Quotient of sum of exps

Percentage Accurate: 99.2% → 99.2%

Time: 3.3s

Alternatives: 17

Speedup: 1.0×

• could not determine a ground truth

  1. a = 1.1093727613178446e+21
  2. b = 7.216202367690328e+60

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b)
  :precision binary64
  (/ (exp a) (+ (exp a) (/ 1 (exp (- b))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[(1 / N[Exp[(-b)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. lift-exp.f64 N/A

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

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

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

   3. add-flip N/A

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

   4. cosh-neg-rev N/A

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

   5. sinh-neg-rev N/A

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

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

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

   7. exp-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 99.2%

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

3. Applied rewrites 99.2%

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

Alternative 2: 97.7% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (a b)
  :precision binary64
  (/ (exp a) (+ 1 (exp b))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(1 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

   2. lower-exp.f64 97.7%

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

4. Applied rewrites 97.7%

   \[\leadsto \frac{e^{a}}{\color{blue}{1 + e^{b}}} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (a b)
  :precision binary64
  (/ 1 (- (+ (+ (exp b) a) (* (sqrt (* (* a a) (* a a))) 1/2)) -1)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (((exp(b) + a) + (sqrt(((a * a) * (a * a))) * 0.5d0)) - (-1.0d0))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (((Math.exp(b) + a) + (Math.sqrt(((a * a) * (a * a))) * 0.5)) - -1.0);
}

Python

def code(a, b):
	return 1.0 / (((math.exp(b) + a) + (math.sqrt(((a * a) * (a * a))) * 0.5)) - -1.0)

Julia

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

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (((exp(b) + a) + (sqrt(((a * a) * (a * a))) * 0.5)) - -1.0);
end

Wolfram

code[a_, b_] := N[(1 / N[(N[(N[(N[Exp[b], $MachinePrecision] + a), $MachinePrecision] + N[(N[Sqrt[N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in a around 0

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

4. Applied rewrites 80.7%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 81.5%

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

8. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 89.6%

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

10. Applied rewrites 89.6%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. add-flip N/A

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

    4. metadata-eval N/A

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

    5. lower--.f64 89.6%

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

12. Applied rewrites 89.6%

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

    1. rem-square-sqrt N/A

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

    2. sqrt-unprod N/A

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

    3. lower-sqrt.f64 N/A

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

    4. lower-*.f64 94.0%

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

14. Applied rewrites 94.0%

    \[\leadsto \frac{1}{\left(\left(e^{b} + a\right) + \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \cdot \frac{1}{2}\right) - -1} \]
15. Add Preprocessing

Alternative 4: 93.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := 1 + a \cdot \left(1 + \frac{1}{2} \cdot a\right)\\ \mathbf{if}\;a \leq -70000000000000000134097256001426811535116858908563440780696246021956552694279212917286530157572622647296:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_0 + e^{b}}\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (let* ((t_0 (+ 1 (* a (+ 1 (* 1/2 a))))))
  (if (<=
       a
       -70000000000000000134097256001426811535116858908563440780696246021956552694279212917286530157572622647296)
    (/ 1 (+ 2 (* a (+ 1 (* a (+ 1/2 (* 1/6 a)))))))
    (/ t_0 (+ t_0 (exp b))))))

C

double code(double a, double b) {
	double t_0 = 1.0 + (a * (1.0 + (0.5 * a)));
	double tmp;
	if (a <= -7e+103) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	} else {
		tmp = t_0 / (t_0 + exp(b));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (a * (1.0d0 + (0.5d0 * a)))
    if (a <= (-7d+103)) then
        tmp = 1.0d0 / (2.0d0 + (a * (1.0d0 + (a * (0.5d0 + (0.16666666666666666d0 * a))))))
    else
        tmp = t_0 / (t_0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = 1.0 + (a * (1.0 + (0.5 * a)));
	double tmp;
	if (a <= -7e+103) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	} else {
		tmp = t_0 / (t_0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = 1.0 + (a * (1.0 + (0.5 * a)))
	tmp = 0
	if a <= -7e+103:
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))))
	else:
		tmp = t_0 / (t_0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(1.0 + Float64(a * Float64(1.0 + Float64(0.5 * a))))
	tmp = 0.0
	if (a <= -7e+103)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(1.0 + Float64(a * Float64(0.5 + Float64(0.16666666666666666 * a)))))));
	else
		tmp = Float64(t_0 / Float64(t_0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = 1.0 + (a * (1.0 + (0.5 * a)));
	tmp = 0.0;
	if (a <= -7e+103)
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	else
		tmp = t_0 / (t_0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(1 + N[(a * N[(1 + N[(1/2 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -70000000000000000134097256001426811535116858908563440780696246021956552694279212917286530157572622647296], N[(1 / N[(2 + N[(a * N[(1 + N[(a * N[(1/2 + N[(1/6 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(t$95$0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -7e103

