Quotient of sum of exps

Percentage Accurate: 98.9% → 98.9%

Time: 3.7s

Alternatives: 13

Speedup: 2.5×

• could not determine a ground truth

  1. a = 5.023784659602923e+278
  2. b = 6.218336726532799e-53

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

Alternative 2: 57.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.0)
   (/ a (+ (+ 1.0 a) (fma (fma 0.5 b 1.0) b 1.0)))
   (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 = a / ((1.0 + a) + fma(fma(0.5, b, 1.0), b, 1.0));
	} 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(a / Float64(Float64(1.0 + a) + fma(fma(0.5, b, 1.0), b, 1.0)));
	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[(a / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision]), $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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 61.4

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

   8. Applied rewrites 61.4%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 33.3

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

   11. Applied rewrites 33.3%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 42.9%

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

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

   1. Initial program 98.1%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lift-exp.f64 N/A

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   8. Applied rewrites 65.6%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + a \cdot \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   10. 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 69.0

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

   11. Applied rewrites 69.0%

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

Alternative 3: 53.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 0.500000002:\\ \;\;\;\;\frac{1}{\left(1 + a\right) + \mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.020833333333333332, 0.25\right), a, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 0.500000002)
   (/ 1.0 (+ (+ 1.0 a) (fma (fma 0.5 b 1.0) b 1.0)))
   (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))) <= 0.500000002) {
		tmp = 1.0 / ((1.0 + a) + fma(fma(0.5, b, 1.0), b, 1.0));
	} 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))) <= 0.500000002)
		tmp = Float64(1.0 / Float64(Float64(1.0 + a) + fma(fma(0.5, b, 1.0), b, 1.0)));
	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], 0.500000002], N[(1.0 / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision]), $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))) < 0.500000002000000054

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 77.9

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 78.1

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

   8. Applied rewrites 78.1%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 62.2

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

   11. Applied rewrites 62.2%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 62.5%

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

   if 0.500000002000000054 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 94.9%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lift-exp.f64 N/A

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

   5. Applied rewrites 10.7%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   8. Applied rewrites 8.8%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + a \cdot \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   10. 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 20.0

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

   11. Applied rewrites 20.0%

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

Alternative 4: 90.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{e^{b} - -1}\\ \mathbf{if}\;b \leq -3.05 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.0035:\\ \;\;\;\;\frac{a - -1}{\frac{1 - a \cdot a}{1 - a} + \mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (/ 1.0 (- (exp b) -1.0))))
   (if (<= b -3.05e-37)
     t_0
     (if (<= b 0.0035)
       (/
        (- a -1.0)
        (+ (/ (- 1.0 (* a a)) (- 1.0 a)) (fma (fma 0.5 b 1.0) b 1.0)))
       t_0))))

C

double code(double a, double b) {
	double t_0 = 1.0 / (exp(b) - -1.0);
	double tmp;
	if (b <= -3.05e-37) {
		tmp = t_0;
	} else if (b <= 0.0035) {
		tmp = (a - -1.0) / (((1.0 - (a * a)) / (1.0 - a)) + fma(fma(0.5, b, 1.0), b, 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(1.0 / Float64(exp(b) - -1.0))
	tmp = 0.0
	if (b <= -3.05e-37)
		tmp = t_0;
	elseif (b <= 0.0035)
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(Float64(1.0 - Float64(a * a)) / Float64(1.0 - a)) + fma(fma(0.5, b, 1.0), b, 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.05e-37], t$95$0, If[LessEqual[b, 0.0035], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(N[(1.0 - N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.0500000000000002e-37 or 0.00350000000000000007 < b

   1. Initial program 98.7%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 96.6

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

   5. Applied rewrites 96.6%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 96.6

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

   7. Applied rewrites 96.6%

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

   if -3.0500000000000002e-37 < b < 0.00350000000000000007

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 67.4

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

   8. Applied rewrites 67.4%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 67.3

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

   11. Applied rewrites 67.3%

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

       1. lift-+.f64 N/A

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

       2. flip-+ N/A

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

       3. lower-/.f64 N/A

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

       4. metadata-eval N/A

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

       5. pow2 N/A

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

       6. lower--.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. lower--.f64 83.7

