Quotient of sum of exps

Percentage Accurate: 98.9% → 99.0%

Time: 3.5s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. a = 2.1022997422643012e+282
  2. b = 4.6264925735568105e+214

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: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{b} - -1\\ t_1 := {t\_0}^{-1}\\ \mathbf{if}\;a \leq -21500:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 - {t\_0}^{-2}, a, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (- (exp b) -1.0)) (t_1 (pow t_0 -1.0)))
   (if (<= a -21500.0)
     (/ (exp a) (+ 1.0 1.0))
     (fma (- t_1 (pow t_0 -2.0)) a t_1))))

C

double code(double a, double b) {
	double t_0 = exp(b) - -1.0;
	double t_1 = pow(t_0, -1.0);
	double tmp;
	if (a <= -21500.0) {
		tmp = exp(a) / (1.0 + 1.0);
	} else {
		tmp = fma((t_1 - pow(t_0, -2.0)), a, t_1);
	}
	return tmp;
}

Julia

function code(a, b)
	t_0 = Float64(exp(b) - -1.0)
	t_1 = t_0 ^ -1.0
	tmp = 0.0
	if (a <= -21500.0)
		tmp = Float64(exp(a) / Float64(1.0 + 1.0));
	else
		tmp = fma(Float64(t_1 - (t_0 ^ -2.0)), a, t_1);
	end
	return tmp
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, -1.0], $MachinePrecision]}, If[LessEqual[a, -21500.0], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Power[t$95$0, -2.0], $MachinePrecision]), $MachinePrecision] * a + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < -21500

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if -21500 < a

   1. Initial program 98.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

Alternative 2: 57.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{a} + e^{b}} \leq 2 \cdot 10^{-159}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left(b \cdot b\right) \cdot 0.16666666666666666, b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp a) (exp b))) 2e-159)
   (/ 1.0 (fma (* (* b b) 0.16666666666666666) b 2.0))
   (fma 0.25 a 0.5)))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(a) + exp(b))) <= 2e-159) {
		tmp = 1.0 / fma(((b * b) * 0.16666666666666666), b, 2.0);
	} else {
		tmp = fma(0.25, a, 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(a) + exp(b))) <= 2e-159)
		tmp = Float64(1.0 / fma(Float64(Float64(b * b) * 0.16666666666666666), b, 2.0));
	else
		tmp = fma(0.25, a, 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-159], N[(1.0 / N[(N[(N[(b * b), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.25 * a + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 1.99999999999999998e-159

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 57.3

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

   5. Applied rewrites 57.3%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 57.3

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

   7. Applied rewrites 57.3%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 40.5

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

   10. Applied rewrites 40.5%

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

   11. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 40.5

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

   13. Applied rewrites 40.5%

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

   if 1.99999999999999998e-159 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 97.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 72.7

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

   8. Applied rewrites 72.7%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

Alternative 4: 98.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -21500:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -21500.0) (/ (exp a) (+ 1.0 1.0)) (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -21500.0) {
		tmp = exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (exp(b) - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-21500.0d0)) then
        tmp = exp(a) / (1.0d0 + 1.0d0)
    else
        tmp = 1.0d0 / (exp(b) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -21500.0) {
		tmp = Math.exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -21500.0:
		tmp = math.exp(a) / (1.0 + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -21500.0)
		tmp = Float64(exp(a) / Float64(1.0 + 1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -21500.0)
		tmp = exp(a) / (1.0 + 1.0);
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -21500.0], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -21500

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if -21500 < a

   1. Initial program 98.3%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 99.6

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

   5. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 99.6

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

   7. Applied rewrites 99.6%

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

Alternative 5: 83.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{+25}:\\ \;\;\;\;{b}^{3} \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.8e+25)
   (* (pow b 3.0) 0.020833333333333332)
   (/ 1.0 (- (exp b) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.8e+25) {
		tmp = pow(b, 3.0) * 0.020833333333333332;
	} else {
		tmp = 1.0 / (exp(b) - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-3.8d+25)) then
        tmp = (b ** 3.0d0) * 0.020833333333333332d0
    else
        tmp = 1.0d0 / (exp(b) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -3.8e+25) {
		tmp = Math.pow(b, 3.0) * 0.020833333333333332;
	} else {
		tmp = 1.0 / (Math.exp(b) - -1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -3.8e+25:
		tmp = math.pow(b, 3.0) * 0.020833333333333332
	else:
		tmp = 1.0 / (math.exp(b) - -1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.8e+25)
		tmp = Float64((b ^ 3.0) * 0.020833333333333332);
	else
		tmp = Float64(1.0 / Float64(exp(b) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -3.8e+25)
		tmp = (b ^ 3.0) * 0.020833333333333332;
	else
		tmp = 1.0 / (exp(b) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.8e+25], N[(N[Power[b, 3.0], $MachinePrecision] * 0.020833333333333332), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.8e25

