Quotient of sum of exps

Percentage Accurate: 98.8% → 98.4%

Time: 4.0s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. a = 5.457386410833273e+209
  2. b = 4.2016472728279424e-106

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e-8) (/ (exp a) (+ (exp a) 1.0)) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.4e-8) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = 1.0 / (exp(b) - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5.4d-8)) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = 1.0d0 / (exp(b) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.4e-8) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.4e-8:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.4e-8)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.4e-8)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.4e-8], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.40000000000000005e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   if -5.40000000000000005e-8 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 99.7

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

   7. Applied rewrites 99.7%

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

Alternative 2: 98.8% 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 99.6%

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

Alternative 3: 98.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -5.4e-8) (/ (exp a) (+ 1.0 1.0)) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -5.4e-8) {
		tmp = exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (exp(b) - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-5.4d-8)) then
        tmp = exp(a) / (1.0d0 + 1.0d0)
    else
        tmp = 1.0d0 / (exp(b) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -5.4e-8) {
		tmp = Math.exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -5.4e-8:
		tmp = math.exp(a) / (1.0 + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -5.4e-8)
		tmp = Float64(exp(a) / Float64(1.0 + 1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -5.4e-8)
		tmp = exp(a) / (1.0 + 1.0);
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -5.4e-8], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.40000000000000005e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 99.4%

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

   if -5.40000000000000005e-8 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 99.7

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

   7. Applied rewrites 99.7%

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

Alternative 4: 84.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{+65}:\\ \;\;\;\;{b}^{5} \cdot -0.0020833333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.5e+65)
   (* (pow b 5.0) -0.0020833333333333333)
   (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.5e+65) {
		tmp = pow(b, 5.0) * -0.0020833333333333333;
	} 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 <= (-3.5d+65)) then
        tmp = (b ** 5.0d0) * (-0.0020833333333333333d0)
    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 <= -3.5e+65) {
		tmp = Math.pow(b, 5.0) * -0.0020833333333333333;
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.5e+65:
		tmp = math.pow(b, 5.0) * -0.0020833333333333333
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.5e+65)
		tmp = Float64((b ^ 5.0) * -0.0020833333333333333);
	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 <= -3.5e+65)
		tmp = (b ^ 5.0) * -0.0020833333333333333;
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.5e+65], N[(N[Power[b, 5.0], $MachinePrecision] * -0.0020833333333333333), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.5000000000000001e65

   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. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 36.7

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}\right) \cdot b + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}, b, \frac{1}{2}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}, b, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{480} \cdot {b}^{2} + \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, {b}^{2}, \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, b \cdot b, \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, b \cdot b, \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, b \cdot b, \frac{1}{48}\right) \cdot \left(b \cdot b\right) - \frac{1}{4}, b, \frac{1}{2}\right) \]

      12. lower-*.f64 2.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0020833333333333333, b \cdot b, 0.020833333333333332\right) \cdot \left(b \cdot b\right) - 0.25, b, 0.5\right) \]

   8. Applied rewrites 2.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0020833333333333333, b \cdot b, 0.020833333333333332\right) \cdot \left(b \cdot b\right) - 0.25, \color{blue}{b}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{-1}{480} \cdot {b}^{\color{blue}{5}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {b}^{5} \cdot \frac{-1}{480} \]

       2. lower-*.f64 N/A

          \[\leadsto {b}^{5} \cdot \frac{-1}{480} \]

       3. lower-pow.f64 49.5

          \[\leadsto {b}^{5} \cdot -0.0020833333333333333 \]

   11. Applied rewrites 49.5%

       \[\leadsto {b}^{5} \cdot -0.0020833333333333333 \]

   if -3.5000000000000001e65 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 95.7

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

   5. Applied rewrites 95.7%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 95.7

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

   7. Applied rewrites 95.7%

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

Alternative 5: 62.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7000000000000:\\ \;\;\;\;{b}^{5} \cdot -0.0020833333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7000000000000.0)
   (* (pow b 5.0) -0.0020833333333333333)
   (/ 1.0 (- (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7000000000000.0) {
		tmp = pow(b, 5.0) * -0.0020833333333333333;
	} else {
		tmp = 1.0 / (fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) - -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7000000000000.0)
		tmp = Float64((b ^ 5.0) * -0.0020833333333333333);
	else
		tmp = Float64(1.0 / Float64(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) - -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7000000000000.0], N[(N[Power[b, 5.0], $MachinePrecision] * -0.0020833333333333333), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7e12

