Octave 3.8, jcobi/1

Percentage Accurate: 74.9% → 99.9%

Time: 9.0s

Alternatives: 19

Speedup: 0.6×

Specification

\[\alpha > -1 \land \beta > -1\]

Math

\[\begin{array}{l} \\ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

C

double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta):
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))

C

double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = (((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta) {
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta):
	return (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta)
	return Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta)
	tmp = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
end

Wolfram

code[alpha_, beta_] := N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\alpha}{2 + \left(\beta + \alpha\right)}\\ t_1 := {t\_0}^{3} + 1\\ t_2 := \left(\alpha + \beta\right) + 2\\ \mathbf{if}\;\frac{\frac{\beta - \alpha}{t\_2} + 1}{2} \leq 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot \mathsf{fma}\left(\frac{2}{\alpha} - \frac{10}{\alpha \cdot \alpha}, \beta, \frac{6}{\alpha} - \frac{16}{\alpha \cdot \alpha}\right) - 1, \beta, \frac{4 - \frac{8}{\alpha}}{\alpha} \cdot 0.5\right) - 1}{-\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \frac{\frac{{t\_0}^{2} \cdot t\_1 - t\_1}{\mathsf{fma}\left(t\_0, t\_0 - 1, 1\right)}}{{\left(1 + \frac{\alpha}{t\_2}\right)}^{2}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ alpha (+ 2.0 (+ beta alpha))))
        (t_1 (+ (pow t_0 3.0) 1.0))
        (t_2 (+ (+ alpha beta) 2.0)))
   (if (<= (/ (+ (/ (- beta alpha) t_2) 1.0) 2.0) 1e-6)
     (/
      (-
       (fma
        (-
         (*
          0.5
          (fma
           (- (/ 2.0 alpha) (/ 10.0 (* alpha alpha)))
           beta
           (- (/ 6.0 alpha) (/ 16.0 (* alpha alpha)))))
         1.0)
        beta
        (* (/ (- 4.0 (/ 8.0 alpha)) alpha) 0.5))
       1.0)
      (- alpha))
     (/
      (-
       (/ beta (+ 2.0 (+ alpha beta)))
       (/
        (/ (- (* (pow t_0 2.0) t_1) t_1) (fma t_0 (- t_0 1.0) 1.0))
        (pow (+ 1.0 (/ alpha t_2)) 2.0)))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = alpha / (2.0 + (beta + alpha));
	double t_1 = pow(t_0, 3.0) + 1.0;
	double t_2 = (alpha + beta) + 2.0;
	double tmp;
	if (((((beta - alpha) / t_2) + 1.0) / 2.0) <= 1e-6) {
		tmp = (fma(((0.5 * fma(((2.0 / alpha) - (10.0 / (alpha * alpha))), beta, ((6.0 / alpha) - (16.0 / (alpha * alpha))))) - 1.0), beta, (((4.0 - (8.0 / alpha)) / alpha) * 0.5)) - 1.0) / -alpha;
	} else {
		tmp = ((beta / (2.0 + (alpha + beta))) - ((((pow(t_0, 2.0) * t_1) - t_1) / fma(t_0, (t_0 - 1.0), 1.0)) / pow((1.0 + (alpha / t_2)), 2.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(alpha / Float64(2.0 + Float64(beta + alpha)))
	t_1 = Float64((t_0 ^ 3.0) + 1.0)
	t_2 = Float64(Float64(alpha + beta) + 2.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / t_2) + 1.0) / 2.0) <= 1e-6)
		tmp = Float64(Float64(fma(Float64(Float64(0.5 * fma(Float64(Float64(2.0 / alpha) - Float64(10.0 / Float64(alpha * alpha))), beta, Float64(Float64(6.0 / alpha) - Float64(16.0 / Float64(alpha * alpha))))) - 1.0), beta, Float64(Float64(Float64(4.0 - Float64(8.0 / alpha)) / alpha) * 0.5)) - 1.0) / Float64(-alpha));
	else
		tmp = Float64(Float64(Float64(beta / Float64(2.0 + Float64(alpha + beta))) - Float64(Float64(Float64(Float64((t_0 ^ 2.0) * t_1) - t_1) / fma(t_0, Float64(t_0 - 1.0), 1.0)) / (Float64(1.0 + Float64(alpha / t_2)) ^ 2.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(alpha / N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[t$95$0, 3.0], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 1e-6], N[(N[(N[(N[(N[(0.5 * N[(N[(N[(2.0 / alpha), $MachinePrecision] - N[(10.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * beta + N[(N[(6.0 / alpha), $MachinePrecision] - N[(16.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * beta + N[(N[(N[(4.0 - N[(8.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / (-alpha)), $MachinePrecision], N[(N[(N[(beta / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(t$95$0 * N[(t$95$0 - 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Power[N[(1.0 + N[(alpha / t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.99999999999999955e-7

   1. Initial program 8.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{2} \cdot \left(\beta \cdot \left(2 + \beta\right) + {\left(2 + \beta\right)}^{2}\right) + \frac{1}{2} \cdot \frac{\beta \cdot {\left(2 + \beta\right)}^{2} + {\left(2 + \beta\right)}^{3}}{\alpha}}{\alpha} + \frac{1}{2} \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right)}{\alpha}} \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(-\beta\right) - \beta\right) - 2, 0.5, \frac{-0.5 \cdot \left(\mathsf{fma}\left(2, \beta, 2\right) \cdot \left(2 + \beta\right) - \frac{\mathsf{fma}\left(2, \beta, 2\right) \cdot {\left(2 + \beta\right)}^{2}}{\alpha}\right)}{-\alpha}\right)}{-\alpha}} \]

