Quotient of sum of exps

Percentage Accurate: 99.1% → 99.1%

Time: 5.4s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. a = 1.3817989107491465e+259
  2. b = 1.112543076641457e+243

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

Alternative 2: 98.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -100000000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -100000000.0) (/ (exp a) 2.0) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -100000000.0) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 1.0 / (exp(b) - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-100000000.0d0)) then
        tmp = exp(a) / 2.0d0
    else
        tmp = 1.0d0 / (exp(b) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -100000000.0) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -100000000.0:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -100000000.0)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(1.0 / Float64(exp(b) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -100000000.0)
		tmp = exp(a) / 2.0;
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -100000000.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1e8

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if -1e8 < a

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lower-exp.f64 98.3

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

   5. Applied rewrites 98.3%

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

Alternative 3: 76.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 23000:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot 0.5}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 23000.0)
   (/ (exp a) 2.0)
   (if (<= b 7e+98)
     (/ (* (* a a) 0.5) (+ 2.0 a))
     (/ 1.0 (* (* (fma 0.16666666666666666 b 0.5) b) b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 23000.0) {
		tmp = exp(a) / 2.0;
	} else if (b <= 7e+98) {
		tmp = ((a * a) * 0.5) / (2.0 + a);
	} else {
		tmp = 1.0 / ((fma(0.16666666666666666, b, 0.5) * b) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 23000.0)
		tmp = Float64(exp(a) / 2.0);
	elseif (b <= 7e+98)
		tmp = Float64(Float64(Float64(a * a) * 0.5) / Float64(2.0 + a));
	else
		tmp = Float64(1.0 / Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 23000.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[b, 7e+98], N[(N[(N[(a * a), $MachinePrecision] * 0.5), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 23000

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 79.6

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 78.5%

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

   if 23000 < b < 7e98

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 18.4

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

   5. Applied rewrites 18.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 18.4%

      \[\leadsto \frac{e^{a}}{2 + \color{blue}{a}} \]

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 2.9

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

   11. Applied rewrites 2.9%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 49.3%

       \[\leadsto \frac{\left(a \cdot a\right) \cdot \color{blue}{0.5}}{2 + a} \]

   if 7e98 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 96.5%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{{b}^{3} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{2} \cdot \frac{1}{b}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.5%

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

Alternative 4: 71.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -740000000:\\ \;\;\;\;\frac{1 + a}{2 + a}\\ \mathbf{elif}\;b \leq 720:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 2\right)\right)}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot 0.5}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -740000000.0)
   (/ (+ 1.0 a) (+ 2.0 a))
   (if (<= b 720.0)
     (/
      1.0
      (fma
       (fma 0.5 b 1.0)
       b
       (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 2.0)))
     (if (<= b 7e+98)
       (/ (* (* a a) 0.5) (+ 2.0 a))
       (/ 1.0 (* (* (fma 0.16666666666666666 b 0.5) b) b))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -740000000.0) {
		tmp = (1.0 + a) / (2.0 + a);
	} else if (b <= 720.0) {
		tmp = 1.0 / fma(fma(0.5, b, 1.0), b, fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 2.0));
	} else if (b <= 7e+98) {
		tmp = ((a * a) * 0.5) / (2.0 + a);
	} else {
		tmp = 1.0 / ((fma(0.16666666666666666, b, 0.5) * b) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -740000000.0)
		tmp = Float64(Float64(1.0 + a) / Float64(2.0 + a));
	elseif (b <= 720.0)
		tmp = Float64(1.0 / fma(fma(0.5, b, 1.0), b, fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 2.0)));
	elseif (b <= 7e+98)
		tmp = Float64(Float64(Float64(a * a) * 0.5) / Float64(2.0 + a));
	else
		tmp = Float64(1.0 / Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -740000000.0], N[(N[(1.0 + a), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 720.0], N[(1.0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e+98], N[(N[(N[(a * a), $MachinePrecision] * 0.5), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -7.4e8

   1. Initial program 95.3%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 17.7

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

   5. Applied rewrites 17.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 17.6%

      \[\leadsto \frac{e^{a}}{2 + \color{blue}{a}} \]