   1. Initial program 98.6%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)} \]
   7. 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 2.8

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

   8. Applied rewrites 2.8%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{48} \cdot {b}^{\color{blue}{3}} \]
   10. 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. lower-pow.f64 48.8

          \[\leadsto {b}^{3} \cdot 0.020833333333333332 \]

   11. Applied rewrites 48.8%

       \[\leadsto {b}^{3} \cdot 0.020833333333333332 \]

   if -3.8e25 < a

   1. Initial program 98.4%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 98.6

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

   5. Applied rewrites 98.6%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 98.6

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

   7. Applied rewrites 98.6%

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

Alternative 6: 59.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -21500:\\ \;\;\;\;{b}^{3} \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -21500.0)
   (* (pow b 3.0) 0.020833333333333332)
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -21500.0) {
		tmp = pow(b, 3.0) * 0.020833333333333332;
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -21500.0)
		tmp = Float64((b ^ 3.0) * 0.020833333333333332);
	else
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -21500.0], N[(N[Power[b, 3.0], $MachinePrecision] * 0.020833333333333332), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -21500

   1. Initial program 98.7%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 31.5

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)} \]
   7. 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 2.7

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

   8. Applied rewrites 2.7%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{48} \cdot {b}^{\color{blue}{3}} \]
   10. 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. lower-pow.f64 47.0

          \[\leadsto {b}^{3} \cdot 0.020833333333333332 \]

   11. Applied rewrites 47.0%

       \[\leadsto {b}^{3} \cdot 0.020833333333333332 \]

   if -21500 < a

   1. Initial program 98.3%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 99.6

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

   5. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 68.7

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

   10. Applied rewrites 68.7%

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

Alternative 7: 57.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -10000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -10000000.0)
   0.5
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -10000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -10000000.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -10000000.0], 0.5, N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1e7

   1. Initial program 93.2%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -1e7 < b

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 75.4

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 66.1

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

   10. Applied rewrites 66.1%

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

Alternative 8: 57.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -10000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -10000000.0)
   0.5
   (/ 1.0 (fma (fma (* 0.16666666666666666 b) b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -10000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / fma(fma((0.16666666666666666 * b), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -10000000.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / fma(fma(Float64(0.16666666666666666 * b), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -10000000.0], 0.5, N[(1.0 / N[(N[(N[(0.16666666666666666 * b), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1e7

   1. Initial program 93.2%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -1e7 < b

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 75.4

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 66.1

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

   10. Applied rewrites 66.1%

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

   11. Taylor expanded in b around inf

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

       1. lower-*.f64 66.0

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

   13. Applied rewrites 66.0%

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

Alternative 9: 53.1% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -10000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -10000000.0) 0.5 (/ 1.0 (fma (fma 0.5 b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -10000000.0) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / fma(fma(0.5, b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -10000000.0)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / fma(fma(0.5, b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, -10000000.0], 0.5, N[(1.0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1e7

   1. Initial program 93.2%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 18.8%

      \[\leadsto 0.5 \]

   if -1e7 < b

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower--.f64 N/A

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

      9. lift-exp.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-pow.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. unpow-1 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift--.f64 75.4

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2 + \color{blue}{b \cdot \left(1 + \frac{1}{2} \cdot b\right)}} \]
   9. 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 61.6

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

   10. Applied rewrites 61.6%

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

Alternative 10: 40.0% accurate, 315.0× speedup

Math

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

FPCore

(FPCore (a b) :precision binary64 0.5)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

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

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in a around 0

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

   1. inv-pow N/A

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

   2. lower-pow.f64 N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower--.f64 N/A

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

   9. lift-exp.f64 79.7

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

5. Applied rewrites 79.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 40.5%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025056 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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