Quotient of sum of exps

Percentage Accurate: 98.9% → 99.2%

Time: 3.1s

Alternatives: 13

Speedup: 1.2×

• could not determine a ground truth

  1. a = 8.193500386274809e+25
  2. b = 5.1963104218374484e+163

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right)\\ \mathbf{if}\;a \leq -0.135:\\ \;\;\;\;\frac{e^{a}}{e^{a} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{t\_0 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 a 1.0) a 1.0)))
   (if (<= a -0.135) (/ (exp a) (- (exp a) -1.0)) (/ t_0 (+ t_0 (exp b))))))

C

double code(double a, double b) {
	double t_0 = fma(fma(0.5, a, 1.0), a, 1.0);
	double tmp;
	if (a <= -0.135) {
		tmp = exp(a) / (exp(a) - -1.0);
	} else {
		tmp = t_0 / (t_0 + exp(b));
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = fma(fma(0.5, a, 1.0), a, 1.0)
	tmp = 0.0
	if (a <= -0.135)
		tmp = Float64(exp(a) / Float64(exp(a) - -1.0));
	else
		tmp = Float64(t_0 / Float64(t_0 + exp(b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision]}, If[LessEqual[a, -0.135], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(t$95$0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.13500000000000001

   1. Initial program 98.9%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-exp.f64 66.4

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

   4. Applied rewrites 66.4%

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

   if -0.13500000000000001 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 76.0

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 76.9

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

   7. Applied rewrites 76.9%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

Alternative 3: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{e^{a}}{e^{a} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - -1}{\left(a - -1\right) + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.8e-8)
   (/ (exp a) (- (exp a) -1.0))
   (/ (- a -1.0) (+ (- a -1.0) (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-8) {
		tmp = exp(a) / (exp(a) - -1.0);
	} else {
		tmp = (a - -1.0) / ((a - -1.0) + exp(b));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-6.8d-8)) then
        tmp = exp(a) / (exp(a) - (-1.0d0))
    else
        tmp = (a - (-1.0d0)) / ((a - (-1.0d0)) + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-8) {
		tmp = Math.exp(a) / (Math.exp(a) - -1.0);
	} else {
		tmp = (a - -1.0) / ((a - -1.0) + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.8e-8:
		tmp = math.exp(a) / (math.exp(a) - -1.0)
	else:
		tmp = (a - -1.0) / ((a - -1.0) + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.8e-8)
		tmp = Float64(exp(a) / Float64(exp(a) - -1.0));
	else
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(a - -1.0) + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6.8e-8)
		tmp = exp(a) / (exp(a) - -1.0);
	else
		tmp = (a - -1.0) / ((a - -1.0) + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.8e-8], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(a - -1.0), $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.8e-8

   1. Initial program 98.9%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-exp.f64 66.4

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

   4. Applied rewrites 66.4%

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

   if -6.8e-8 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 81.5

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

   7. Applied rewrites 81.5%

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

Alternative 4: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - -1}{\left(a - -1\right) + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.8e-8)
   (/ (exp a) (fma (fma 0.5 a 1.0) a 2.0))
   (/ (- a -1.0) (+ (- a -1.0) (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-8) {
		tmp = exp(a) / fma(fma(0.5, a, 1.0), a, 2.0);
	} else {
		tmp = (a - -1.0) / ((a - -1.0) + exp(b));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.8e-8)
		tmp = Float64(exp(a) / fma(fma(0.5, a, 1.0), a, 2.0));
	else
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(a - -1.0) + exp(b)));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.8e-8], N[(N[Exp[a], $MachinePrecision] / N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(a - -1.0), $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.8e-8

   1. Initial program 98.9%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-exp.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-fma.f64 65.8

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

   7. Applied rewrites 65.8%

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

   if -6.8e-8 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 81.5

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

   7. Applied rewrites 81.5%

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

Alternative 5: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -230:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - -1}{\left(a - -1\right) + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -230.0) (/ (exp a) 2.0) (/ (- a -1.0) (+ (- a -1.0) (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -230.0) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = (a - -1.0) / ((a - -1.0) + exp(b));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-230.0d0)) then
        tmp = exp(a) / 2.0d0
    else
        tmp = (a - (-1.0d0)) / ((a - (-1.0d0)) + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -230.0) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = (a - -1.0) / ((a - -1.0) + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -230.0:
		tmp = math.exp(a) / 2.0
	else:
		tmp = (a - -1.0) / ((a - -1.0) + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -230.0)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(Float64(a - -1.0) / Float64(Float64(a - -1.0) + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -230.0)
		tmp = exp(a) / 2.0;
	else
		tmp = (a - -1.0) / ((a - -1.0) + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -230.0], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(a - -1.0), $MachinePrecision] / N[(N[(a - -1.0), $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -230

