Quotient of sum of exps

Percentage Accurate: 98.7% → 98.7%

Time: 3.3s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. a = 7.886610844271556e+59
  2. b = 4.136192739954696e+25

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.7% 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, 1.0× 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 98.7%

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

Alternative 2: 57.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   3. 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 62.5

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

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{{\left(e^{b} - -1\right)}^{-1}} \]
   5. 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 62.5

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

   6. Applied rewrites 62.5%

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

   7. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2 + \color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}} \]
   8. 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 44.0

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

   9. Applied rewrites 44.0%

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

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

   1. Initial program 97.6%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \frac{-1 \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}}{1 + e^{a}} + \left(\frac{-1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{6} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right)\right) - \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right) + \frac{e^{a}}{1 + e^{a}}} \]

   4. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(-b\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, 0.5, -e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}\right)}{e^{a} - -1}, -1, \mathsf{fma}\left(-0.5, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2} \cdot 0.16666666666666666\right)\right) - \mathsf{fma}\left(e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, 0.5, -e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}\right)\right) \cdot b - e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, b, \frac{e^{a}}{e^{a} - -1}\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\left(a \cdot \left(\frac{1}{4} + \left(a \cdot \left(a \cdot \left({b}^{2} \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{-1}{12} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \left(\frac{1}{24} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \left(\frac{1}{4} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \left(\frac{1}{8} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right)\right)\right)\right)\right) + \frac{-1}{12} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{6} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \frac{1}{48}\right) + b \cdot \left(b \cdot \left(\left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \left(\frac{1}{8} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right)\right) + \left(\frac{-1}{24} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} + \frac{1}{8} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right) + \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + {b}^{2} \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{-1}{2} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right) + b \cdot \left(b \cdot \left(\left(e^{\mathsf{neg}\left(3 \cdot \log 2\right)} + -1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} + \left(\frac{-1}{2} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{6} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)\right)\right) - \frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) - e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)} \]

   7. Applied rewrites 64.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-b, 0, 0.020833333333333332\right) \cdot \left(b \cdot b\right) - 0.020833333333333332, a, \mathsf{fma}\left(\mathsf{fma}\left(-b, 0.020833333333333332, 0\right), b, 0.0625\right) \cdot b\right), a, \mathsf{fma}\left(-b, 0, -0.0625\right) \cdot \left(b \cdot b\right)\right) + 0.25, a, \left(\left(\mathsf{fma}\left(-b, -0.020833333333333332, 0.125\right) - 0.125\right) \cdot b - 0.25\right) \cdot b\right) + \color{blue}{0.5} \]

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 68.6

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

   10. Applied rewrites 68.6%

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

Alternative 3: 53.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   3. 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 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 34.3

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

   7. Applied rewrites 34.3%

      \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 34.3

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

   9. Applied rewrites 34.3%

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

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

   1. Initial program 97.6%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \frac{-1 \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}}{1 + e^{a}} + \left(\frac{-1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{6} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right)\right) - \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right) + \frac{e^{a}}{1 + e^{a}}} \]

   4. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(-b\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, 0.5, -e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}\right)}{e^{a} - -1}, -1, \mathsf{fma}\left(-0.5, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2} \cdot 0.16666666666666666\right)\right) - \mathsf{fma}\left(e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, 0.5, -e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}\right)\right) \cdot b - e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, b, \frac{e^{a}}{e^{a} - -1}\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\left(a \cdot \left(\frac{1}{4} + \left(a \cdot \left(a \cdot \left({b}^{2} \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{-1}{12} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \left(\frac{1}{24} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \left(\frac{1}{4} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \left(\frac{1}{8} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right)\right)\right)\right)\right) + \frac{-1}{12} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{6} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \frac{1}{48}\right) + b \cdot \left(b \cdot \left(\left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \left(\frac{1}{8} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right)\right) + \left(\frac{-1}{24} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} + \frac{1}{8} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right) + \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + {b}^{2} \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{-1}{2} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right) + b \cdot \left(b \cdot \left(\left(e^{\mathsf{neg}\left(3 \cdot \log 2\right)} + -1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} + \left(\frac{-1}{2} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{6} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)\right)\right) - \frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) - e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)} \]