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 89.6%

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

   10. Applied rewrites 89.6%

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

   11. Taylor expanded in a around 0

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

       1. lower-+.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-*.f64 86.9%

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

   13. Applied rewrites 86.9%

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

   14. Taylor expanded in b around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 56.3%

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

   16. Applied rewrites 56.3%

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

   if -7e103 < a

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 77.0%

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 77.7%

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

   7. Applied rewrites 77.7%

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

Alternative 5: 93.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -70000000000000000134097256001426811535116858908563440780696246021956552694279212917286530157572622647296:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + e^{b}}\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<=
     a
     -70000000000000000134097256001426811535116858908563440780696246021956552694279212917286530157572622647296)
  (/ 1 (+ 2 (* a (+ 1 (* a (+ 1/2 (* 1/6 a)))))))
  (/ (+ 1 a) (+ (+ 1 a) (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7e+103) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + exp(b));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-7d+103)) then
        tmp = 1.0d0 / (2.0d0 + (a * (1.0d0 + (a * (0.5d0 + (0.16666666666666666d0 * a))))))
    else
        tmp = (1.0d0 + a) / ((1.0d0 + a) + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -7e+103) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	} else {
		tmp = (1.0 + a) / ((1.0 + a) + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -7e+103:
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))))
	else:
		tmp = (1.0 + a) / ((1.0 + a) + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7e+103)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(1.0 + Float64(a * Float64(0.5 + Float64(0.16666666666666666 * a)))))));
	else
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7e+103)
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	else
		tmp = (1.0 + a) / ((1.0 + a) + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -70000000000000000134097256001426811535116858908563440780696246021956552694279212917286530157572622647296], N[(1 / N[(2 + N[(a * N[(1 + N[(a * N[(1/2 + N[(1/6 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1 + a), $MachinePrecision] / N[(N[(1 + a), $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7e103

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 89.6%

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

   10. Applied rewrites 89.6%

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

   11. Taylor expanded in a around 0

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

       1. lower-+.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-*.f64 86.9%

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

   13. Applied rewrites 86.9%

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

   14. Taylor expanded in b around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 56.3%

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

   16. Applied rewrites 56.3%

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

   if -7e103 < a

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

      1. lower-+.f64 81.1%

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

   4. Applied rewrites 81.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 82.0%

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

   7. Applied rewrites 82.0%

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

Alternative 6: 93.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -94999999999999992224817350588207255454399766741059799452217921129377556751955301525765481584743122206720:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<=
     a
     -94999999999999992224817350588207255454399766741059799452217921129377556751955301525765481584743122206720)
  (/ 1 (+ 2 (* a (+ 1 (* a (+ 1/2 (* 1/6 a)))))))
  (/ 1 (+ 1 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -9.5e+103) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-9.5d+103)) then
        tmp = 1.0d0 / (2.0d0 + (a * (1.0d0 + (a * (0.5d0 + (0.16666666666666666d0 * a))))))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -9.5e+103) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -9.5e+103:
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -9.5e+103)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(1.0 + Float64(a * Float64(0.5 + Float64(0.16666666666666666 * a)))))));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -9.5e+103)
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -94999999999999992224817350588207255454399766741059799452217921129377556751955301525765481584743122206720], N[(1 / N[(2 + N[(a * N[(1 + N[(a * N[(1/2 + N[(1/6 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1 / N[(1 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.4999999999999992e103

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 89.6%

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

   10. Applied rewrites 89.6%

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

   11. Taylor expanded in a around 0

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

       1. lower-+.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-*.f64 86.9%

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

   13. Applied rewrites 86.9%

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

   14. Taylor expanded in b around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 56.3%

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

   16. Applied rewrites 56.3%

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

   if -9.4999999999999992e103 < a

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 81.5%

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

Alternative 7: 87.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\frac{1}{2} \cdot b\right) \cdot b\\ \mathbf{if}\;b \leq \frac{-4436777100798803}{158456325028528675187087900672}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right)}}\\ \mathbf{elif}\;b \leq 9599999999999999937557116997815619882469347802379932726602603143008612253696:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right)}\\ \mathbf{elif}\;b \leq 13500000000000000275507010685175621526490118987092636456657125042259125821644957267949903389666459196246900088209596760608108317076954234449082739494748160:\\ \;\;\;\;\frac{1}{2 + \frac{b \cdot b - t\_0 \cdot t\_0}{b - t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)}\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (let* ((t_0 (* (* 1/2 b) b)))
  (if (<= b -4436777100798803/158456325028528675187087900672)
    (/ 1 (+ 1 (/ 1 (+ 1 (* b (- (* b (+ 1/2 (* -1/6 b))) 1))))))
    (if (<=
         b
         9599999999999999937557116997815619882469347802379932726602603143008612253696)
      (/ 1 (+ 2 (* a (+ 1 (* a (+ 1/2 (* 1/6 a)))))))
      (if (<=
           b
           13500000000000000275507010685175621526490118987092636456657125042259125821644957267949903389666459196246900088209596760608108317076954234449082739494748160)
        (/ 1 (+ 2 (/ (- (* b b) (* t_0 t_0)) (- b t_0))))
        (/ 1 (+ 2 (* b (+ 1 (* 1/2 b))))))))))