   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. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto b \cdot \left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2} \]

      2. *-commutative N/A

         \[\leadsto \left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}\right) \cdot b + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}, b, \frac{1}{2}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}, b, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{480} \cdot {b}^{2} + \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, {b}^{2}, \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, b \cdot b, \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, b \cdot b, \frac{1}{48}\right) \cdot {b}^{2} - \frac{1}{4}, b, \frac{1}{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{480}, b \cdot b, \frac{1}{48}\right) \cdot \left(b \cdot b\right) - \frac{1}{4}, b, \frac{1}{2}\right) \]

      12. lower-*.f64 2.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0020833333333333333, b \cdot b, 0.020833333333333332\right) \cdot \left(b \cdot b\right) - 0.25, b, 0.5\right) \]

   8. Applied rewrites 2.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0020833333333333333, b \cdot b, 0.020833333333333332\right) \cdot \left(b \cdot b\right) - 0.25, \color{blue}{b}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{-1}{480} \cdot {b}^{\color{blue}{5}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {b}^{5} \cdot \frac{-1}{480} \]

       2. lower-*.f64 N/A

          \[\leadsto {b}^{5} \cdot \frac{-1}{480} \]

       3. lower-pow.f64 46.5

          \[\leadsto {b}^{5} \cdot -0.0020833333333333333 \]

   11. Applied rewrites 46.5%

       \[\leadsto {b}^{5} \cdot -0.0020833333333333333 \]

   if -7e12 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 99.0

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

   5. Applied rewrites 99.0%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 66.4

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

   10. Applied rewrites 66.4%

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

Alternative 6: 57.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-263}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.5e-263)
   (fma 0.25 a 0.5)
   (/ 1.0 (- (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.5e-263) {
		tmp = fma(0.25, a, 0.5);
	} else {
		tmp = 1.0 / (fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) - -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.5e-263)
		tmp = fma(0.25, a, 0.5);
	else
		tmp = Float64(1.0 / Float64(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) - -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.5e-263], N[(0.25 * a + 0.5), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.49999999999999969e-263

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 46.8

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

   8. Applied rewrites 46.8%

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

   if 3.49999999999999969e-263 < b

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 91.0

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

   5. Applied rewrites 91.0%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 91.0

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

   7. Applied rewrites 91.0%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 71.1

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

   10. Applied rewrites 71.1%

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

Alternative 7: 57.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-263}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.5e-263)
   (fma 0.25 a 0.5)
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.5e-263) {
		tmp = fma(0.25, a, 0.5);
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.5e-263)
		tmp = fma(0.25, a, 0.5);
	else
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.5e-263], N[(0.25 * a + 0.5), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.49999999999999969e-263

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 46.8

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

   8. Applied rewrites 46.8%

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

   if 3.49999999999999969e-263 < b

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 91.0

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

   5. Applied rewrites 91.0%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 91.0

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

   7. Applied rewrites 91.0%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 71.1

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

   10. Applied rewrites 71.1%

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

Alternative 8: 53.4% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-263}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.5e-263) (fma 0.25 a 0.5) (/ 1.0 (fma (fma 0.5 b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.5e-263) {
		tmp = fma(0.25, a, 0.5);
	} else {
		tmp = 1.0 / fma(fma(0.5, b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.5e-263)
		tmp = fma(0.25, a, 0.5);
	else
		tmp = Float64(1.0 / fma(fma(0.5, b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.5e-263], N[(0.25 * a + 0.5), $MachinePrecision], N[(1.0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.49999999999999969e-263