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{4 - 8 \cdot \frac{1}{\alpha}}{\alpha} + \beta \cdot \left(\left(\frac{1}{2} \cdot \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 10 \cdot \frac{1}{{\alpha}^{2}}\right)\right) + \frac{1}{2} \cdot \left(6 \cdot \frac{1}{\alpha} - 16 \cdot \frac{1}{{\alpha}^{2}}\right)\right) - 1\right)\right) - 1}{-\color{blue}{\alpha}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot \mathsf{fma}\left(\frac{2}{\alpha} - \frac{10}{\alpha \cdot \alpha}, \beta, \frac{6}{\alpha} - \frac{16}{\alpha \cdot \alpha}\right) - 1, \beta, \frac{4 - \frac{8}{\alpha}}{\alpha} \cdot 0.5\right) - 1}{-\color{blue}{\alpha}} \]

   if 9.99999999999999955e-7 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)}}{2} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. div-sub N/A

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

      5. frac-sub N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. *-rgt-identity N/A

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

   6. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

         \[\leadsto \frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2} \cdot \color{blue}{\frac{{1}^{3} + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}}{1 \cdot 1 + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}} - \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}{{\left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2}}}{2} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \frac{\color{blue}{\frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2} \cdot \left({1}^{3} + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}\right)}{1 \cdot 1 + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}} - \left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}{{\left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2}}}{2} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \frac{\frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2} \cdot \left({1}^{3} + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}\right)}{1 \cdot 1 + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} - \color{blue}{\left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}}{{\left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2}}}{2} \]

      7. flip3-+ N/A

         \[\leadsto \frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \frac{\frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2} \cdot \left({1}^{3} + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}\right)}{1 \cdot 1 + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} - \color{blue}{\frac{{1}^{3} + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}}{1 \cdot 1 + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}}}{{\left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2}}}{2} \]

      8. sub-div N/A

         \[\leadsto \frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \frac{\color{blue}{\frac{{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2} \cdot \left({1}^{3} + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}\right) - \left({1}^{3} + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}\right)}{1 \cdot 1 + \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1 \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}}}{{\left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2}}}{2} \]

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \frac{\color{blue}{\frac{{\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)}\right)}^{2} \cdot \left({\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)}\right)}^{3} + 1\right) - \left({\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)}\right)}^{3} + 1\right)}{\mathsf{fma}\left(\frac{\alpha}{2 + \left(\beta + \alpha\right)}, \frac{\alpha}{2 + \left(\beta + \alpha\right)} - 1, 1\right)}}}{{\left(1 + \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2}}}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2\\ t_1 := \left(\beta + \alpha\right) + 2\\ t_2 := \frac{\alpha}{t\_1}\\ \mathbf{if}\;\frac{\frac{\beta - \alpha}{t\_0} + 1}{2} \leq 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot \mathsf{fma}\left(\frac{2}{\alpha} - \frac{10}{\alpha \cdot \alpha}, \beta, \frac{6}{\alpha} - \frac{16}{\alpha \cdot \alpha}\right) - 1, \beta, \frac{4 - \frac{8}{\alpha}}{\alpha} \cdot 0.5\right) - 1}{-\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \frac{\mathsf{fma}\left(\alpha, \frac{t\_2}{t\_1}, {t\_2}^{3} - \left(t\_2 + 1\right)\right)}{{\left(1 + \frac{\alpha}{t\_0}\right)}^{2}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) 2.0))
        (t_1 (+ (+ beta alpha) 2.0))
        (t_2 (/ alpha t_1)))
   (if (<= (/ (+ (/ (- beta alpha) t_0) 1.0) 2.0) 1e-6)
     (/
      (-
       (fma
        (-
         (*
          0.5
          (fma
           (- (/ 2.0 alpha) (/ 10.0 (* alpha alpha)))
           beta
           (- (/ 6.0 alpha) (/ 16.0 (* alpha alpha)))))
         1.0)
        beta
        (* (/ (- 4.0 (/ 8.0 alpha)) alpha) 0.5))
       1.0)
      (- alpha))
     (/
      (-
       (/ beta (+ 2.0 (+ alpha beta)))
       (/
        (fma alpha (/ t_2 t_1) (- (pow t_2 3.0) (+ t_2 1.0)))
        (pow (+ 1.0 (/ alpha t_0)) 2.0)))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = (alpha + beta) + 2.0;
	double t_1 = (beta + alpha) + 2.0;
	double t_2 = alpha / t_1;
	double tmp;
	if (((((beta - alpha) / t_0) + 1.0) / 2.0) <= 1e-6) {
		tmp = (fma(((0.5 * fma(((2.0 / alpha) - (10.0 / (alpha * alpha))), beta, ((6.0 / alpha) - (16.0 / (alpha * alpha))))) - 1.0), beta, (((4.0 - (8.0 / alpha)) / alpha) * 0.5)) - 1.0) / -alpha;
	} else {
		tmp = ((beta / (2.0 + (alpha + beta))) - (fma(alpha, (t_2 / t_1), (pow(t_2, 3.0) - (t_2 + 1.0))) / pow((1.0 + (alpha / t_0)), 2.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(alpha + beta) + 2.0)
	t_1 = Float64(Float64(beta + alpha) + 2.0)
	t_2 = Float64(alpha / t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / t_0) + 1.0) / 2.0) <= 1e-6)
		tmp = Float64(Float64(fma(Float64(Float64(0.5 * fma(Float64(Float64(2.0 / alpha) - Float64(10.0 / Float64(alpha * alpha))), beta, Float64(Float64(6.0 / alpha) - Float64(16.0 / Float64(alpha * alpha))))) - 1.0), beta, Float64(Float64(Float64(4.0 - Float64(8.0 / alpha)) / alpha) * 0.5)) - 1.0) / Float64(-alpha));
	else
		tmp = Float64(Float64(Float64(beta / Float64(2.0 + Float64(alpha + beta))) - Float64(fma(alpha, Float64(t_2 / t_1), Float64((t_2 ^ 3.0) - Float64(t_2 + 1.0))) / (Float64(1.0 + Float64(alpha / t_0)) ^ 2.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(alpha / t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 1e-6], N[(N[(N[(N[(N[(0.5 * N[(N[(N[(2.0 / alpha), $MachinePrecision] - N[(10.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * beta + N[(N[(6.0 / alpha), $MachinePrecision] - N[(16.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * beta + N[(N[(N[(4.0 - N[(8.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / (-alpha)), $MachinePrecision], N[(N[(N[(beta / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(alpha * N[(t$95$2 / t$95$1), $MachinePrecision] + N[(N[Power[t$95$2, 3.0], $MachinePrecision] - N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(1.0 + N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.99999999999999955e-7