C

double code(double a, double b) {
	double t_0 = (0.5 * b) * b;
	double tmp;
	if (b <= -2.8e-14) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * (0.5 + (-0.16666666666666666 * b))) - 1.0)))));
	} else if (b <= 9.6e+75) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	} else if (b <= 1.35e+154) {
		tmp = 1.0 / (2.0 + (((b * b) - (t_0 * t_0)) / (b - t_0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * 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) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * b) * b
    if (b <= (-2.8d-14)) then
        tmp = 1.0d0 / (1.0d0 + (1.0d0 / (1.0d0 + (b * ((b * (0.5d0 + ((-0.16666666666666666d0) * b))) - 1.0d0)))))
    else if (b <= 9.6d+75) then
        tmp = 1.0d0 / (2.0d0 + (a * (1.0d0 + (a * (0.5d0 + (0.16666666666666666d0 * a))))))
    else if (b <= 1.35d+154) then
        tmp = 1.0d0 / (2.0d0 + (((b * b) - (t_0 * t_0)) / (b - t_0)))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (0.5d0 * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = (0.5 * b) * b;
	double tmp;
	if (b <= -2.8e-14) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * (0.5 + (-0.16666666666666666 * b))) - 1.0)))));
	} else if (b <= 9.6e+75) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	} else if (b <= 1.35e+154) {
		tmp = 1.0 / (2.0 + (((b * b) - (t_0 * t_0)) / (b - t_0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))));
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = (0.5 * b) * b
	tmp = 0
	if b <= -2.8e-14:
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * (0.5 + (-0.16666666666666666 * b))) - 1.0)))))
	elif b <= 9.6e+75:
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))))
	elif b <= 1.35e+154:
		tmp = 1.0 / (2.0 + (((b * b) - (t_0 * t_0)) / (b - t_0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))))
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(0.5 * b) * b)
	tmp = 0.0
	if (b <= -2.8e-14)
		tmp = Float64(1.0 / Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(b * Float64(Float64(b * Float64(0.5 + Float64(-0.16666666666666666 * b))) - 1.0))))));
	elseif (b <= 9.6e+75)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(1.0 + Float64(a * Float64(0.5 + Float64(0.16666666666666666 * a)))))));
	elseif (b <= 1.35e+154)
		tmp = Float64(1.0 / Float64(2.0 + Float64(Float64(Float64(b * b) - Float64(t_0 * t_0)) / Float64(b - t_0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(0.5 * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = (0.5 * b) * b;
	tmp = 0.0;
	if (b <= -2.8e-14)
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * (0.5 + (-0.16666666666666666 * b))) - 1.0)))));
	elseif (b <= 9.6e+75)
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	elseif (b <= 1.35e+154)
		tmp = 1.0 / (2.0 + (((b * b) - (t_0 * t_0)) / (b - t_0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(1/2 * b), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -4436777100798803/158456325028528675187087900672], N[(1 / N[(1 + N[(1 / N[(1 + N[(b * N[(N[(b * N[(1/2 + N[(-1/6 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9599999999999999937557116997815619882469347802379932726602603143008612253696], N[(1 / N[(2 + N[(a * N[(1 + N[(a * N[(1/2 + N[(1/6 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 13500000000000000275507010685175621526490118987092636456657125042259125821644957267949903389666459196246900088209596760608108317076954234449082739494748160], N[(1 / N[(2 + N[(N[(N[(b * b), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1 / N[(2 + N[(b * N[(1 + N[(1/2 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.8000000000000001e-14

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

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

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

      7. exp-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 54.9%

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

   9. Applied rewrites 54.9%

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

   if -2.8000000000000001e-14 < b < 9.5999999999999999e75

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 89.6%

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

   10. Applied rewrites 89.6%

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

   11. Taylor expanded in a around 0

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

       1. lower-+.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-*.f64 86.9%

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

   13. Applied rewrites 86.9%

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

   14. Taylor expanded in b around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 56.3%

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

   16. Applied rewrites 56.3%

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

   if 9.5999999999999999e75 < b < 1.35e154

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

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

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

      7. exp-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 50.2%

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

   9. Applied rewrites 50.2%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. distribute-rgt-in N/A

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

       4. flip-+ N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-unsound-/.f64 N/A

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

       8. lower-unsound--.f64 N/A

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

       9. *-lft-identity N/A

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

       10. *-lft-identity N/A

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

       11. lower-unsound-*.f64 N/A

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

       12. lower-unsound-*.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. *-rgt-identity N/A