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

   13. Applied rewrites 83.7%

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

Alternative 5: 99.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -700:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - -1}{\left(1 + a\right) + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -700.0)
   (/ (exp a) (+ 1.0 1.0))
   (/ (- a -1.0) (+ (+ 1.0 a) (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -700.0) {
		tmp = exp(a) / (1.0 + 1.0);
	} else {
		tmp = (a - -1.0) / ((1.0 + a) + exp(b));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-700.0d0)) then
        tmp = exp(a) / (1.0d0 + 1.0d0)
    else
        tmp = (a - (-1.0d0)) / ((1.0d0 + a) + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -700.0) {
		tmp = Math.exp(a) / (1.0 + 1.0);
	} else {
		tmp = (a - -1.0) / ((1.0 + a) + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -700.0:
		tmp = math.exp(a) / (1.0 + 1.0)
	else:
		tmp = (a - -1.0) / ((1.0 + a) + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -700.0)
		tmp = Float64(exp(a) / Float64(1.0 + 1.0));
	else
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(1.0 + a) + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -700.0)
		tmp = exp(a) / (1.0 + 1.0);
	else
		tmp = (a - -1.0) / ((1.0 + a) + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -700.0], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -700

   1. Initial program 98.8%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.5%

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

   if -700 < a

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 98.8

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

   8. Applied rewrites 98.8%

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

Alternative 6: 69.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -105:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.020833333333333332, a, 0.5\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{+34}:\\ \;\;\;\;\frac{a - -1}{\frac{1 - a \cdot a}{1 - a} + \mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1\right)}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;{a}^{3} \cdot -0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{a - -1}{\left(1 + a\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -105.0)
   (fma (* (* a a) -0.020833333333333332) a 0.5)
   (if (<= b 1.7e+34)
     (/
      (- a -1.0)
      (+ (/ (- 1.0 (* a a)) (- 1.0 a)) (fma (fma 0.5 b 1.0) b 1.0)))
     (if (<= b 1.05e+103)
       (* (pow a 3.0) -0.020833333333333332)
       (/
        (- a -1.0)
        (+
         (+ 1.0 a)
         (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -105.0) {
		tmp = fma(((a * a) * -0.020833333333333332), a, 0.5);
	} else if (b <= 1.7e+34) {
		tmp = (a - -1.0) / (((1.0 - (a * a)) / (1.0 - a)) + fma(fma(0.5, b, 1.0), b, 1.0));
	} else if (b <= 1.05e+103) {
		tmp = pow(a, 3.0) * -0.020833333333333332;
	} else {
		tmp = (a - -1.0) / ((1.0 + a) + fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -105.0)
		tmp = fma(Float64(Float64(a * a) * -0.020833333333333332), a, 0.5);
	elseif (b <= 1.7e+34)
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(Float64(1.0 - Float64(a * a)) / Float64(1.0 - a)) + fma(fma(0.5, b, 1.0), b, 1.0)));
	elseif (b <= 1.05e+103)
		tmp = Float64((a ^ 3.0) * -0.020833333333333332);
	else
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(1.0 + a) + fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -105.0], N[(N[(N[(a * a), $MachinePrecision] * -0.020833333333333332), $MachinePrecision] * a + 0.5), $MachinePrecision], If[LessEqual[b, 1.7e+34], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(N[(1.0 - N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e+103], N[(N[Power[a, 3.0], $MachinePrecision] * -0.020833333333333332), $MachinePrecision], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -105

   1. Initial program 97.4%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lift-exp.f64 N/A

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   8. Applied rewrites 5.2%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + a \cdot \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   10. 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 18.3

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

   11. Applied rewrites 18.3%

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

   12. Taylor expanded in a around inf

       \[\leadsto \mathsf{fma}\left(\frac{-1}{48} \cdot {a}^{2}, a, \frac{1}{2}\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 18.3