   7. Applied rewrites 64.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-b, 0, 0.020833333333333332\right) \cdot \left(b \cdot b\right) - 0.020833333333333332, a, \mathsf{fma}\left(\mathsf{fma}\left(-b, 0.020833333333333332, 0\right), b, 0.0625\right) \cdot b\right), a, \mathsf{fma}\left(-b, 0, -0.0625\right) \cdot \left(b \cdot b\right)\right) + 0.25, a, \left(\left(\mathsf{fma}\left(-b, -0.020833333333333332, 0.125\right) - 0.125\right) \cdot b - 0.25\right) \cdot b\right) + \color{blue}{0.5} \]

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 68.6

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

   10. Applied rewrites 68.6%

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

Alternative 4: 53.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   3. 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 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 34.3

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

   7. Applied rewrites 34.3%

      \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 34.3

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

   9. Applied rewrites 34.3%

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

   10. Taylor expanded in b around inf

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

       1. lower-*.f64 34.3

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

   12. Applied rewrites 34.3%

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

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

   1. Initial program 97.6%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \frac{-1 \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}}{1 + e^{a}} + \left(\frac{-1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{6} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right)\right) - \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right) + \frac{e^{a}}{1 + e^{a}}} \]

   4. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(-b\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, 0.5, -e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}\right)}{e^{a} - -1}, -1, \mathsf{fma}\left(-0.5, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2} \cdot 0.16666666666666666\right)\right) - \mathsf{fma}\left(e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, 0.5, -e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}\right)\right) \cdot b - e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, b, \frac{e^{a}}{e^{a} - -1}\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\left(a \cdot \left(\frac{1}{4} + \left(a \cdot \left(a \cdot \left({b}^{2} \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{-1}{12} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \left(\frac{1}{24} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \left(\frac{1}{4} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \left(\frac{1}{8} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right)\right)\right)\right)\right) + \frac{-1}{12} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{6} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \frac{1}{48}\right) + b \cdot \left(b \cdot \left(\left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \left(\frac{1}{8} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right)\right) + \left(\frac{-1}{24} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} + \frac{1}{8} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right) + \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + {b}^{2} \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{-1}{2} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right) + b \cdot \left(b \cdot \left(\left(e^{\mathsf{neg}\left(3 \cdot \log 2\right)} + -1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} + \left(\frac{-1}{2} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{6} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)\right)\right) - \frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) - e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)} \]

   7. Applied rewrites 64.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-b, 0, 0.020833333333333332\right) \cdot \left(b \cdot b\right) - 0.020833333333333332, a, \mathsf{fma}\left(\mathsf{fma}\left(-b, 0.020833333333333332, 0\right), b, 0.0625\right) \cdot b\right), a, \mathsf{fma}\left(-b, 0, -0.0625\right) \cdot \left(b \cdot b\right)\right) + 0.25, a, \left(\left(\mathsf{fma}\left(-b, -0.020833333333333332, 0.125\right) - 0.125\right) \cdot b - 0.25\right) \cdot b\right) + \color{blue}{0.5} \]

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 68.6

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

   10. Applied rewrites 68.6%

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

Alternative 5: 53.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   3. 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 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 34.3

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

   7. Applied rewrites 34.3%

      \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 34.3

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

   9. Applied rewrites 34.3%

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

   10. Taylor expanded in b around inf

       \[\leadsto \frac{1}{\frac{1}{2} \cdot {b}^{\color{blue}{2}}} \]
   11. 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. lift-*.f64 33.8

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

   12. Applied rewrites 33.8%

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

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

   1. Initial program 97.6%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \frac{-1 \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}}{1 + e^{a}} + \left(\frac{-1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{6} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right)\right) - \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right) + \frac{e^{a}}{1 + e^{a}}} \]