   1. Initial program 8.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{2} \cdot \left(\beta \cdot \left(2 + \beta\right) + {\left(2 + \beta\right)}^{2}\right) + \frac{1}{2} \cdot \frac{\beta \cdot {\left(2 + \beta\right)}^{2} + {\left(2 + \beta\right)}^{3}}{\alpha}}{\alpha} + \frac{1}{2} \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right)}{\alpha}} \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(-\beta\right) - \beta\right) - 2, 0.5, \frac{-0.5 \cdot \left(\mathsf{fma}\left(2, \beta, 2\right) \cdot \left(2 + \beta\right) - \frac{\mathsf{fma}\left(2, \beta, 2\right) \cdot {\left(2 + \beta\right)}^{2}}{\alpha}\right)}{-\alpha}\right)}{-\alpha}} \]

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{4 - 8 \cdot \frac{1}{\alpha}}{\alpha} + \beta \cdot \left(\left(\frac{1}{2} \cdot \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 10 \cdot \frac{1}{{\alpha}^{2}}\right)\right) + \frac{1}{2} \cdot \left(6 \cdot \frac{1}{\alpha} - 16 \cdot \frac{1}{{\alpha}^{2}}\right)\right) - 1\right)\right) - 1}{-\color{blue}{\alpha}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot \mathsf{fma}\left(\frac{2}{\alpha} - \frac{10}{\alpha \cdot \alpha}, \beta, \frac{6}{\alpha} - \frac{16}{\alpha \cdot \alpha}\right) - 1, \beta, \frac{4 - \frac{8}{\alpha}}{\alpha} \cdot 0.5\right) - 1}{-\color{blue}{\alpha}} \]

   if 9.99999999999999955e-7 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)}}{2} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. div-sub N/A

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

      5. frac-sub N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. *-rgt-identity N/A

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

   6. Applied rewrites 99.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate--l+ N/A

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

   10. Applied rewrites 99.9%

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

Alternative 3: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\beta + \alpha\right)\\ t_1 := \frac{\alpha}{\left(\beta + \alpha\right) + 2} + 1\\ t_2 := \left(\alpha + \beta\right) + 2\\ t_3 := 1 + \frac{\alpha}{t\_2}\\ \mathbf{if}\;\frac{\frac{\beta - \alpha}{t\_2} + 1}{2} \leq 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot \mathsf{fma}\left(\frac{2}{\alpha} - \frac{10}{\alpha \cdot \alpha}, \beta, \frac{6}{\alpha} - \frac{16}{\alpha \cdot \alpha}\right) - 1, \beta, \frac{4 - \frac{8}{\alpha}}{\alpha} \cdot 0.5\right) - 1}{-\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \frac{\frac{\frac{\alpha}{t\_0} \cdot \alpha}{t\_0} \cdot t\_3 - t\_3}{t\_1 \cdot t\_1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ beta alpha)))
        (t_1 (+ (/ alpha (+ (+ beta alpha) 2.0)) 1.0))
        (t_2 (+ (+ alpha beta) 2.0))
        (t_3 (+ 1.0 (/ alpha t_2))))
   (if (<= (/ (+ (/ (- beta alpha) t_2) 1.0) 2.0) 1e-6)
     (/
      (-
       (fma
        (-
         (*
          0.5
          (fma
           (- (/ 2.0 alpha) (/ 10.0 (* alpha alpha)))
           beta
           (- (/ 6.0 alpha) (/ 16.0 (* alpha alpha)))))
         1.0)
        beta
        (* (/ (- 4.0 (/ 8.0 alpha)) alpha) 0.5))
       1.0)
      (- alpha))
     (/
      (-
       (/ beta (+ 2.0 (+ alpha beta)))
       (/ (- (* (/ (* (/ alpha t_0) alpha) t_0) t_3) t_3) (* t_1 t_1)))
      2.0))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (beta + alpha);
	double t_1 = (alpha / ((beta + alpha) + 2.0)) + 1.0;
	double t_2 = (alpha + beta) + 2.0;
	double t_3 = 1.0 + (alpha / t_2);
	double tmp;
	if (((((beta - alpha) / t_2) + 1.0) / 2.0) <= 1e-6) {
		tmp = (fma(((0.5 * fma(((2.0 / alpha) - (10.0 / (alpha * alpha))), beta, ((6.0 / alpha) - (16.0 / (alpha * alpha))))) - 1.0), beta, (((4.0 - (8.0 / alpha)) / alpha) * 0.5)) - 1.0) / -alpha;
	} else {
		tmp = ((beta / (2.0 + (alpha + beta))) - ((((((alpha / t_0) * alpha) / t_0) * t_3) - t_3) / (t_1 * t_1))) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(beta + alpha))
	t_1 = Float64(Float64(alpha / Float64(Float64(beta + alpha) + 2.0)) + 1.0)
	t_2 = Float64(Float64(alpha + beta) + 2.0)
	t_3 = Float64(1.0 + Float64(alpha / t_2))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / t_2) + 1.0) / 2.0) <= 1e-6)
		tmp = Float64(Float64(fma(Float64(Float64(0.5 * fma(Float64(Float64(2.0 / alpha) - Float64(10.0 / Float64(alpha * alpha))), beta, Float64(Float64(6.0 / alpha) - Float64(16.0 / Float64(alpha * alpha))))) - 1.0), beta, Float64(Float64(Float64(4.0 - Float64(8.0 / alpha)) / alpha) * 0.5)) - 1.0) / Float64(-alpha));
	else
		tmp = Float64(Float64(Float64(beta / Float64(2.0 + Float64(alpha + beta))) - Float64(Float64(Float64(Float64(Float64(Float64(alpha / t_0) * alpha) / t_0) * t_3) - t_3) / Float64(t_1 * t_1))) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha / N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(alpha / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / t$95$2), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 1e-6], N[(N[(N[(N[(N[(0.5 * N[(N[(N[(2.0 / alpha), $MachinePrecision] - N[(10.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * beta + N[(N[(6.0 / alpha), $MachinePrecision] - N[(16.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * beta + N[(N[(N[(4.0 - N[(8.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / (-alpha)), $MachinePrecision], N[(N[(N[(beta / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(N[(alpha / t$95$0), $MachinePrecision] * alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.99999999999999955e-7