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

   14. Applied rewrites 18.3%

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

   if -105 < b < 1.7e34

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 68.7

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

   8. Applied rewrites 68.7%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 63.6

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

   11. Applied rewrites 63.6%

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

       1. lift-+.f64 N/A

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

       2. flip-+ N/A

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

       3. lower-/.f64 N/A

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

       4. metadata-eval N/A

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

       5. pow2 N/A

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

       6. lower--.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. lower--.f64 80.0

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

   13. Applied rewrites 80.0%

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

   if 1.7e34 < b < 1.0500000000000001e103

   1. Initial program 98.9%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lift-exp.f64 N/A

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   8. Applied rewrites 2.4%

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

   9. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 44.2

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

   11. Applied rewrites 44.2%

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

   if 1.0500000000000001e103 < b

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 7: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -700:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -700.0) (/ (exp a) (+ 1.0 1.0)) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -700.0) {
		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 <= (-700.0d0)) 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 <= -700.0) {
		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 <= -700.0:
		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 <= -700.0)
		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 <= -700.0)
		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, -700.0], 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 < -700

   1. Initial program 98.8%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.5%

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

   if -700 < a

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 98.0

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

   7. Applied rewrites 98.0%

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

Alternative 8: 69.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1\right)\\ \mathbf{if}\;b \leq -105:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.020833333333333332, a, 0.5\right)\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+35}:\\ \;\;\;\;\frac{a - -1}{\frac{1 - a \cdot a}{1 - a} + t\_0}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{a}{\left(1 + a\right) + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - -1}{\left(1 + a\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 b 1.0) b 1.0)))
   (if (<= b -105.0)
     (fma (* (* a a) -0.020833333333333332) a 0.5)
     (if (<= b 1.35e+35)
       (/ (- a -1.0) (+ (/ (- 1.0 (* a a)) (- 1.0 a)) t_0))
       (if (<= b 1.05e+103)
         (/ a (+ (+ 1.0 a) t_0))
         (/
          (- a -1.0)
          (+
           (+ 1.0 a)
           (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0))))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(0.5, b, 1.0), b, 1.0);
	double tmp;
	if (b <= -105.0) {
		tmp = fma(((a * a) * -0.020833333333333332), a, 0.5);
	} else if (b <= 1.35e+35) {
		tmp = (a - -1.0) / (((1.0 - (a * a)) / (1.0 - a)) + t_0);
	} else if (b <= 1.05e+103) {
		tmp = a / ((1.0 + a) + t_0);
	} else {
		tmp = (a - -1.0) / ((1.0 + a) + fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(0.5, b, 1.0), b, 1.0)
	tmp = 0.0
	if (b <= -105.0)
		tmp = fma(Float64(Float64(a * a) * -0.020833333333333332), a, 0.5);
	elseif (b <= 1.35e+35)
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(Float64(1.0 - Float64(a * a)) / Float64(1.0 - a)) + t_0));
	elseif (b <= 1.05e+103)
		tmp = Float64(a / Float64(Float64(1.0 + a) + t_0));
	else
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(1.0 + a) + fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision]}, If[LessEqual[b, -105.0], N[(N[(N[(a * a), $MachinePrecision] * -0.020833333333333332), $MachinePrecision] * a + 0.5), $MachinePrecision], If[LessEqual[b, 1.35e+35], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(N[(1.0 - N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(1.0 - a), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e+103], N[(a / N[(N[(1.0 + a), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -105

   1. Initial program 97.4%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. pow-to-exp N/A

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

      7. div-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lift-exp.f64 N/A

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   8. Applied rewrites 5.2%

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

   9. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + a \cdot \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
   10. 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 18.3

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

   11. Applied rewrites 18.3%

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

   12. Taylor expanded in a around inf

       \[\leadsto \mathsf{fma}\left(\frac{-1}{48} \cdot {a}^{2}, a, \frac{1}{2}\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 18.3

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

   14. Applied rewrites 18.3%

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

   if -105 < b < 1.35000000000000001e35

   1. Initial program 99.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 68.8

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

   8. Applied rewrites 68.8%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 63.5

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

   11. Applied rewrites 63.5%

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

       1. lift-+.f64 N/A

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

       2. flip-+ N/A

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

       3. lower-/.f64 N/A

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

       4. metadata-eval N/A

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

       5. pow2 N/A

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

       6. lower--.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. lower--.f64 79.9