   4. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(-b\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, 0.5, -e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}\right)}{e^{a} - -1}, -1, \mathsf{fma}\left(-0.5, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2} \cdot 0.16666666666666666\right)\right) - \mathsf{fma}\left(e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, 0.5, -e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}\right)\right) \cdot b - e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, b, \frac{e^{a}}{e^{a} - -1}\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\left(a \cdot \left(\frac{1}{4} + \left(a \cdot \left(a \cdot \left({b}^{2} \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{-1}{12} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \left(\frac{1}{24} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \left(\frac{1}{4} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \left(\frac{1}{8} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right)\right)\right)\right)\right) + \frac{-1}{12} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{6} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \frac{1}{48}\right) + b \cdot \left(b \cdot \left(\left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \left(\frac{1}{8} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right)\right) + \left(\frac{-1}{24} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} + \frac{1}{8} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right) + \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + {b}^{2} \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{-1}{2} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right) + b \cdot \left(b \cdot \left(\left(e^{\mathsf{neg}\left(3 \cdot \log 2\right)} + -1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} + \left(\frac{-1}{2} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{6} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)\right)\right) - \frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) - e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)} \]

   7. Applied rewrites 64.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-b, 0, 0.020833333333333332\right) \cdot \left(b \cdot b\right) - 0.020833333333333332, a, \mathsf{fma}\left(\mathsf{fma}\left(-b, 0.020833333333333332, 0\right), b, 0.0625\right) \cdot b\right), a, \mathsf{fma}\left(-b, 0, -0.0625\right) \cdot \left(b \cdot b\right)\right) + 0.25, a, \left(\left(\mathsf{fma}\left(-b, -0.020833333333333332, 0.125\right) - 0.125\right) \cdot b - 0.25\right) \cdot b\right) + \color{blue}{0.5} \]

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 68.6

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

   10. Applied rewrites 68.6%

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

Alternative 6: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -850000:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -850000.0) (/ (exp a) (+ 1.0 1.0)) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -850000.0) {
		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 <= (-850000.0d0)) 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 <= -850000.0) {
		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 <= -850000.0:
		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 <= -850000.0)
		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 <= -850000.0)
		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, -850000.0], 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 < -8.5e5

   1. Initial program 98.8%

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

   2. Taylor expanded in b around 0

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 99.8%

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

   if -8.5e5 < a

   1. Initial program 98.6%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   3. 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 97.9

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

   4. Applied rewrites 97.9%

      \[\leadsto \color{blue}{{\left(e^{b} - -1\right)}^{-1}} \]
   5. 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 97.9

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

   6. Applied rewrites 97.9%

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

Alternative 7: 81.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{b} - -1} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (- (exp b) -1.0)))

C

double code(double a, double b) {
	return 1.0 / (exp(b) - -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 = 1.0d0 / (exp(b) - (-1.0d0))
end function

Java

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

Python

def code(a, b):
	return 1.0 / (math.exp(b) - -1.0)

Julia

function code(a, b)
	return Float64(1.0 / Float64(exp(b) - -1.0))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (exp(b) - -1.0);
end

Wolfram

code[a_, b_] := N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
3. 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 81.7

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

4. Applied rewrites 81.7%

   \[\leadsto \color{blue}{{\left(e^{b} - -1\right)}^{-1}} \]
5. 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 81.7

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

6. Applied rewrites 81.7%

   \[\leadsto \frac{1}{\color{blue}{e^{b} - -1}} \]
7. Add Preprocessing

Alternative 8: 59.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, b, 1\right) \cdot b\\ \mathbf{if}\;b \leq 4.5 \cdot 10^{-72}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.020833333333333332, 0.25\right), a, 0.5\right)\\ \mathbf{elif}\;b \leq 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{t\_0 \cdot t\_0 - 4}{t\_0 - 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (fma 0.5 b 1.0) b)))
   (if (<= b 4.5e-72)
     (fma (fma (* a a) -0.020833333333333332 0.25) a 0.5)
     (if (<= b 1e+154)
       (/ 1.0 (/ (- (* t_0 t_0) 4.0) (- t_0 2.0)))
       (/ 1.0 (* (* b b) 0.5))))))