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 20.8

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

   11. Applied rewrites 20.8%

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

   if -7.4e8 < b < 720

   1. Initial program 99.3%

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

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-exp.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 87.3%

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

   if 720 < b < 7e98

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 18.4

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

   5. Applied rewrites 18.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 18.4%

      \[\leadsto \frac{e^{a}}{2 + \color{blue}{a}} \]

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 2.9

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

   11. Applied rewrites 2.9%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 49.3%

       \[\leadsto \frac{\left(a \cdot a\right) \cdot \color{blue}{0.5}}{2 + a} \]

   if 7e98 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 96.5%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{{b}^{3} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{2} \cdot \frac{1}{b}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.5%

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

Alternative 5: 59.6% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 190:\\ \;\;\;\;\frac{1 + a}{2 + a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot 0.5}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 190.0)
   (/ (+ 1.0 a) (+ 2.0 a))
   (if (<= b 7e+98)
     (/ (* (* a a) 0.5) (+ 2.0 a))
     (/ 1.0 (* (* (fma 0.16666666666666666 b 0.5) b) b)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 190.0) {
		tmp = (1.0 + a) / (2.0 + a);
	} else if (b <= 7e+98) {
		tmp = ((a * a) * 0.5) / (2.0 + a);
	} else {
		tmp = 1.0 / ((fma(0.16666666666666666, b, 0.5) * b) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 190.0)
		tmp = Float64(Float64(1.0 + a) / Float64(2.0 + a));
	elseif (b <= 7e+98)
		tmp = Float64(Float64(Float64(a * a) * 0.5) / Float64(2.0 + a));
	else
		tmp = Float64(1.0 / Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 190.0], N[(N[(1.0 + a), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e+98], N[(N[(N[(a * a), $MachinePrecision] * 0.5), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 190

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 79.6

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.1%

      \[\leadsto \frac{e^{a}}{2 + \color{blue}{a}} \]

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 59.2

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

   11. Applied rewrites 59.2%

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

   if 190 < b < 7e98

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 18.4

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

   5. Applied rewrites 18.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 18.4%

      \[\leadsto \frac{e^{a}}{2 + \color{blue}{a}} \]

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 2.9

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

   11. Applied rewrites 2.9%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 49.3%

       \[\leadsto \frac{\left(a \cdot a\right) \cdot \color{blue}{0.5}}{2 + a} \]

   if 7e98 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 96.5%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{{b}^{3} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{2} \cdot \frac{1}{b}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.5%

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

Alternative 6: 56.7% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 190:\\ \;\;\;\;\frac{1 + a}{2 + a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot 0.5}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 1\right) - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 190.0)
   (/ (+ 1.0 a) (+ 2.0 a))
   (if (<= b 1.9e+154)
     (/ (* (* a a) 0.5) (+ 2.0 a))
     (/ 1.0 (- (fma (fma 0.5 b 1.0) b 1.0) -1.0)))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 190.0) {
		tmp = (1.0 + a) / (2.0 + a);
	} else if (b <= 1.9e+154) {
		tmp = ((a * a) * 0.5) / (2.0 + a);
	} else {
		tmp = 1.0 / (fma(fma(0.5, b, 1.0), b, 1.0) - -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 190.0)
		tmp = Float64(Float64(1.0 + a) / Float64(2.0 + a));
	elseif (b <= 1.9e+154)
		tmp = Float64(Float64(Float64(a * a) * 0.5) / Float64(2.0 + a));
	else
		tmp = Float64(1.0 / Float64(fma(fma(0.5, b, 1.0), b, 1.0) - -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 190.0], N[(N[(1.0 + a), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.9e+154], N[(N[(N[(a * a), $MachinePrecision] * 0.5), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 190

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 79.6

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.1%

      \[\leadsto \frac{e^{a}}{2 + \color{blue}{a}} \]

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 59.2

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

   11. Applied rewrites 59.2%

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

   if 190 < b < 1.8999999999999999e154

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 31.6

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

   5. Applied rewrites 31.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 31.6%

      \[\leadsto \frac{e^{a}}{2 + \color{blue}{a}} \]