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

       16. *-commutative N/A

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

       17. lower-unsound--.f64 N/A

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

       18. lower-*.f64 43.1%

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

   11. Applied rewrites 43.1%

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

   if 1.35e154 < b

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

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

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

      7. exp-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 50.2%

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

   9. Applied rewrites 50.2%

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

Alternative 8: 85.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq \frac{-4436777100798803}{158456325028528675187087900672}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right)}}\\ \mathbf{elif}\;b \leq 2900000000000000220958584518619378718576206097660833200853200506439430156592527384137291964367241216:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<= b -4436777100798803/158456325028528675187087900672)
  (/ 1 (+ 1 (/ 1 (+ 1 (* b (- (* b (+ 1/2 (* -1/6 b))) 1))))))
  (if (<=
       b
       2900000000000000220958584518619378718576206097660833200853200506439430156592527384137291964367241216)
    (/ 1 (+ 2 (* a (+ 1 (* a (+ 1/2 (* 1/6 a)))))))
    (/ 1 (+ 2 (* b (+ 1 (* b (+ 1/2 (* 1/6 b))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.8e-14) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * (0.5 + (-0.16666666666666666 * b))) - 1.0)))));
	} else if (b <= 2.9e+99) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (0.16666666666666666 * 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 (b <= (-2.8d-14)) then
        tmp = 1.0d0 / (1.0d0 + (1.0d0 / (1.0d0 + (b * ((b * (0.5d0 + ((-0.16666666666666666d0) * b))) - 1.0d0)))))
    else if (b <= 2.9d+99) then
        tmp = 1.0d0 / (2.0d0 + (a * (1.0d0 + (a * (0.5d0 + (0.16666666666666666d0 * a))))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (0.16666666666666666d0 * b))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -2.8e-14) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * (0.5 + (-0.16666666666666666 * b))) - 1.0)))));
	} else if (b <= 2.9e+99) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (0.16666666666666666 * b))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -2.8e-14:
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * (0.5 + (-0.16666666666666666 * b))) - 1.0)))))
	elif b <= 2.9e+99:
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (0.16666666666666666 * b))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -2.8e-14)
		tmp = Float64(1.0 / Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(b * Float64(Float64(b * Float64(0.5 + Float64(-0.16666666666666666 * b))) - 1.0))))));
	elseif (b <= 2.9e+99)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(1.0 + Float64(a * Float64(0.5 + Float64(0.16666666666666666 * a)))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(0.16666666666666666 * b)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -2.8e-14)
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * (0.5 + (-0.16666666666666666 * b))) - 1.0)))));
	elseif (b <= 2.9e+99)
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (0.16666666666666666 * b))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -4436777100798803/158456325028528675187087900672], N[(1 / N[(1 + N[(1 / N[(1 + N[(b * N[(N[(b * N[(1/2 + N[(-1/6 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2900000000000000220958584518619378718576206097660833200853200506439430156592527384137291964367241216], N[(1 / N[(2 + N[(a * N[(1 + N[(a * N[(1/2 + N[(1/6 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1 / N[(2 + N[(b * N[(1 + N[(b * N[(1/2 + N[(1/6 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8000000000000001e-14

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

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

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

      7. exp-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 54.9%

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

   9. Applied rewrites 54.9%

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

   if -2.8000000000000001e-14 < b < 2.9000000000000002e99

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 89.6%

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

   10. Applied rewrites 89.6%

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

   11. Taylor expanded in a around 0

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

       1. lower-+.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-*.f64 86.9%

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

   13. Applied rewrites 86.9%

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

   14. Taylor expanded in b around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 56.3%

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

   16. Applied rewrites 56.3%

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

   if 2.9000000000000002e99 < b

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

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

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

      7. exp-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 50.2%

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

   9. Applied rewrites 50.2%

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

   10. Taylor expanded in b around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 54.6%

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

   12. Applied rewrites 54.6%

       \[\leadsto \frac{1}{2 + \color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 85.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq \frac{-4436777100798803}{158456325028528675187087900672}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + -1 \cdot b}}\\ \mathbf{elif}\;b \leq 2900000000000000220958584518619378718576206097660833200853200506439430156592527384137291964367241216:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<= b -4436777100798803/158456325028528675187087900672)
  (/ 1 (+ 1 (/ 1 (+ 1 (* -1 b)))))
  (if (<=
       b
       2900000000000000220958584518619378718576206097660833200853200506439430156592527384137291964367241216)
    (/ 1 (+ 2 (* a (+ 1 (* a (+ 1/2 (* 1/6 a)))))))
    (/ 1 (+ 2 (* b (+ 1 (* b (+ 1/2 (* 1/6 b))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.8e-14) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))));
	} else if (b <= 2.9e+99) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (0.16666666666666666 * 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 (b <= (-2.8d-14)) then
        tmp = 1.0d0 / (1.0d0 + (1.0d0 / (1.0d0 + ((-1.0d0) * b))))
    else if (b <= 2.9d+99) then
        tmp = 1.0d0 / (2.0d0 + (a * (1.0d0 + (a * (0.5d0 + (0.16666666666666666d0 * a))))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (0.16666666666666666d0 * b))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -2.8e-14) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))));
	} else if (b <= 2.9e+99) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (0.16666666666666666 * b))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -2.8e-14:
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))))
	elif b <= 2.9e+99:
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (0.16666666666666666 * b))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -2.8e-14)
		tmp = Float64(1.0 / Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(-1.0 * b)))));
	elseif (b <= 2.9e+99)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(1.0 + Float64(a * Float64(0.5 + Float64(0.16666666666666666 * a)))))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(0.16666666666666666 * b)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -2.8e-14)
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))));
	elseif (b <= 2.9e+99)
		tmp = 1.0 / (2.0 + (a * (1.0 + (a * (0.5 + (0.16666666666666666 * a))))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (0.16666666666666666 * b))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -4436777100798803/158456325028528675187087900672], N[(1 / N[(1 + N[(1 / N[(1 + N[(-1 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2900000000000000220958584518619378718576206097660833200853200506439430156592527384137291964367241216], N[(1 / N[(2 + N[(a * N[(1 + N[(a * N[(1/2 + N[(1/6 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1 / N[(2 + N[(b * N[(1 + N[(b * N[(1/2 + N[(1/6 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8000000000000001e-14

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

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

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

      7. exp-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in b around 0

      \[\leadsto \frac{1}{1 + \frac{1}{1 + \color{blue}{-1 \cdot b}}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 54.5%

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

   9. Applied rewrites 54.5%

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

   if -2.8000000000000001e-14 < b < 2.9000000000000002e99

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 89.6%

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

   10. Applied rewrites 89.6%

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

   11. Taylor expanded in a around 0

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

       1. lower-+.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. lower-*.f64 86.9%

