} \]
12. Add Preprocessing

Alternative 10: 40.0% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-/r* N/A

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

   4. div-add-rev N/A

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

   5. lower-/.f64 N/A

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

5. Applied rewrites 62.3%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 34.4%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 37.0%

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

Alternative 11: 39.8% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0
end function

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5;
end

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 80.3

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

5. Applied rewrites 80.3%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 36.9%

   \[\leadsto 0.5 \]
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 2024351 
(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))))