   1. Initial program 8.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{2} \cdot \left(\beta \cdot \left(2 + \beta\right) + {\left(2 + \beta\right)}^{2}\right) + \frac{1}{2} \cdot \frac{\beta \cdot {\left(2 + \beta\right)}^{2} + {\left(2 + \beta\right)}^{3}}{\alpha}}{\alpha} + \frac{1}{2} \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right)}{\alpha}} \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(-\beta\right) - \beta\right) - 2, 0.5, \frac{-0.5 \cdot \left(\mathsf{fma}\left(2, \beta, 2\right) \cdot \left(2 + \beta\right) - \frac{\mathsf{fma}\left(2, \beta, 2\right) \cdot {\left(2 + \beta\right)}^{2}}{\alpha}\right)}{-\alpha}\right)}{-\alpha}} \]

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{4 - 8 \cdot \frac{1}{\alpha}}{\alpha} + \beta \cdot \left(\left(\frac{1}{2} \cdot \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 10 \cdot \frac{1}{{\alpha}^{2}}\right)\right) + \frac{1}{2} \cdot \left(6 \cdot \frac{1}{\alpha} - 16 \cdot \frac{1}{{\alpha}^{2}}\right)\right) - 1\right)\right) - 1}{-\color{blue}{\alpha}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot \mathsf{fma}\left(\frac{2}{\alpha} - \frac{10}{\alpha \cdot \alpha}, \beta, \frac{6}{\alpha} - \frac{16}{\alpha \cdot \alpha}\right) - 1, \beta, \frac{4 - \frac{8}{\alpha}}{\alpha} \cdot 0.5\right) - 1}{-\color{blue}{\alpha}} \]

   if 9.99999999999999955e-7 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\frac{\beta}{2 + \left(\alpha + \beta\right)} - \left(\frac{\alpha}{2 + \left(\alpha + \beta\right)} - 1\right)}}{2} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. div-sub N/A

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

      5. frac-sub N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. *-rgt-identity N/A

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

   6. Applied rewrites 99.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.9

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 99.9

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

   10. Applied rewrites 99.9%

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

Alternative 4: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5 \cdot \mathsf{fma}\left(\frac{2}{\alpha} - \frac{10}{\alpha \cdot \alpha}, \beta, \frac{6}{\alpha} - \frac{16}{\alpha \cdot \alpha}\right) - 1, \beta, \frac{4 - \frac{8}{\alpha}}{\alpha} \cdot 0.5\right) - 1}{-\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} - \left(\frac{\alpha}{t\_0} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 1e-6)
     (/
      (-
       (fma
        (-
         (*
          0.5
          (fma
           (- (/ 2.0 alpha) (/ 10.0 (* alpha alpha)))
           beta
           (- (/ 6.0 alpha) (/ 16.0 (* alpha alpha)))))
         1.0)
        beta
        (* (/ (- 4.0 (/ 8.0 alpha)) alpha) 0.5))
       1.0)
      (- alpha))
     (/ (- (/ beta t_0) (- (/ alpha t_0) 1.0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-6) {
		tmp = (fma(((0.5 * fma(((2.0 / alpha) - (10.0 / (alpha * alpha))), beta, ((6.0 / alpha) - (16.0 / (alpha * alpha))))) - 1.0), beta, (((4.0 - (8.0 / alpha)) / alpha) * 0.5)) - 1.0) / -alpha;
	} else {
		tmp = ((beta / t_0) - ((alpha / t_0) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-6)
		tmp = Float64(Float64(fma(Float64(Float64(0.5 * fma(Float64(Float64(2.0 / alpha) - Float64(10.0 / Float64(alpha * alpha))), beta, Float64(Float64(6.0 / alpha) - Float64(16.0 / Float64(alpha * alpha))))) - 1.0), beta, Float64(Float64(Float64(4.0 - Float64(8.0 / alpha)) / alpha) * 0.5)) - 1.0) / Float64(-alpha));
	else
		tmp = Float64(Float64(Float64(beta / t_0) - Float64(Float64(alpha / t_0) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 1e-6], N[(N[(N[(N[(N[(0.5 * N[(N[(N[(2.0 / alpha), $MachinePrecision] - N[(10.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * beta + N[(N[(6.0 / alpha), $MachinePrecision] - N[(16.0 / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * beta + N[(N[(N[(4.0 - N[(8.0 / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / (-alpha)), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] - N[(N[(alpha / t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.99999999999999955e-7

   1. Initial program 8.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{2} \cdot \left(\beta \cdot \left(2 + \beta\right) + {\left(2 + \beta\right)}^{2}\right) + \frac{1}{2} \cdot \frac{\beta \cdot {\left(2 + \beta\right)}^{2} + {\left(2 + \beta\right)}^{3}}{\alpha}}{\alpha} + \frac{1}{2} \cdot \left(-1 \cdot \beta - \left(2 + \beta\right)\right)}{\alpha}} \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(-\beta\right) - \beta\right) - 2, 0.5, \frac{-0.5 \cdot \left(\mathsf{fma}\left(2, \beta, 2\right) \cdot \left(2 + \beta\right) - \frac{\mathsf{fma}\left(2, \beta, 2\right) \cdot {\left(2 + \beta\right)}^{2}}{\alpha}\right)}{-\alpha}\right)}{-\alpha}} \]