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 46.8

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

   8. Applied rewrites 46.8%

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

   if 3.49999999999999969e-263 < b

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 61.7

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

   8. Applied rewrites 61.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 61.7

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

   10. Applied rewrites 61.7%

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

Alternative 9: 53.1% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-263}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5 \cdot b, b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.5e-263) (fma 0.25 a 0.5) (/ 1.0 (fma (* 0.5 b) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.5e-263) {
		tmp = fma(0.25, a, 0.5);
	} else {
		tmp = 1.0 / fma((0.5 * b), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.5e-263)
		tmp = fma(0.25, a, 0.5);
	else
		tmp = Float64(1.0 / fma(Float64(0.5 * b), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.5e-263], N[(0.25 * a + 0.5), $MachinePrecision], N[(1.0 / N[(N[(0.5 * b), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.49999999999999969e-263

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 46.8

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

   8. Applied rewrites 46.8%

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

   if 3.49999999999999969e-263 < b

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 61.7

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

   8. Applied rewrites 61.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 61.7

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

   10. Applied rewrites 61.7%

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

   11. Taylor expanded in b around inf

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

       1. lower-*.f64 61.2

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

   13. Applied rewrites 61.2%

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

Alternative 10: 53.3% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.12:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b \cdot b, 0.5, b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.12) (fma 0.25 a 0.5) (/ 1.0 (fma (* b b) 0.5 b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.12) {
		tmp = fma(0.25, a, 0.5);
	} else {
		tmp = 1.0 / fma((b * b), 0.5, b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.12)
		tmp = fma(0.25, a, 0.5);
	else
		tmp = Float64(1.0 / fma(Float64(b * b), 0.5, b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.12], N[(0.25 * a + 0.5), $MachinePrecision], N[(1.0 / N[(N[(b * b), $MachinePrecision] * 0.5 + b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1200000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 55.0

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

   8. Applied rewrites 55.0%

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

   if 1.1200000000000001 < b

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 54.0

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

   8. Applied rewrites 54.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 54.0

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

   10. Applied rewrites 54.0%

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

   11. Taylor expanded in b around inf

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

       1. distribute-lft-in N/A

          \[\leadsto \frac{1}{{b}^{2} \cdot \frac{1}{2} + {b}^{2} \cdot \frac{1}{\color{blue}{b}}} \]

       2. inv-pow N/A

          \[\leadsto \frac{1}{{b}^{2} \cdot \frac{1}{2} + {b}^{2} \cdot {b}^{-1}} \]

       3. pow-prod-up N/A

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

       4. metadata-eval N/A

          \[\leadsto \frac{1}{{b}^{2} \cdot \frac{1}{2} + {b}^{1}} \]

       5. unpow1 N/A

          \[\leadsto \frac{1}{{b}^{2} \cdot \frac{1}{2} + b} \]

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 54.0

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

   13. Applied rewrites 54.0%

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

Alternative 11: 53.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.8) (fma 0.25 a 0.5) (/ 1.0 (* (* b b) 0.5))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.8) {
		tmp = fma(0.25, a, 0.5);
	} else {
		tmp = 1.0 / ((b * b) * 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.8)
		tmp = fma(0.25, a, 0.5);
	else
		tmp = Float64(1.0 / Float64(Float64(b * b) * 0.5));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.8], N[(0.25 * a + 0.5), $MachinePrecision], N[(1.0 / N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 55.0

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

   8. Applied rewrites 55.0%

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

   if 1.80000000000000004 < b

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 54.0

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

   8. Applied rewrites 54.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lower-/.f64 54.0

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

   10. Applied rewrites 54.0%

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

   11. Taylor expanded in b around inf

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

       1. *-commutative N/A

          \[\leadsto \frac{1}{{b}^{2} \cdot \frac{1}{2}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{{b}^{2} \cdot \frac{1}{2}} \]

       3. unpow2 N/A

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

       4. lower-*.f64 54.0

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

   13. Applied rewrites 54.0%

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

Alternative 12: 39.2% accurate, 45.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

5. Applied rewrites 84.0%

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

6. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 37.1

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

8. Applied rewrites 37.1%

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

Alternative 13: 39.0% 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 99.6%

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

3. Taylor expanded in a around 0

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

   1. inv-pow N/A

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

   2. lower-pow.f64 N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower--.f64 N/A

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

   9. lift-exp.f64 83.7

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

5. Applied rewrites 83.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 36.9%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025084 
(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))))