C

double code(double a, double b) {
	double t_0 = fma(0.5, b, 1.0) * b;
	double tmp;
	if (b <= 4.5e-72) {
		tmp = fma(fma((a * a), -0.020833333333333332, 0.25), a, 0.5);
	} else if (b <= 1e+154) {
		tmp = 1.0 / (((t_0 * t_0) - 4.0) / (t_0 - 2.0));
	} else {
		tmp = 1.0 / ((b * b) * 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(fma(0.5, b, 1.0) * b)
	tmp = 0.0
	if (b <= 4.5e-72)
		tmp = fma(fma(Float64(a * a), -0.020833333333333332, 0.25), a, 0.5);
	elseif (b <= 1e+154)
		tmp = Float64(1.0 / Float64(Float64(Float64(t_0 * t_0) - 4.0) / Float64(t_0 - 2.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(b * b) * 0.5));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.5 * b + 1.0), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, 4.5e-72], N[(N[(N[(a * a), $MachinePrecision] * -0.020833333333333332 + 0.25), $MachinePrecision] * a + 0.5), $MachinePrecision], If[LessEqual[b, 1e+154], N[(1.0 / N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 4.0), $MachinePrecision] / N[(t$95$0 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.5e-72

   1. Initial program 98.4%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \frac{-1 \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}}{1 + e^{a}} + \left(\frac{-1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{6} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right)\right) - \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right) + \frac{e^{a}}{1 + e^{a}}} \]

   4. Applied rewrites 71.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(-b\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, 0.5, -e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}\right)}{e^{a} - -1}, -1, \mathsf{fma}\left(-0.5, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2} \cdot 0.16666666666666666\right)\right) - \mathsf{fma}\left(e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, 0.5, -e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}\right)\right) \cdot b - e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, b, \frac{e^{a}}{e^{a} - -1}\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\left(a \cdot \left(\frac{1}{4} + \left(a \cdot \left(a \cdot \left({b}^{2} \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{-1}{12} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \left(\frac{1}{24} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \left(\frac{1}{4} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \left(\frac{1}{8} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right)\right)\right)\right)\right) + \frac{-1}{12} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{6} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \frac{1}{48}\right) + b \cdot \left(b \cdot \left(\left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \left(\frac{1}{8} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right)\right) + \left(\frac{-1}{24} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} + \frac{1}{8} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right) + \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + {b}^{2} \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{-1}{2} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right) + b \cdot \left(b \cdot \left(\left(e^{\mathsf{neg}\left(3 \cdot \log 2\right)} + -1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} + \left(\frac{-1}{2} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{6} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)\right)\right) - \frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) - e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)} \]

   7. Applied rewrites 47.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-b, 0, 0.020833333333333332\right) \cdot \left(b \cdot b\right) - 0.020833333333333332, a, \mathsf{fma}\left(\mathsf{fma}\left(-b, 0.020833333333333332, 0\right), b, 0.0625\right) \cdot b\right), a, \mathsf{fma}\left(-b, 0, -0.0625\right) \cdot \left(b \cdot b\right)\right) + 0.25, a, \left(\left(\mathsf{fma}\left(-b, -0.020833333333333332, 0.125\right) - 0.125\right) \cdot b - 0.25\right) \cdot b\right) + \color{blue}{0.5} \]

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 52.2

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

   10. Applied rewrites 52.2%

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

   if 4.5e-72 < b < 1.00000000000000004e154

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   3. 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 89.5

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 24.3

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

   7. Applied rewrites 24.3%

      \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 24.3

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

   9. Applied rewrites 24.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}} \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. flip-+ N/A

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

       4. lower-/.f64 N/A

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

   11. Applied rewrites 56.1%

       \[\leadsto \frac{1}{\frac{\left(\mathsf{fma}\left(0.5, b, 1\right) \cdot b\right) \cdot \left(\mathsf{fma}\left(0.5, b, 1\right) \cdot b\right) - 4}{\mathsf{fma}\left(0.5, b, 1\right) \cdot b - \color{blue}{2}}} \]

   if 1.00000000000000004e154 < b

   1. Initial program 99.3%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   3. 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

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

      \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)\right)}^{-1} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in b around inf

       \[\leadsto \frac{1}{\frac{1}{2} \cdot {b}^{\color{blue}{2}}} \]
   11. 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. lift-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