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

   13. Applied rewrites 86.9%

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

   14. Taylor expanded in b around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 56.3%

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

   16. Applied rewrites 56.3%

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

   if 2.9000000000000002e99 < b

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

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

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

      7. exp-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 50.2%

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

   9. Applied rewrites 50.2%

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

   10. Taylor expanded in b around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 54.6%

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

   12. Applied rewrites 54.6%

       \[\leadsto \frac{1}{2 + \color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 82.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq \frac{-3422735716801577}{4503599627370496}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + -1 \cdot b}}\\ \mathbf{elif}\;b \leq 319999999999999989553922400211056615127884915412799357224512787796256103101104128:\\ \;\;\;\;\frac{1}{\left(\left(\left(1 + b\right) + a\right) + \left(a \cdot a\right) \cdot \frac{1}{2}\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<= b -3422735716801577/4503599627370496)
  (/ 1 (+ 1 (/ 1 (+ 1 (* -1 b)))))
  (if (<=
       b
       319999999999999989553922400211056615127884915412799357224512787796256103101104128)
    (/ 1 (- (+ (+ (+ 1 b) a) (* (* a a) 1/2)) -1))
    (/ 1 (+ 2 (* b (+ 1 (* b (+ 1/2 (* 1/6 b))))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.76) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))));
	} else if (b <= 3.2e+80) {
		tmp = 1.0 / ((((1.0 + b) + a) + ((a * a) * 0.5)) - -1.0);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (0.16666666666666666 * 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 (b <= (-0.76d0)) then
        tmp = 1.0d0 / (1.0d0 + (1.0d0 / (1.0d0 + ((-1.0d0) * b))))
    else if (b <= 3.2d+80) then
        tmp = 1.0d0 / ((((1.0d0 + b) + a) + ((a * a) * 0.5d0)) - (-1.0d0))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (0.5d0 + (0.16666666666666666d0 * b))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.76) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))));
	} else if (b <= 3.2e+80) {
		tmp = 1.0 / ((((1.0 + b) + a) + ((a * a) * 0.5)) - -1.0);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (0.16666666666666666 * b))))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.76:
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))))
	elif b <= 3.2e+80:
		tmp = 1.0 / ((((1.0 + b) + a) + ((a * a) * 0.5)) - -1.0)
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (0.16666666666666666 * b))))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.76)
		tmp = Float64(1.0 / Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(-1.0 * b)))));
	elseif (b <= 3.2e+80)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(1.0 + b) + a) + Float64(Float64(a * a) * 0.5)) - -1.0));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(0.16666666666666666 * b)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.76)
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))));
	elseif (b <= 3.2e+80)
		tmp = 1.0 / ((((1.0 + b) + a) + ((a * a) * 0.5)) - -1.0);
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (0.5 + (0.16666666666666666 * b))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -3422735716801577/4503599627370496], N[(1 / N[(1 + N[(1 / N[(1 + N[(-1 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 319999999999999989553922400211056615127884915412799357224512787796256103101104128], N[(1 / N[(N[(N[(N[(1 + b), $MachinePrecision] + a), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision], N[(1 / N[(2 + N[(b * N[(1 + N[(b * N[(1/2 + N[(1/6 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.76000000000000001

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

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

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

      7. exp-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in b around 0

      \[\leadsto \frac{1}{1 + \frac{1}{1 + \color{blue}{-1 \cdot b}}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 54.5%

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

   9. Applied rewrites 54.5%

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

   if -0.76000000000000001 < b < 3.1999999999999999e80

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 89.6%

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

   10. Applied rewrites 89.6%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. add-flip N/A

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

       4. metadata-eval N/A

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

       5. lower--.f64 89.6%

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

   12. Applied rewrites 89.6%

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

   13. Taylor expanded in b around 0

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

       1. lower-+.f64 49.8%

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

   15. Applied rewrites 49.8%

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

   if 3.1999999999999999e80 < b

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

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

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

      7. exp-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 50.2%

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

   9. Applied rewrites 50.2%

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

   10. Taylor expanded in b around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 54.6%

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

   12. Applied rewrites 54.6%

       \[\leadsto \frac{1}{2 + \color{blue}{b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 78.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq \frac{-3422735716801577}{4503599627370496}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + -1 \cdot b}}\\ \mathbf{elif}\;b \leq 18999999999999999064581089979963101249120767266604493066770021892695906750618596520567186646577434152905173284039181572611431631767743819258237216708624384:\\ \;\;\;\;\frac{1}{\left(\left(\left(1 + b\right) + a\right) + \left(a \cdot a\right) \cdot \frac{1}{2}\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)}\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<= b -3422735716801577/4503599627370496)
  (/ 1 (+ 1 (/ 1 (+ 1 (* -1 b)))))
  (if (<=
       b
       18999999999999999064581089979963101249120767266604493066770021892695906750618596520567186646577434152905173284039181572611431631767743819258237216708624384)
    (/ 1 (- (+ (+ (+ 1 b) a) (* (* a a) 1/2)) -1))
    (/ 1 (+ 2 (* b (+ 1 (* 1/2 b))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.76) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))));
	} else if (b <= 1.9e+154) {
		tmp = 1.0 / ((((1.0 + b) + a) + ((a * a) * 0.5)) - -1.0);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * 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 (b <= (-0.76d0)) then
        tmp = 1.0d0 / (1.0d0 + (1.0d0 / (1.0d0 + ((-1.0d0) * b))))
    else if (b <= 1.9d+154) then
        tmp = 1.0d0 / ((((1.0d0 + b) + a) + ((a * a) * 0.5d0)) - (-1.0d0))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (0.5d0 * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.76) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))));
	} else if (b <= 1.9e+154) {
		tmp = 1.0 / ((((1.0 + b) + a) + ((a * a) * 0.5)) - -1.0);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.76:
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))))
	elif b <= 1.9e+154:
		tmp = 1.0 / ((((1.0 + b) + a) + ((a * a) * 0.5)) - -1.0)
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.76)
		tmp = Float64(1.0 / Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(-1.0 * b)))));
	elseif (b <= 1.9e+154)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(1.0 + b) + a) + Float64(Float64(a * a) * 0.5)) - -1.0));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(0.5 * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.76)
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))));
	elseif (b <= 1.9e+154)
		tmp = 1.0 / ((((1.0 + b) + a) + ((a * a) * 0.5)) - -1.0);
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -3422735716801577/4503599627370496], N[(1 / N[(1 + N[(1 / N[(1 + N[(-1 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 18999999999999999064581089979963101249120767266604493066770021892695906750618596520567186646577434152905173284039181572611431631767743819258237216708624384], N[(1 / N[(N[(N[(N[(1 + b), $MachinePrecision] + a), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision], N[(1 / N[(2 + N[(b * N[(1 + N[(1/2 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.76000000000000001