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{\left(\frac{1}{2} \cdot \frac{4 - 8 \cdot \frac{1}{\alpha}}{\alpha} + \beta \cdot \left(\left(\frac{1}{2} \cdot \left(\beta \cdot \left(2 \cdot \frac{1}{\alpha} - 10 \cdot \frac{1}{{\alpha}^{2}}\right)\right) + \frac{1}{2} \cdot \left(6 \cdot \frac{1}{\alpha} - 16 \cdot \frac{1}{{\alpha}^{2}}\right)\right) - 1\right)\right) - 1}{-\color{blue}{\alpha}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot \mathsf{fma}\left(\frac{2}{\alpha} - \frac{10}{\alpha \cdot \alpha}, \beta, \frac{6}{\alpha} - \frac{16}{\alpha \cdot \alpha}\right) - 1, \beta, \frac{4 - \frac{8}{\alpha}}{\alpha} \cdot 0.5\right) - 1}{-\color{blue}{\alpha}} \]

   if 9.99999999999999955e-7 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 5: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0.002:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{1 + \alpha}{\beta}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 0.002)
     (/ (+ 1.0 beta) alpha)
     (if (<= t_0 0.6)
       (fma (fma -0.125 beta 0.25) beta 0.5)
       (fma -1.0 (/ (+ 1.0 alpha) beta) 1.0)))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 0.002) {
		tmp = (1.0 + beta) / alpha;
	} else if (t_0 <= 0.6) {
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	} else {
		tmp = fma(-1.0, ((1.0 + alpha) / beta), 1.0);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 0.002)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.6)
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	else
		tmp = fma(-1.0, Float64(Float64(1.0 + alpha) / beta), 1.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.002], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(N[(-0.125 * beta + 0.25), $MachinePrecision] * beta + 0.5), $MachinePrecision], N[(-1.0 * N[(N[(1.0 + alpha), $MachinePrecision] / beta), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2e-3

   1. Initial program 9.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{1 + \color{blue}{1} \cdot \beta}{\alpha} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{1 + \color{blue}{\beta}}{\alpha} \]

      8. lower-+.f64 97.1

         \[\leadsto \frac{\color{blue}{1 + \beta}}{\alpha} \]

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha}} \]

   if 2e-3 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.599999999999999978

   1. Initial program 100.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\beta \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \color{blue}{\beta}, 0.5\right) \]

   if 0.599999999999999978 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{2 + 2 \cdot \alpha}{\beta} + 1} \]

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{\alpha}{\beta} + \frac{1}{\beta}, 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\frac{1}{\beta} + \frac{\alpha}{\beta}}, 1\right) \]

      12. div-add-rev N/A

          \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\frac{1 + \alpha}{\beta}}, 1\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\frac{1 + \alpha}{\beta}}, 1\right) \]

      14. lower-+.f64 98.1

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{1 + \alpha}}{\beta}, 1\right) \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{1 + \alpha}{\beta}, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0.002:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 0.002)
     (/ (+ 1.0 beta) alpha)
     (if (<= t_0 0.6)
       (fma (fma -0.125 beta 0.25) beta 0.5)
       (- 1.0 (/ 1.0 beta))))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 0.002) {
		tmp = (1.0 + beta) / alpha;
	} else if (t_0 <= 0.6) {
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 0.002)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	elseif (t_0 <= 0.6)
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.002], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(N[(-0.125 * beta + 0.25), $MachinePrecision] * beta + 0.5), $MachinePrecision], N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2e-3

   1. Initial program 9.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{1 + \color{blue}{1} \cdot \beta}{\alpha} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{1 + \color{blue}{\beta}}{\alpha} \]

      8. lower-+.f64 97.1

         \[\leadsto \frac{\color{blue}{1 + \beta}}{\alpha} \]

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha}} \]

   if 2e-3 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.599999999999999978

   1. Initial program 100.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\beta \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \color{blue}{\beta}, 0.5\right) \]

   if 0.599999999999999978 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in beta around inf

      \[\leadsto 1 - \color{blue}{\frac{1}{\beta}} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.5%

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

Alternative 7: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0.002:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 0.002)
     (/ 1.0 alpha)
     (if (<= t_0 0.6)
       (fma (fma -0.125 beta 0.25) beta 0.5)
       (- 1.0 (/ 1.0 beta))))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 0.002) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	} else {
		tmp = 1.0 - (1.0 / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 0.002)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.6)
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	else
		tmp = Float64(1.0 - Float64(1.0 / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.002], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(N[(-0.125 * beta + 0.25), $MachinePrecision] * beta + 0.5), $MachinePrecision], N[(1.0 - N[(1.0 / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2e-3

   1. Initial program 9.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{1 + \color{blue}{1} \cdot \beta}{\alpha} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{1 + \color{blue}{\beta}}{\alpha} \]

      8. lower-+.f64 97.1

         \[\leadsto \frac{\color{blue}{1 + \beta}}{\alpha} \]

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha}} \]

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{\color{blue}{\alpha}} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.9%

      \[\leadsto \frac{1}{\color{blue}{\alpha}} \]

   if 2e-3 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.599999999999999978

   1. Initial program 100.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\beta \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \color{blue}{\beta}, 0.5\right) \]

   if 0.599999999999999978 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in beta around inf

      \[\leadsto 1 - \color{blue}{\frac{1}{\beta}} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.5%

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

Alternative 8: 92.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0.002:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 0.002)
     (/ 1.0 alpha)
     (if (<= t_0 0.6) (fma (fma -0.125 beta 0.25) beta 0.5) 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 0.002) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 0.002)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.6)
		tmp = fma(fma(-0.125, beta, 0.25), beta, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.002], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(N[(-0.125 * beta + 0.25), $MachinePrecision] * beta + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2e-3

   1. Initial program 9.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{1 + \color{blue}{1} \cdot \beta}{\alpha} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{1 + \color{blue}{\beta}}{\alpha} \]