Alternative 9: 39.1% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.020833333333333332, 0.25\right), a, 0.5\right) \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (fma (fma (* a a) -0.020833333333333332 0.25) a 0.5))

C

double code(double a, double b) {
	return fma(fma((a * a), -0.020833333333333332, 0.25), a, 0.5);
}

Julia

function code(a, b)
	return fma(fma(Float64(a * a), -0.020833333333333332, 0.25), a, 0.5)
end

Wolfram

code[a_, b_] := N[(N[(N[(a * a), $MachinePrecision] * -0.020833333333333332 + 0.25), $MachinePrecision] * a + 0.5), $MachinePrecision]

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in b around 0

   \[\leadsto \color{blue}{b \cdot \left(b \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \frac{-1 \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}}{1 + e^{a}} + \left(\frac{-1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{6} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{2} \cdot \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right)\right) - \frac{e^{a}}{{\left(1 + e^{a}\right)}^{2}}\right) + \frac{e^{a}}{1 + e^{a}}} \]

4. Applied rewrites 62.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(-b\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, 0.5, -e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}\right)}{e^{a} - -1}, -1, \mathsf{fma}\left(-0.5, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}, e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2} \cdot 0.16666666666666666\right)\right) - \mathsf{fma}\left(e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, 0.5, -e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 3}\right)\right) \cdot b - e^{a - \mathsf{log1p}\left(e^{a}\right) \cdot 2}, b, \frac{e^{a}}{e^{a} - -1}\right)} \]

5. Taylor expanded in a around 0

   \[\leadsto \frac{1}{2} + \color{blue}{\left(a \cdot \left(\frac{1}{4} + \left(a \cdot \left(a \cdot \left({b}^{2} \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{-1}{12} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \left(\frac{1}{24} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \left(\frac{1}{4} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \left(\frac{1}{8} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right)\right)\right)\right)\right) + \frac{-1}{12} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{6} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \frac{1}{48}\right) + b \cdot \left(b \cdot \left(\left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{1}{2} \cdot \left(\frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \left(\frac{1}{8} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{2} \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right)\right) + \left(\frac{-1}{24} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} + \frac{1}{8} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right) + \frac{-1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) - \frac{-1}{8} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) - \frac{-1}{4} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right) + {b}^{2} \cdot \left(-1 \cdot \left(b \cdot \left(-1 \cdot \left(\frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} - \frac{1}{4} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{1}{4} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right) + \frac{-1}{2} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right)\right)\right) + b \cdot \left(b \cdot \left(\left(e^{\mathsf{neg}\left(3 \cdot \log 2\right)} + -1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot e^{\mathsf{neg}\left(3 \cdot \log 2\right)} + \left(\frac{-1}{2} \cdot \left(\frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)} - e^{\mathsf{neg}\left(3 \cdot \log 2\right)}\right) + \frac{1}{6} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)\right)\right) - \frac{1}{2} \cdot e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right) - e^{\mathsf{neg}\left(2 \cdot \log 2\right)}\right)\right)} \]

7. Applied rewrites 35.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-b, 0, 0.020833333333333332\right) \cdot \left(b \cdot b\right) - 0.020833333333333332, a, \mathsf{fma}\left(\mathsf{fma}\left(-b, 0.020833333333333332, 0\right), b, 0.0625\right) \cdot b\right), a, \mathsf{fma}\left(-b, 0, -0.0625\right) \cdot \left(b \cdot b\right)\right) + 0.25, a, \left(\left(\mathsf{fma}\left(-b, -0.020833333333333332, 0.125\right) - 0.125\right) \cdot b - 0.25\right) \cdot b\right) + \color{blue}{0.5} \]

8. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 39.1

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

10. Applied rewrites 39.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.020833333333333332, 0.25\right), a, 0.5\right) \]
11. Add Preprocessing

Alternative 10: 39.2% 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 98.7%

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

2. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

4. Applied rewrites 82.0%

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

5. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 39.2

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

7. Applied rewrites 39.2%

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

Alternative 11: 39.1% 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 98.7%

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

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
3. 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 81.7

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

4. Applied rewrites 81.7%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 39.1%

   \[\leadsto 0.5 \]
8. 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 2025097 
(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))))