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

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

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

      7. exp-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in b around 0

      \[\leadsto \frac{1}{1 + \frac{1}{1 + \color{blue}{-1 \cdot b}}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 54.5%

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

   9. Applied rewrites 54.5%

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

   if -0.76000000000000001 < b < 1.8999999999999999e154

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 89.6%

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

   10. Applied rewrites 89.6%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. add-flip N/A

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

       4. metadata-eval N/A

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

       5. lower--.f64 89.6%

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

   12. Applied rewrites 89.6%

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

   13. Taylor expanded in b around 0

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

       1. lower-+.f64 49.8%

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

   15. Applied rewrites 49.8%

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

   if 1.8999999999999999e154 < b

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

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

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

      7. exp-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 50.2%

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

   9. Applied rewrites 50.2%

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

Alternative 12: 77.6% accurate, 7.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq \frac{-4436777100798803}{158456325028528675187087900672}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + -1 \cdot b}}\\ \mathbf{elif}\;b \leq 2799999999999999877191140408237791714304266252051693240387236978082789331230448443084335838169194391935053968040682242044884434993644427170305585304288886784:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(1 + \frac{1}{2} \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)}\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<= b -4436777100798803/158456325028528675187087900672)
  (/ 1 (+ 1 (/ 1 (+ 1 (* -1 b)))))
  (if (<=
       b
       2799999999999999877191140408237791714304266252051693240387236978082789331230448443084335838169194391935053968040682242044884434993644427170305585304288886784)
    (/ 1 (+ 2 (* a (+ 1 (* 1/2 a)))))
    (/ 1 (+ 2 (* b (+ 1 (* 1/2 b))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.8e-14) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))));
	} else if (b <= 2.8e+156) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (0.5 * a))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * 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 (b <= (-2.8d-14)) then
        tmp = 1.0d0 / (1.0d0 + (1.0d0 / (1.0d0 + ((-1.0d0) * b))))
    else if (b <= 2.8d+156) then
        tmp = 1.0d0 / (2.0d0 + (a * (1.0d0 + (0.5d0 * a))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (0.5d0 * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -2.8e-14) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))));
	} else if (b <= 2.8e+156) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (0.5 * a))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -2.8e-14:
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))))
	elif b <= 2.8e+156:
		tmp = 1.0 / (2.0 + (a * (1.0 + (0.5 * a))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -2.8e-14)
		tmp = Float64(1.0 / Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(-1.0 * b)))));
	elseif (b <= 2.8e+156)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(1.0 + Float64(0.5 * a)))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(0.5 * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -2.8e-14)
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (-1.0 * b))));
	elseif (b <= 2.8e+156)
		tmp = 1.0 / (2.0 + (a * (1.0 + (0.5 * a))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -4436777100798803/158456325028528675187087900672], N[(1 / N[(1 + N[(1 / N[(1 + N[(-1 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2799999999999999877191140408237791714304266252051693240387236978082789331230448443084335838169194391935053968040682242044884434993644427170305585304288886784], N[(1 / N[(2 + N[(a * N[(1 + N[(1/2 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1 / N[(2 + N[(b * N[(1 + N[(1/2 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8000000000000001e-14

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

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

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

      7. exp-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in b around 0

      \[\leadsto \frac{1}{1 + \frac{1}{1 + \color{blue}{-1 \cdot b}}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 54.5%