      8. lower-+.f64 97.1

         \[\leadsto \frac{\color{blue}{1 + \beta}}{\alpha} \]

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha}} \]

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{\color{blue}{\alpha}} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.9%

      \[\leadsto \frac{1}{\color{blue}{\alpha}} \]

   if 2e-3 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.599999999999999978

   1. Initial program 100.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\beta \cdot \left(\frac{1}{4} + \frac{-1}{8} \cdot \beta\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, \beta, 0.25\right), \color{blue}{\beta}, 0.5\right) \]

   if 0.599999999999999978 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \left(\alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 10^{-6}:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\mathsf{fma}\left(2, \beta, 2\right) \cdot \frac{2 + \beta}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{t\_0} - \left(\frac{\alpha}{t\_0} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (+ 2.0 (+ alpha beta))))
   (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 1e-6)
     (/
      (*
       -0.5
       (- (- (* (fma 2.0 beta 2.0) (/ (+ 2.0 beta) alpha)) beta) (+ 2.0 beta)))
      alpha)
     (/ (- (/ beta t_0) (- (/ alpha t_0) 1.0)) 2.0))))

C

double code(double alpha, double beta) {
	double t_0 = 2.0 + (alpha + beta);
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-6) {
		tmp = (-0.5 * (((fma(2.0, beta, 2.0) * ((2.0 + beta) / alpha)) - beta) - (2.0 + beta))) / alpha;
	} else {
		tmp = ((beta / t_0) - ((alpha / t_0) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(2.0 + Float64(alpha + beta))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-6)
		tmp = Float64(Float64(-0.5 * Float64(Float64(Float64(fma(2.0, beta, 2.0) * Float64(Float64(2.0 + beta) / alpha)) - beta) - Float64(2.0 + beta))) / alpha);
	else
		tmp = Float64(Float64(Float64(beta / t_0) - Float64(Float64(alpha / t_0) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 1e-6], N[(N[(-0.5 * N[(N[(N[(N[(2.0 * beta + 2.0), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] - beta), $MachinePrecision] - N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta / t$95$0), $MachinePrecision] - N[(N[(alpha / t$95$0), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.99999999999999955e-7

   1. Initial program 8.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around -inf

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

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\left(\frac{\mathsf{fma}\left(2, \beta, 2\right) \cdot \left(2 + \beta\right)}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

      \[\leadsto \frac{-0.5 \cdot \left(\left(\mathsf{fma}\left(2, \beta, 2\right) \cdot \frac{2 + \beta}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha} \]

   if 9.99999999999999955e-7 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. associate-+l- N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 10: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 10^{-6}:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\mathsf{fma}\left(2, \beta, 2\right) \cdot \frac{2 + \beta}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{2 + \left(\alpha + \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 1e-6)
   (/
    (*
     -0.5
     (- (- (* (fma 2.0 beta 2.0) (/ (+ 2.0 beta) alpha)) beta) (+ 2.0 beta)))
    alpha)
   (/ (fma (- beta alpha) (/ 1.0 (+ 2.0 (+ alpha beta))) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-6) {
		tmp = (-0.5 * (((fma(2.0, beta, 2.0) * ((2.0 + beta) / alpha)) - beta) - (2.0 + beta))) / alpha;
	} else {
		tmp = fma((beta - alpha), (1.0 / (2.0 + (alpha + beta))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-6)
		tmp = Float64(Float64(-0.5 * Float64(Float64(Float64(fma(2.0, beta, 2.0) * Float64(Float64(2.0 + beta) / alpha)) - beta) - Float64(2.0 + beta))) / alpha);
	else
		tmp = Float64(fma(Float64(beta - alpha), Float64(1.0 / Float64(2.0 + Float64(alpha + beta))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 1e-6], N[(N[(-0.5 * N[(N[(N[(N[(2.0 * beta + 2.0), $MachinePrecision] * N[(N[(2.0 + beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] - beta), $MachinePrecision] - N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.99999999999999955e-7

   1. Initial program 8.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around -inf

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

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\left(\frac{\mathsf{fma}\left(2, \beta, 2\right) \cdot \left(2 + \beta\right)}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

      \[\leadsto \frac{-0.5 \cdot \left(\left(\mathsf{fma}\left(2, \beta, 2\right) \cdot \frac{2 + \beta}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha} \]

   if 9.99999999999999955e-7 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, 1\right)}}{2} \]

      7. lower-/.f64 99.9

         \[\leadsto \frac{\mathsf{fma}\left(\beta - \alpha, \color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2}}, 1\right)}{2} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\color{blue}{\left(\alpha + \beta\right) + 2}}, 1\right)}{2} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\color{blue}{2 + \left(\alpha + \beta\right)}}, 1\right)}{2} \]

      10. lower-+.f64 99.9

          \[\leadsto \frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\color{blue}{2 + \left(\alpha + \beta\right)}}, 1\right)}{2} \]

   4. Applied rewrites 99.9%

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

Alternative 11: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 10^{-6}:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\mathsf{fma}\left(2, \beta, 2\right) \cdot \frac{2}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{2 + \left(\alpha + \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 1e-6)
   (/
    (* -0.5 (- (- (* (fma 2.0 beta 2.0) (/ 2.0 alpha)) beta) (+ 2.0 beta)))
    alpha)
   (/ (fma (- beta alpha) (/ 1.0 (+ 2.0 (+ alpha beta))) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-6) {
		tmp = (-0.5 * (((fma(2.0, beta, 2.0) * (2.0 / alpha)) - beta) - (2.0 + beta))) / alpha;
	} else {
		tmp = fma((beta - alpha), (1.0 / (2.0 + (alpha + beta))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-6)
		tmp = Float64(Float64(-0.5 * Float64(Float64(Float64(fma(2.0, beta, 2.0) * Float64(2.0 / alpha)) - beta) - Float64(2.0 + beta))) / alpha);
	else
		tmp = Float64(fma(Float64(beta - alpha), Float64(1.0 / Float64(2.0 + Float64(alpha + beta))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 1e-6], N[(N[(-0.5 * N[(N[(N[(N[(2.0 * beta + 2.0), $MachinePrecision] * N[(2.0 / alpha), $MachinePrecision]), $MachinePrecision] - beta), $MachinePrecision] - N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.99999999999999955e-7