   1. Initial program 98.9%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-exp.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 65.4%

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

   if -230 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 81.5

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

   7. Applied rewrites 81.5%

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

Alternative 6: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{e^{a}}{2 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.8e-8) (/ (exp a) (+ 2.0 a)) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-8) {
		tmp = exp(a) / (2.0 + a);
	} 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 <= (-6.8d-8)) then
        tmp = exp(a) / (2.0d0 + a)
    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 <= -6.8e-8) {
		tmp = Math.exp(a) / (2.0 + a);
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6.8e-8:
		tmp = math.exp(a) / (2.0 + a)
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6.8e-8)
		tmp = Float64(exp(a) / Float64(2.0 + a));
	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 <= -6.8e-8)
		tmp = exp(a) / (2.0 + a);
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6.8e-8], N[(N[Exp[a], $MachinePrecision] / N[(2.0 + a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.8e-8

   1. Initial program 98.9%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-exp.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 65.7

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

   7. Applied rewrites 65.7%

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

   if -6.8e-8 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-exp.f64 81.0

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

   4. Applied rewrites 81.0%

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

Alternative 7: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6.8e-8) (/ (exp a) 2.0) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6.8e-8) {
		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 <= (-6.8d-8)) 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 <= -6.8e-8) {
		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 <= -6.8e-8:
		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 <= -6.8e-8)
		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 <= -6.8e-8)
		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, -6.8e-8], 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 < -6.8e-8

   1. Initial program 98.9%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-exp.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 65.4%

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

   if -6.8e-8 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-exp.f64 81.0

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

   4. Applied rewrites 81.0%

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

Alternative 8: 74.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(b \cdot b\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.35e+154) (/ (exp a) 2.0) (/ 1.0 (* (* b b) 0.5))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+154) {
		tmp = exp(a) / 2.0;
	} else {
		tmp = 1.0 / ((b * b) * 0.5);
	}
	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 <= 1.35d+154) then
        tmp = exp(a) / 2.0d0
    else
        tmp = 1.0d0 / ((b * b) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.35e+154) {
		tmp = Math.exp(a) / 2.0;
	} else {
		tmp = 1.0 / ((b * b) * 0.5);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.35e+154:
		tmp = math.exp(a) / 2.0
	else:
		tmp = 1.0 / ((b * b) * 0.5)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.35e+154)
		tmp = Float64(exp(a) / 2.0);
	else
		tmp = Float64(1.0 / Float64(Float64(b * b) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.35e+154)
		tmp = exp(a) / 2.0;
	else
		tmp = 1.0 / ((b * b) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.35e+154], N[(N[Exp[a], $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(N[(b * b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35000000000000003e154

   1. Initial program 98.9%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-exp.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 65.4%

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

   if 1.35000000000000003e154 < b

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-exp.f64 81.0

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

   4. Applied rewrites 81.0%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 50.4

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

   7. Applied rewrites 50.4%

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

   8. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 16.9

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

   10. Applied rewrites 16.9%

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

Alternative 9: 64.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 2\right)}\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{+29}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.9e+152)
   (/ 1.0 (fma (fma 0.5 a 1.0) a 2.0))
   (if (<= a -2.2e+29)
     (* (* (* b b) b) 0.020833333333333332)
     (/ 1.0 (fma (fma 0.5 b 1.0) b 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.9e+152) {
		tmp = 1.0 / fma(fma(0.5, a, 1.0), a, 2.0);
	} else if (a <= -2.2e+29) {
		tmp = ((b * b) * b) * 0.020833333333333332;
	} 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 (a <= -2.9e+152)
		tmp = Float64(1.0 / fma(fma(0.5, a, 1.0), a, 2.0));
	elseif (a <= -2.2e+29)
		tmp = Float64(Float64(Float64(b * b) * b) * 0.020833333333333332);
	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[a, -2.9e+152], N[(1.0 / N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -2.2e+29], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * 0.020833333333333332), $MachinePrecision], N[(1.0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.8999999999999998e152

   1. Initial program 98.9%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-exp.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-fma.f64 65.8

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

   7. Applied rewrites 65.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 53.1%

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

   if -2.8999999999999998e152 < a < -2.2000000000000001e29

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-exp.f64 81.0

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

   4. Applied rewrites 81.0%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 39.3%

      \[\leadsto 0.5 \]

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 36.1

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

   10. Applied rewrites 36.1%

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

   11. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow3 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \frac{1}{48} \]

       4. pow2 N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \frac{1}{48} \]

       7. lift-*.f64 13.5

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332 \]

   13. Applied rewrites 13.5%

       \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332 \]

   if -2.2000000000000001e29 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-exp.f64 81.0