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

   9. Applied rewrites 54.5%

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

   if -2.8000000000000001e-14 < b < 2.7999999999999999e156

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 89.6%

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

   10. Applied rewrites 89.6%

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

   11. Taylor expanded in b around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 52.1%

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

   13. Applied rewrites 52.1%

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

   if 2.7999999999999999e156 < b

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

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

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

      7. exp-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 50.2%

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

   9. Applied rewrites 50.2%

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

Alternative 13: 62.8% accurate, 9.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq 2799999999999999877191140408237791714304266252051693240387236978082789331230448443084335838169194391935053968040682242044884434993644427170305585304288886784:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(1 + \frac{1}{2} \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)}\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<=
     b
     2799999999999999877191140408237791714304266252051693240387236978082789331230448443084335838169194391935053968040682242044884434993644427170305585304288886784)
  (/ 1 (+ 2 (* a (+ 1 (* 1/2 a)))))
  (/ 1 (+ 2 (* b (+ 1 (* 1/2 b)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.8e+156) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (0.5 * a))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * 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 (b <= 2.8d+156) then
        tmp = 1.0d0 / (2.0d0 + (a * (1.0d0 + (0.5d0 * a))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (0.5d0 * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.8e+156) {
		tmp = 1.0 / (2.0 + (a * (1.0 + (0.5 * a))));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.8e+156:
		tmp = 1.0 / (2.0 + (a * (1.0 + (0.5 * a))))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.8e+156)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(1.0 + Float64(0.5 * a)))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(0.5 * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.8e+156)
		tmp = 1.0 / (2.0 + (a * (1.0 + (0.5 * a))));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2799999999999999877191140408237791714304266252051693240387236978082789331230448443084335838169194391935053968040682242044884434993644427170305585304288886784], N[(1 / N[(2 + N[(a * N[(1 + N[(1/2 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1 / N[(2 + N[(b * N[(1 + N[(1/2 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7999999999999999e156

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 89.6%

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

   10. Applied rewrites 89.6%

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

   11. Taylor expanded in b around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 52.1%

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

   13. Applied rewrites 52.1%

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

   if 2.7999999999999999e156 < b

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

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

      4. cosh-neg-rev N/A

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

      5. sinh-neg-rev N/A

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

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

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

      7. exp-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 99.2%

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

   3. Applied rewrites 99.2%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 81.5%

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

   6. Applied rewrites 81.5%

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

   7. Taylor expanded in b around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 50.2%

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

   9. Applied rewrites 50.2%

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

Alternative 14: 53.0% accurate, 9.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;b \leq \frac{5342339453620755}{242833611528216133864932738352939863330300854881517440156476551217363035650651062272}:\\ \;\;\;\;\frac{1}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)}\\ \end{array} \]

FPCore

(FPCore (a b)
  :precision binary64
  (if (<=
     b
     5342339453620755/242833611528216133864932738352939863330300854881517440156476551217363035650651062272)
  (/ 1 (+ 2 a))
  (/ 1 (+ 2 (* b (+ 1 (* 1/2 b)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.2e-68) {
		tmp = 1.0 / (2.0 + a);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * 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 (b <= 2.2d-68) then
        tmp = 1.0d0 / (2.0d0 + a)
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (0.5d0 * b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2.2e-68) {
		tmp = 1.0 / (2.0 + a);
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.2e-68:
		tmp = 1.0 / (2.0 + a)
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.2e-68)
		tmp = Float64(1.0 / Float64(2.0 + a));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(0.5 * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2.2e-68)
		tmp = 1.0 / (2.0 + a);
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (0.5 * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 5342339453620755/242833611528216133864932738352939863330300854881517440156476551217363035650651062272], N[(1 / N[(2 + a), $MachinePrecision]), $MachinePrecision], N[(1 / N[(2 + N[(b * N[(1 + N[(1/2 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2e-68

   1. Initial program 99.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 81.1%

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

   10. Applied rewrites 81.1%

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

   11. Taylor expanded in b around 0

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

       1. lower-+.f64 39.7%

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

   13. Applied rewrites 39.7%

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

   if 2.2e-68 < b

   1. Initial program 99.2%

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

      1. lift-exp.f64 N/A

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

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

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

      3. add-flip N/A

         \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{\left(\cosh b - \left(\mathsf{neg}\left(\sinh b\right)\right)\right)}} \]

      4. cosh-neg-rev N/A

         \[\leadsto \frac{e^{a}}{e^{a} + \left(\color{blue}{\cosh \left(\mathsf{neg}\left(b\right)\right)} - \left(\mathsf{neg}\left(\sinh b\right)\right)\right)} \]

      5. sinh-neg-rev N/A

         \[\leadsto \frac{e^{a}}{e^{a} + \left(\cosh \left(\mathsf{neg}\left(b\right)\right) - \color{blue}{\sinh \left(\mathsf{neg}\left(b\right)\right)}\right)} \]

      6. sinh---cosh-rev N/A

         \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(b\right)\right)\right)}}} \]

      7. exp-neg N/A

         \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{\frac{1}{e^{\mathsf{neg}\left(b\right)}}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{\frac{1}{e^{\mathsf{neg}\left(b\right)}}}} \]

      9. lower-exp.f64 N/A

         \[\leadsto \frac{e^{a}}{e^{a} + \frac{1}{\color{blue}{e^{\mathsf{neg}\left(b\right)}}}} \]

      10. lower-neg.f64 99.2%

          \[\leadsto \frac{e^{a}}{e^{a} + \frac{1}{e^{\color{blue}{-b}}}} \]

   3. Applied rewrites 99.2%

      \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{\frac{1}{e^{-b}}}} \]

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{e^{\mathsf{neg}\left(b\right)}}}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{1 + \frac{1}{e^{\mathsf{neg}\left(b\right)}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{e^{\mathsf{neg}\left(b\right)}}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{e^{\mathsf{neg}\left(b\right)}}}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \frac{1}{1 + \frac{1}{e^{\mathsf{neg}\left(b\right)}}} \]

      5. lower-neg.f64 81.5%

         \[\leadsto \frac{1}{1 + \frac{1}{e^{-b}}} \]

   6. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{e^{-b}}}} \]

