Octave 3.8, jcobi/1

Percentage Accurate: 74.9% → 99.7%

Time: 3.3s

Alternatives: 8

Speedup: 0.5×

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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-2 - \beta\right) - \alpha\\ \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-1}{t\_0}, \beta, \frac{2 + \beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t\_0 - \left(\beta - \alpha\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (- -2.0 beta) alpha)))
   (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.0)
     (/ (fma (/ -1.0 t_0) beta (/ (+ 2.0 beta) alpha)) 2.0)
     (* 0.5 (/ (- t_0 (- beta alpha)) t_0)))))

C

double code(double alpha, double beta) {
	double t_0 = (-2.0 - beta) - alpha;
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.0) {
		tmp = fma((-1.0 / t_0), beta, ((2.0 + beta) / alpha)) / 2.0;
	} else {
		tmp = 0.5 * ((t_0 - (beta - alpha)) / t_0);
	}
	return tmp;
}

Julia

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

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(-2.0 - beta), $MachinePrecision] - alpha), $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], 0.0], N[(N[(N[(-1.0 / t$95$0), $MachinePrecision] * beta + N[(N[(2.0 + beta), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.5 * N[(N[(t$95$0 - N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $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)) < 0.0

   1. Initial program 74.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 75.7

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. add-flip-rev N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. associate-/r/ N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 29.1

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

   8. Applied rewrites 29.1%

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

   if 0.0 < (/.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 74.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 75.7

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 75.7

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

   3. Applied rewrites 75.7%

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

   5. Applied rewrites 74.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. add-to-fraction-rev N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. add-flip N/A

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

      14. lift--.f64 N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. lower--.f64 75.3

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

   7. Applied rewrites 75.3%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-2 - \beta\right) - \alpha\\ \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0:\\ \;\;\;\;0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{t\_0 - \left(\beta - \alpha\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (let* ((t_0 (- (- -2.0 beta) alpha)))
   (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.0)
     (* 0.5 (/ (+ 2.0 (* 2.0 beta)) alpha))
     (* 0.5 (/ (- t_0 (- beta alpha)) t_0)))))

C

double code(double alpha, double beta) {
	double t_0 = (-2.0 - beta) - alpha;
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.0) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = 0.5 * ((t_0 - (beta - alpha)) / t_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) :: t_0
    real(8) :: tmp
    t_0 = ((-2.0d0) - beta) - alpha
    if (((((beta - alpha) / ((alpha + beta) + 2.0d0)) + 1.0d0) / 2.0d0) <= 0.0d0) then
        tmp = 0.5d0 * ((2.0d0 + (2.0d0 * beta)) / alpha)
    else
        tmp = 0.5d0 * ((t_0 - (beta - alpha)) / t_0)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta) {
	double t_0 = (-2.0 - beta) - alpha;
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.0) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = 0.5 * ((t_0 - (beta - alpha)) / t_0);
	}
	return tmp;
}

Python

def code(alpha, beta):
	t_0 = (-2.0 - beta) - alpha
	tmp = 0
	if ((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.0:
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha)
	else:
		tmp = 0.5 * ((t_0 - (beta - alpha)) / t_0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(alpha, beta)
	t_0 = (-2.0 - beta) - alpha;
	tmp = 0.0;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.0)
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	else
		tmp = 0.5 * ((t_0 - (beta - alpha)) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_] := Block[{t$95$0 = N[(N[(-2.0 - beta), $MachinePrecision] - alpha), $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], 0.0], N[(0.5 * N[(N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(t$95$0 - N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $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)) < 0.0

   1. Initial program 74.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 75.7

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. add-flip-rev N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. associate-/r/ N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 29.1

         \[\leadsto 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} \]

   8. Applied rewrites 29.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   if 0.0 < (/.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 74.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 75.7

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 75.7

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

   3. Applied rewrites 75.7%

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

   5. Applied rewrites 74.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. add-to-fraction-rev N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-lft-out N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. add-flip N/A

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

      14. lift--.f64 N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f64 N/A

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

      17. lower--.f64 75.3

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

   7. Applied rewrites 75.3%

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

Alternative 3: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0:\\ \;\;\;\;0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta - \left(-2 - \alpha\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.0)
   (* 0.5 (/ (+ 2.0 (* 2.0 beta)) alpha))
   (fma (/ (- beta alpha) (- beta (- -2.0 alpha))) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.0) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = fma(((beta - alpha) / (beta - (-2.0 - alpha))), 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) <= 0.0)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(2.0 * beta)) / alpha));
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(beta - Float64(-2.0 - alpha))), 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], 0.0], N[(0.5 * N[(N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta - N[(-2.0 - alpha), $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)) < 0.0

   1. Initial program 74.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 75.7

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. add-flip-rev N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. associate-/r/ N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 29.1

         \[\leadsto 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} \]

   8. Applied rewrites 29.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   if 0.0 < (/.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 74.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 74.9

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

   3. Applied rewrites 74.9%

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

   5. Applied rewrites 74.9%

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

Alternative 4: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0:\\ \;\;\;\;0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\alpha - \beta, \frac{0.5}{\left(-2 - \beta\right) - \alpha}, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.0)
   (* 0.5 (/ (+ 2.0 (* 2.0 beta)) alpha))
   (fma (- alpha beta) (/ 0.5 (- (- -2.0 beta) alpha)) 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.0) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = fma((alpha - beta), (0.5 / ((-2.0 - beta) - alpha)), 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) <= 0.0)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(2.0 * beta)) / alpha));
	else
		tmp = fma(Float64(alpha - beta), Float64(0.5 / Float64(Float64(-2.0 - beta) - alpha)), 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], 0.0], N[(0.5 * N[(N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(alpha - beta), $MachinePrecision] * N[(0.5 / N[(N[(-2.0 - beta), $MachinePrecision] - alpha), $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)) < 0.0