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

   4. Applied rewrites 81.0%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 50.4

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

   7. Applied rewrites 50.4%

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

Alternative 10: 56.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{+29}:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -2.2e+29)
   (* (* (* b b) b) 0.020833333333333332)
   (/ 1.0 (fma (fma 0.5 b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -2.2e+29) {
		tmp = ((b * b) * b) * 0.020833333333333332;
	} 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 (a <= -2.2e+29)
		tmp = Float64(Float64(Float64(b * b) * b) * 0.020833333333333332);
	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[a, -2.2e+29], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * 0.020833333333333332), $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 a < -2.2000000000000001e29

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-exp.f64 81.0

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

   4. Applied rewrites 81.0%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 39.3%

      \[\leadsto 0.5 \]

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 36.1

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

   10. Applied rewrites 36.1%

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

   11. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow3 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \frac{1}{48} \]

       4. pow2 N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \frac{1}{48} \]

       7. lift-*.f64 13.5

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332 \]

   13. Applied rewrites 13.5%

       \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332 \]

   if -2.2000000000000001e29 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-exp.f64 81.0

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

   4. Applied rewrites 81.0%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 50.4

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

   7. Applied rewrites 50.4%

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

Alternative 11: 50.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6000000:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6000000.0)
   (* (* (* b b) b) 0.020833333333333332)
   (/ (+ 1.0 a) (- (+ 1.0 a) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6000000.0) {
		tmp = ((b * b) * b) * 0.020833333333333332;
	} else {
		tmp = (1.0 + a) / ((1.0 + a) - -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 <= (-6000000.0d0)) then
        tmp = ((b * b) * b) * 0.020833333333333332d0
    else
        tmp = (1.0d0 + a) / ((1.0d0 + a) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6000000.0) {
		tmp = ((b * b) * b) * 0.020833333333333332;
	} else {
		tmp = (1.0 + a) / ((1.0 + a) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6000000.0:
		tmp = ((b * b) * b) * 0.020833333333333332
	else:
		tmp = (1.0 + a) / ((1.0 + a) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6000000.0)
		tmp = Float64(Float64(Float64(b * b) * b) * 0.020833333333333332);
	else
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6000000.0)
		tmp = ((b * b) * b) * 0.020833333333333332;
	else
		tmp = (1.0 + a) / ((1.0 + a) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6000000.0], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * 0.020833333333333332), $MachinePrecision], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6e6

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-exp.f64 81.0

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

   4. Applied rewrites 81.0%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 39.3%

      \[\leadsto 0.5 \]

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 36.1

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

   10. Applied rewrites 36.1%

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

   11. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow3 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \frac{1}{48} \]

       4. pow2 N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \frac{1}{48} \]

       7. lift-*.f64 13.5

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332 \]

   13. Applied rewrites 13.5%

       \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332 \]

   if -6e6 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-exp.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-+.f64 39.3

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

   7. Applied rewrites 39.3%

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

   8. Taylor expanded in a around 0

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

      1. lower-+.f64 39.8

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

   10. Applied rewrites 39.8%

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

Alternative 12: 50.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6000000:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -6000000.0) (* (* (* b b) b) 0.020833333333333332) 0.5))

C

double code(double a, double b) {
	double tmp;
	if (a <= -6000000.0) {
		tmp = ((b * b) * b) * 0.020833333333333332;
	} else {
		tmp = 0.5;
	}
	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 <= (-6000000.0d0)) then
        tmp = ((b * b) * b) * 0.020833333333333332d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -6000000.0) {
		tmp = ((b * b) * b) * 0.020833333333333332;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -6000000.0:
		tmp = ((b * b) * b) * 0.020833333333333332
	else:
		tmp = 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -6000000.0)
		tmp = Float64(Float64(Float64(b * b) * b) * 0.020833333333333332);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -6000000.0)
		tmp = ((b * b) * b) * 0.020833333333333332;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -6000000.0], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * 0.020833333333333332), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if a < -6e6

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-exp.f64 81.0

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

   4. Applied rewrites 81.0%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 39.3%

      \[\leadsto 0.5 \]

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 36.1

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

   10. Applied rewrites 36.1%

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

   11. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow3 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \frac{1}{48} \]

       4. pow2 N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot \frac{1}{48} \]

       7. lift-*.f64 13.5

          \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332 \]

   13. Applied rewrites 13.5%

       \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332 \]

   if -6e6 < a

   1. Initial program 98.9%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lift-exp.f64 81.0

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

   4. Applied rewrites 81.0%

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

   5. Taylor expanded in b around 0

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

   7. Applied rewrites 39.3%

      \[\leadsto 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 39.3% accurate, 37.5× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower--.f64 N/A

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

   8. lift-exp.f64 81.0

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

4. Applied rewrites 81.0%

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

5. Taylor expanded in b around 0

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

7. Applied rewrites 39.3%

   \[\leadsto 0.5 \]
8. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.9× 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 2025140 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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