   7. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2 + \color{blue}{b \cdot \left(1 + \frac{1}{2} \cdot b\right)}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{2 + b \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot b\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + \color{blue}{\frac{1}{2} \cdot b}\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + \frac{1}{2} \cdot \color{blue}{b}\right)} \]

      4. lower-*.f64 50.2%

         \[\leadsto \frac{1}{2 + b \cdot \left(1 + \frac{1}{2} \cdot b\right)} \]

   9. Applied rewrites 50.2%

      \[\leadsto \frac{1}{2 + \color{blue}{b \cdot \left(1 + \frac{1}{2} \cdot b\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 39.7% accurate, 21.0× speedup

Math

\[\frac{1}{2 + a} \]

FPCore

(FPCore (a b)
  :precision binary64
  (/ 1 (+ 2 a)))

C

double code(double a, double b) {
	return 1.0 / (2.0 + a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 + a)
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 + a);
}

Python

def code(a, b):
	return 1.0 / (2.0 + a)

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 + a))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 + a);
end

Wolfram

code[a_, b_] := N[(1 / N[(2 + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]

2. Taylor expanded in a around 0

   \[\leadsto \frac{\color{blue}{1}}{e^{a} + e^{b}} \]
3. Step-by-step derivation

4. Applied rewrites 80.7%

   \[\leadsto \frac{\color{blue}{1}}{e^{a} + e^{b}} \]

5. Taylor expanded in a around 0

   \[\leadsto \frac{1}{\color{blue}{1} + e^{b}} \]
6. Step-by-step derivation

7. Applied rewrites 81.5%

   \[\leadsto \frac{1}{\color{blue}{1} + e^{b}} \]

8. Taylor expanded in a around 0

   \[\leadsto \frac{1}{\color{blue}{1 + \left(a + e^{b}\right)}} \]
9. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{1}{1 + \color{blue}{\left(a + e^{b}\right)}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{1}{1 + \left(a + \color{blue}{e^{b}}\right)} \]

   3. lower-exp.f64 81.1%

      \[\leadsto \frac{1}{1 + \left(a + e^{b}\right)} \]

10. Applied rewrites 81.1%

    \[\leadsto \frac{1}{\color{blue}{1 + \left(a + e^{b}\right)}} \]

11. Taylor expanded in b around 0

    \[\leadsto \frac{1}{2 + \color{blue}{a}} \]
12. Step-by-step derivation

    1. lower-+.f64 39.7%

       \[\leadsto \frac{1}{2 + a} \]

13. Applied rewrites 39.7%

    \[\leadsto \frac{1}{2 + \color{blue}{a}} \]
14. Add Preprocessing

Alternative 16: 39.2% accurate, 315.0× speedup

Math

\[\frac{1}{2} \]

FPCore

(FPCore (a b)
  :precision binary64
  1/2)

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_] := 1/2

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{e^{b}}} \]

   2. sinh-+-cosh-rev N/A

      \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{\left(\cosh b + \sinh b\right)}} \]

   3. add-flip N/A

      \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{\left(\cosh b - \left(\mathsf{neg}\left(\sinh b\right)\right)\right)}} \]

   4. cosh-neg-rev N/A

      \[\leadsto \frac{e^{a}}{e^{a} + \left(\color{blue}{\cosh \left(\mathsf{neg}\left(b\right)\right)} - \left(\mathsf{neg}\left(\sinh b\right)\right)\right)} \]

   5. sinh-neg-rev N/A

      \[\leadsto \frac{e^{a}}{e^{a} + \left(\cosh \left(\mathsf{neg}\left(b\right)\right) - \color{blue}{\sinh \left(\mathsf{neg}\left(b\right)\right)}\right)} \]

   6. sinh---cosh-rev N/A

      \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(b\right)\right)\right)}}} \]

   7. exp-neg N/A

      \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{\frac{1}{e^{\mathsf{neg}\left(b\right)}}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{\frac{1}{e^{\mathsf{neg}\left(b\right)}}}} \]

   9. lower-exp.f64 N/A

      \[\leadsto \frac{e^{a}}{e^{a} + \frac{1}{\color{blue}{e^{\mathsf{neg}\left(b\right)}}}} \]

   10. lower-neg.f64 99.2%

       \[\leadsto \frac{e^{a}}{e^{a} + \frac{1}{e^{\color{blue}{-b}}}} \]

3. Applied rewrites 99.2%

   \[\leadsto \frac{e^{a}}{e^{a} + \color{blue}{\frac{1}{e^{-b}}}} \]

4. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{e^{\mathsf{neg}\left(b\right)}}}} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{1 + \frac{1}{e^{\mathsf{neg}\left(b\right)}}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{e^{\mathsf{neg}\left(b\right)}}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{e^{\mathsf{neg}\left(b\right)}}}} \]

   4. lower-exp.f64 N/A

      \[\leadsto \frac{1}{1 + \frac{1}{e^{\mathsf{neg}\left(b\right)}}} \]

   5. lower-neg.f64 81.5%

      \[\leadsto \frac{1}{1 + \frac{1}{e^{-b}}} \]

6. Applied rewrites 81.5%

   \[\leadsto \color{blue}{\frac{1}{1 + \frac{1}{e^{-b}}}} \]

7. Taylor expanded in b around 0

   \[\leadsto \frac{1}{2} \]
8. Step-by-step derivation

9. Applied rewrites 39.2%

   \[\leadsto \frac{1}{2} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64
  (/ (exp a) (+ (exp a) (exp b))))