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

   13. Applied rewrites 79.9%

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

   if 1.35000000000000001e35 < b < 1.0500000000000001e103

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 4.2

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

   11. Applied rewrites 4.2%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 32.7%

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

   if 1.0500000000000001e103 < b

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 9: 59.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 350:\\ \;\;\;\;\frac{a - -1}{\left(1 + a\right) + 1}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;\frac{a}{\left(1 + a\right) + \mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - -1}{\left(1 + a\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 350.0)
   (/ (- a -1.0) (+ (+ 1.0 a) 1.0))
   (if (<= b 1.05e+103)
     (/ a (+ (+ 1.0 a) (fma (fma 0.5 b 1.0) b 1.0)))
     (/
      (- a -1.0)
      (+ (+ 1.0 a) (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 350.0) {
		tmp = (a - -1.0) / ((1.0 + a) + 1.0);
	} else if (b <= 1.05e+103) {
		tmp = a / ((1.0 + a) + fma(fma(0.5, b, 1.0), b, 1.0));
	} else {
		tmp = (a - -1.0) / ((1.0 + a) + fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 350.0)
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(1.0 + a) + 1.0));
	elseif (b <= 1.05e+103)
		tmp = Float64(a / Float64(Float64(1.0 + a) + fma(fma(0.5, b, 1.0), b, 1.0)));
	else
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(1.0 + a) + fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 350.0], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.05e+103], N[(a / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 350

   1. Initial program 98.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 75.4

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

   8. Applied rewrites 75.4%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 53.6%

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

   if 350 < b < 1.0500000000000001e103

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 3.9

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

   11. Applied rewrites 3.9%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 25.4%

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

   if 1.0500000000000001e103 < b

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 10: 40.2% accurate, 15.0× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (a - -1.0) / ((1.0 + a) + 1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a - (-1.0d0)) / ((1.0d0 + a) + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 80.9

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

5. Applied rewrites 80.9%

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

6. Taylor expanded in a around 0

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

   1. lower-+.f64 81.9

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

8. Applied rewrites 81.9%

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

9. Taylor expanded in b around 0

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

11. Applied rewrites 40.2%

    \[\leadsto \frac{a - -1}{\left(1 + a\right) + \color{blue}{1}} \]
12. Add Preprocessing

Alternative 11: 39.7% 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 98.9%

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

3. Taylor expanded in b around 0

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

   1. mul-1-neg N/A

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

   2. associate-/l* N/A

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

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

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. pow-to-exp N/A

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

   7. div-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-log1p.f64 N/A

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

   12. lift-exp.f64 N/A

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

5. Applied rewrites 63.9%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

8. Applied rewrites 37.6%

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

9. Taylor expanded in b around 0

   \[\leadsto \frac{1}{2} + a \cdot \color{blue}{\left(\frac{1}{4} + \frac{-1}{48} \cdot {a}^{2}\right)} \]
10. 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 39.7

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

11. Applied rewrites 39.7%

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

Alternative 12: 39.8% 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 98.9%

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

3. Taylor expanded in b around 0

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

   1. mul-1-neg N/A

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

   2. associate-/l* N/A

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

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

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. pow-to-exp N/A

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

   7. div-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-log1p.f64 N/A

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

   12. lift-exp.f64 N/A

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

5. Applied rewrites 63.9%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. pow-to-exp N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lift-neg.f64 37.7

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

8. Applied rewrites 37.7%

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

9. Taylor expanded in b around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 39.8

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

11. Applied rewrites 39.8%

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

Alternative 13: 39.5% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in a around 0

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

   1. inv-pow N/A

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

   2. lower-pow.f64 N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower--.f64 N/A

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

   9. lift-exp.f64 81.4

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

5. Applied rewrites 81.4%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 39.5%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return 1.0 / (1.0 + exp((b - a)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (1.0d0 + exp((b - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025088 
(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))))