   1. Initial program 74.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 75.7

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. add-flip-rev N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. associate-/r/ N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 29.1

         \[\leadsto 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} \]

   8. Applied rewrites 29.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   if 0.0 < (/.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 74.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 75.7

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 75.7

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

   3. Applied rewrites 75.7%

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

   5. Applied rewrites 74.6%

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

Alternative 5: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.0002:\\ \;\;\;\;0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\beta - -2}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.0002)
   (* 0.5 (/ (+ 2.0 (* 2.0 beta)) alpha))
   (fma (/ (- beta alpha) (- beta -2.0)) 0.5 0.5)))

C

double code(double alpha, double beta) {
	double tmp;
	if (((((beta - alpha) / ((alpha + beta) + 2.0)) + 1.0) / 2.0) <= 0.0002) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = fma(((beta - alpha) / (beta - -2.0)), 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) <= 0.0002)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(2.0 * beta)) / alpha));
	else
		tmp = fma(Float64(Float64(beta - alpha) / Float64(beta - -2.0)), 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], 0.0002], N[(0.5 * N[(N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(beta - -2.0), $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)) < 2.0000000000000001e-4

   1. Initial program 74.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 75.7

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. add-flip-rev N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. associate-/r/ N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 29.1

         \[\leadsto 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} \]

   8. Applied rewrites 29.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   if 2.0000000000000001e-4 < (/.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 74.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 74.9

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

   3. Applied rewrites 74.9%

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 72.1%

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

Alternative 6: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \leq 0.0002:\\ \;\;\;\;0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta)
 :precision binary64
 (if (<= (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) 0.0002)
   (* 0.5 (/ (+ 2.0 (* 2.0 beta)) alpha))
   (fma (/ beta (+ 2.0 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) <= 0.0002) {
		tmp = 0.5 * ((2.0 + (2.0 * beta)) / alpha);
	} else {
		tmp = fma((beta / (2.0 + 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) <= 0.0002)
		tmp = Float64(0.5 * Float64(Float64(2.0 + Float64(2.0 * beta)) / alpha));
	else
		tmp = fma(Float64(beta / Float64(2.0 + 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], 0.0002], N[(0.5 * N[(N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], N[(N[(beta / N[(2.0 + beta), $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)) < 2.0000000000000001e-4

   1. Initial program 74.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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. add-flip N/A

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

      10. lower--.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 75.7

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 75.7

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

   3. Applied rewrites 75.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. add-flip-rev N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. associate-/r/ N/A

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

      8. sub-negate-rev N/A

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

      9. lift--.f64 N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 29.1

         \[\leadsto 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha} \]

   8. Applied rewrites 29.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}} \]

   if 2.0000000000000001e-4 < (/.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 74.9%

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
   2. 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. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 74.9

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

   3. Applied rewrites 74.9%

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 72.8

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

   8. Applied rewrites 72.8%

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

Alternative 7: 72.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right) \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (fma (/ beta (+ 2.0 beta)) 0.5 0.5))

C

double code(double alpha, double beta) {
	return fma((beta / (2.0 + beta)), 0.5, 0.5);
}

Julia

function code(alpha, beta)
	return fma(Float64(beta / Float64(2.0 + beta)), 0.5, 0.5)
end

Wolfram

code[alpha_, beta_] := N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]

Derivation

1. Initial program 74.9%

   \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
2. 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. mult-flip N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. lift--.f64 N/A

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

   9. sub-negate-rev N/A

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

   10. lower-/.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. add-flip N/A

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

   14. sub-negate N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval 74.9

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

3. Applied rewrites 74.9%

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

5. Applied rewrites 74.9%

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

6. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 72.8

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

8. Applied rewrites 72.8%

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

Alternative 8: 36.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2} \end{array} \]

FPCore

(FPCore (alpha beta) :precision binary64 (/ 2.0 2.0))

C

double code(double alpha, double beta) {
	return 2.0 / 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 = 2.0d0 / 2.0d0
end function

Java

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

Python

def code(alpha, beta):
	return 2.0 / 2.0

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_] := N[(2.0 / 2.0), $MachinePrecision]

Derivation

1. Initial program 74.9%

   \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2} \]
2. 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. add-flip N/A

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

   10. lower--.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. lower-/.f64 75.7

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

   14. lift-+.f64 N/A

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

   15. add-flip N/A

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

   16. lower--.f64 N/A

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

   17. metadata-eval 75.7

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

3. Applied rewrites 75.7%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-negate-rev N/A

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

   4. add-flip-rev N/A

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

   5. lift-/.f64 N/A

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

   6. div-flip N/A

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

   7. associate-/r/ N/A

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

   8. sub-negate-rev N/A

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

   9. lift--.f64 N/A

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

   10. lower-fma.f64 N/A

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

5. Applied rewrites 75.7%

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

6. Taylor expanded in beta around inf

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

8. Applied rewrites 36.6%

   \[\leadsto \frac{\color{blue}{2}}{2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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))