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 2.7

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

   11. Applied rewrites 2.7%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 37.4%

       \[\leadsto \frac{\left(a \cdot a\right) \cdot \color{blue}{0.5}}{2 + a} \]

   if 1.8999999999999999e154 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 100.0%

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

Alternative 7: 48.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 190:\\ \;\;\;\;\frac{1 + a}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot 0.5}{2 + a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 190.0) (/ (+ 1.0 a) (+ 2.0 a)) (/ (* (* a a) 0.5) (+ 2.0 a))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 190.0) {
		tmp = (1.0 + a) / (2.0 + a);
	} else {
		tmp = ((a * a) * 0.5) / (2.0 + a);
	}
	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 <= 190.0d0) then
        tmp = (1.0d0 + a) / (2.0d0 + a)
    else
        tmp = ((a * a) * 0.5d0) / (2.0d0 + a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 190.0) {
		tmp = (1.0 + a) / (2.0 + a);
	} else {
		tmp = ((a * a) * 0.5) / (2.0 + a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 190.0:
		tmp = (1.0 + a) / (2.0 + a)
	else:
		tmp = ((a * a) * 0.5) / (2.0 + a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 190.0)
		tmp = Float64(Float64(1.0 + a) / Float64(2.0 + a));
	else
		tmp = Float64(Float64(Float64(a * a) * 0.5) / Float64(2.0 + a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 190.0)
		tmp = (1.0 + a) / (2.0 + a);
	else
		tmp = ((a * a) * 0.5) / (2.0 + a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 190.0], N[(N[(1.0 + a), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * 0.5), $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 190

   1. Initial program 98.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 79.6

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.1%

      \[\leadsto \frac{e^{a}}{2 + \color{blue}{a}} \]

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 59.2

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

   11. Applied rewrites 59.2%

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

   if 190 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-exp.f64 33.6

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

   5. Applied rewrites 33.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 33.6%

      \[\leadsto \frac{e^{a}}{2 + \color{blue}{a}} \]

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 2.7

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

   11. Applied rewrites 2.7%

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

   12. Taylor expanded in a around inf

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

   14. Applied rewrites 33.4%

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

Alternative 8: 39.9% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \frac{1 + a}{2 + a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (+ 1.0 a) (+ 2.0 a)))

C

double code(double a, double b) {
	return (1.0 + a) / (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 + a) / (2.0d0 + a)
end function

Java

public static double code(double a, double b) {
	return (1.0 + a) / (2.0 + a);
}

Python

def code(a, b):
	return (1.0 + a) / (2.0 + a)

Julia

function code(a, b)
	return Float64(Float64(1.0 + a) / Float64(2.0 + a))
end

MATLAB

function tmp = code(a, b)
	tmp = (1.0 + a) / (2.0 + a);
end

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 N/A

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

   7. lower-exp.f64 67.0

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

5. Applied rewrites 67.0%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 66.6%

   \[\leadsto \frac{e^{a}}{2 + \color{blue}{a}} \]

9. Taylor expanded in a around 0

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

    1. lower-+.f64 43.9

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

11. Applied rewrites 43.9%

    \[\leadsto \frac{\color{blue}{1 + a}}{2 + a} \]
12. Add Preprocessing

Alternative 9: 39.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 + a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 2.0 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[(1.0 / N[(2.0 + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

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

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. lower--.f64 N/A

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

   7. lower-exp.f64 67.0

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

5. Applied rewrites 67.0%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 66.6%

   \[\leadsto \frac{e^{a}}{2 + \color{blue}{a}} \]

9. Taylor expanded in a around 0

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

11. Applied rewrites 43.4%

    \[\leadsto \frac{\color{blue}{1}}{2 + a} \]
12. Add Preprocessing

Alternative 10: 39.4% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower--.f64 N/A

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

   8. lower-exp.f64 83.6

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

5. Applied rewrites 83.6%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 43.0%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025017 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ 1 (exp (- b a)))))

  (/ (exp a) (+ (exp a) (exp b))))