   1. Initial program 8.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around -inf

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

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\left(\frac{\mathsf{fma}\left(2, \beta, 2\right) \cdot \left(2 + \beta\right)}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

      \[\leadsto \frac{-0.5 \cdot \left(\left(\mathsf{fma}\left(2, \beta, 2\right) \cdot \frac{2 + \beta}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha} \]

   8. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{-1}{2} \cdot \left(\left(\mathsf{fma}\left(2, \beta, 2\right) \cdot \frac{2}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.2%

       \[\leadsto \frac{-0.5 \cdot \left(\left(\mathsf{fma}\left(2, \beta, 2\right) \cdot \frac{2}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha} \]

   if 9.99999999999999955e-7 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, 1\right)}}{2} \]

      7. lower-/.f64 99.9

         \[\leadsto \frac{\mathsf{fma}\left(\beta - \alpha, \color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2}}, 1\right)}{2} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\color{blue}{\left(\alpha + \beta\right) + 2}}, 1\right)}{2} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\color{blue}{2 + \left(\alpha + \beta\right)}}, 1\right)}{2} \]

      10. lower-+.f64 99.9

          \[\leadsto \frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\color{blue}{2 + \left(\alpha + \beta\right)}}, 1\right)}{2} \]

   4. Applied rewrites 99.9%

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

Alternative 12: 92.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\\ \mathbf{if}\;t\_0 \leq 0.002:\\ \;\;\;\;\frac{1}{\alpha}\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;\mathsf{fma}\left(0.25, \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)))
   (if (<= t_0 0.002)
     (/ 1.0 alpha)
     (if (<= t_0 0.6) (fma 0.25 beta 0.5) 1.0))))

C

double code(double alpha, double beta) {
	double t_0 = (((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_0 <= 0.002) {
		tmp = 1.0 / alpha;
	} else if (t_0 <= 0.6) {
		tmp = fma(0.25, beta, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	t_0 = Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_0 <= 0.002)
		tmp = Float64(1.0 / alpha);
	elseif (t_0 <= 0.6)
		tmp = fma(0.25, beta, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.002], N[(1.0 / alpha), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(0.25 * beta + 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2e-3

   1. Initial program 9.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{1 + \color{blue}{1} \cdot \beta}{\alpha} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{1 + \color{blue}{\beta}}{\alpha} \]

      8. lower-+.f64 97.1

         \[\leadsto \frac{\color{blue}{1 + \beta}}{\alpha} \]

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha}} \]

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{1}{\color{blue}{\alpha}} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.9%

      \[\leadsto \frac{1}{\color{blue}{\alpha}} \]

   if 2e-3 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.599999999999999978

   1. Initial program 100.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 97.7%

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

   if 0.599999999999999978 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 10^{-6}:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\frac{4}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{2 + \left(\alpha + \beta\right)}, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 1e-6)
   (/ (* -0.5 (- (- (/ 4.0 alpha) beta) (+ 2.0 beta))) alpha)
   (/ (fma (- beta alpha) (/ 1.0 (+ 2.0 (+ alpha beta))) 1.0) 2.0)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-6) {
		tmp = (-0.5 * (((4.0 / alpha) - beta) - (2.0 + beta))) / alpha;
	} else {
		tmp = fma((beta - alpha), (1.0 / (2.0 + (alpha + beta))), 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-6)
		tmp = Float64(Float64(-0.5 * Float64(Float64(Float64(4.0 / alpha) - beta) - Float64(2.0 + beta))) / alpha);
	else
		tmp = Float64(fma(Float64(beta - alpha), Float64(1.0 / Float64(2.0 + Float64(alpha + beta))), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 1e-6], N[(N[(-0.5 * N[(N[(N[(4.0 / alpha), $MachinePrecision] - beta), $MachinePrecision] - N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(1.0 / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.99999999999999955e-7

   1. Initial program 8.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around -inf

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

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\left(\frac{\mathsf{fma}\left(2, \beta, 2\right) \cdot \left(2 + \beta\right)}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha}} \]

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{-1}{2} \cdot \left(\left(\frac{4}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.2%

      \[\leadsto \frac{-0.5 \cdot \left(\left(\frac{4}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha} \]

   if 9.99999999999999955e-7 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\left(\alpha + \beta\right) + 2}, 1\right)}}{2} \]

      7. lower-/.f64 99.9

         \[\leadsto \frac{\mathsf{fma}\left(\beta - \alpha, \color{blue}{\frac{1}{\left(\alpha + \beta\right) + 2}}, 1\right)}{2} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\color{blue}{\left(\alpha + \beta\right) + 2}}, 1\right)}{2} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\color{blue}{2 + \left(\alpha + \beta\right)}}, 1\right)}{2} \]

      10. lower-+.f64 99.9

          \[\leadsto \frac{\mathsf{fma}\left(\beta - \alpha, \frac{1}{\color{blue}{2 + \left(\alpha + \beta\right)}}, 1\right)}{2} \]

   4. Applied rewrites 99.9%

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

Alternative 14: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 10^{-6}:\\ \;\;\;\;\frac{-0.5 \cdot \left(\left(\frac{4}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 1e-6)
   (/ (* -0.5 (- (- (/ 4.0 alpha) beta) (+ 2.0 beta))) alpha)
   (fma (/ (- beta alpha) (+ 2.0 (+ alpha beta))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-6) {
		tmp = (-0.5 * (((4.0 / alpha) - beta) - (2.0 + beta))) / alpha;
	} else {
		tmp = fma(((beta - alpha) / (2.0 + (alpha + beta))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 1e-6)
		tmp = Float64(Float64(-0.5 * Float64(Float64(Float64(4.0 / alpha) - beta) - Float64(2.0 + beta))) / alpha);
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(2.0 + Float64(alpha + beta))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 1e-6], N[(N[(-0.5 * N[(N[(N[(4.0 / alpha), $MachinePrecision] - beta), $MachinePrecision] - N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 9.99999999999999955e-7

   1. Initial program 8.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around -inf

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

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\left(\frac{\mathsf{fma}\left(2, \beta, 2\right) \cdot \left(2 + \beta\right)}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha}} \]

   6. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{-1}{2} \cdot \left(\left(\frac{4}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.2%

      \[\leadsto \frac{-0.5 \cdot \left(\left(\frac{4}{\alpha} - \beta\right) - \left(2 + \beta\right)\right)}{\alpha} \]

   if 9.99999999999999955e-7 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}, \frac{1}{2}, \frac{1}{2}\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + 2}}, \frac{1}{2}, \frac{1}{2}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta - \alpha}{\color{blue}{2 + \left(\alpha + \beta\right)}}, \frac{1}{2}, \frac{1}{2}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta - \alpha}{\color{blue}{2 + \left(\alpha + \beta\right)}}, \frac{1}{2}, \frac{1}{2}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}, \color{blue}{\frac{1}{2}}, \frac{1}{2}\right) \]

      11. metadata-eval 99.9

          \[\leadsto \mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}, 0.5, \color{blue}{0.5}\right) \]

   4. Applied rewrites 99.9%

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

Alternative 15: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 2e-10)
   (/ (+ 1.0 beta) alpha)
   (fma (/ (- beta alpha) (+ 2.0 (+ alpha beta))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 2e-10) {
		tmp = (1.0 + beta) / alpha;
	} else {
		tmp = fma(((beta - alpha) / (2.0 + (alpha + beta))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 2e-10)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(2.0 + Float64(alpha + beta))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 2e-10], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2.00000000000000007e-10

   1. Initial program 7.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{1 + \color{blue}{1} \cdot \beta}{\alpha} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{1 + \color{blue}{\beta}}{\alpha} \]

      8. lower-+.f64 98.6

         \[\leadsto \frac{\color{blue}{1 + \beta}}{\alpha} \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha}} \]

   if 2.00000000000000007e-10 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.8%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}, \frac{1}{2}, \frac{1}{2}\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + 2}}, \frac{1}{2}, \frac{1}{2}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta - \alpha}{\color{blue}{2 + \left(\alpha + \beta\right)}}, \frac{1}{2}, \frac{1}{2}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\beta - \alpha}{\color{blue}{2 + \left(\alpha + \beta\right)}}, \frac{1}{2}, \frac{1}{2}\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}, \color{blue}{\frac{1}{2}}, \frac{1}{2}\right) \]

      11. metadata-eval 99.8

          \[\leadsto \mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}, 0.5, \color{blue}{0.5}\right) \]

   4. Applied rewrites 99.8%

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

Alternative 16: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.002:\\ \;\;\;\;\frac{1 + \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\beta, \frac{0.5}{2 + \beta}, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.002)
   (/ (+ 1.0 beta) alpha)
   (fma beta (/ 0.5 (+ 2.0 beta)) 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.002) {
		tmp = (1.0 + beta) / alpha;
	} else {
		tmp = fma(beta, (0.5 / (2.0 + beta)), 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.002)
		tmp = Float64(Float64(1.0 + beta) / alpha);
	else
		tmp = fma(beta, Float64(0.5 / Float64(2.0 + beta)), 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.002], N[(N[(1.0 + beta), $MachinePrecision] / alpha), $MachinePrecision], N[(beta * N[(0.5 / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2e-3

   1. Initial program 9.7%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{1 + \color{blue}{1} \cdot \beta}{\alpha} \]

      7. *-lft-identity N/A

         \[\leadsto \frac{1 + \color{blue}{\beta}}{\alpha} \]

      8. lower-+.f64 97.1

         \[\leadsto \frac{\color{blue}{1 + \beta}}{\alpha} \]

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\frac{1 + \beta}{\alpha}} \]

   if 2e-3 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 100.0%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 98.3

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

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.4%

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

Alternative 17: 71.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.6:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.6)
   0.5
   1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = 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(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8) :: tmp
    if (((((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0) <= 0.6d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta):
	tmp = 0
	if ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.6:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(alpha, beta)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(beta - alpha) / Float64(Float64(alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.6)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta)
	tmp = 0.0;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.6)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := If[LessEqual[N[(N[(N[(N[(beta - alpha), $MachinePrecision] / N[(N[(alpha + beta), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.6], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.599999999999999978

   1. Initial program 64.6%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 62.3

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

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 61.2%

      \[\leadsto 0.5 \]

   if 0.599999999999999978 < (/.f64 (+.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 99.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 96.0%

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

Alternative 18: 72.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.25, \beta, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= beta 2.0) (fma 0.25 beta 0.5) 1.0))

C

double code(double alpha, double beta) {
	double tmp;
	if (beta <= 2.0) {
		tmp = fma(0.25, beta, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(alpha, beta)
	tmp = 0.0
	if (beta <= 2.0)
		tmp = fma(0.25, beta, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[alpha_, beta_] := If[LessEqual[beta, 2.0], N[(0.25 * beta + 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if beta < 2

   1. Initial program 69.4%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in alpha around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 67.4

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in beta around 0

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

   8. Applied rewrites 66.8%

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

   if 2 < beta

   1. Initial program 80.1%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 75.5%

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

Alternative 19: 36.8% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 1.0)

C

double code(double alpha, double beta) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    code = 1.0d0
end function

Java

public static double code(double alpha, double beta) {
	return 1.0;
}

Python

def code(alpha, beta):
	return 1.0

Julia

function code(alpha, beta)
	return 1.0
end

MATLAB

function tmp = code(alpha, beta)
	tmp = 1.0;
end

Wolfram

code[alpha_, beta_] := 1.0

Derivation

1. Initial program 72.5%

   \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
2. Add Preprocessing

3. Taylor expanded in beta around inf

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 32.0%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025006 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))