Octave 3.8, jcobi/2

Percentage Accurate: 62.4% → 97.5%

Time: 11.1s

Alternatives: 6

Speedup: N/A×

Specification

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 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, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

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

Initial Program: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))))
   (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0) 2.0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 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, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    t_0 = (alpha + beta) + (2.0d0 * i)
    code = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	return (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	return Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
end

Wolfram

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

Alternative 1: 97.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 + 2\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_1} + 1}{2} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(2, i, \beta\right) + 2, -1 \cdot \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{t\_1} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i))) (t_1 (+ t_0 2.0)))
   (if (<=
        (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) 1.0) 2.0)
        2e-16)
     (/
      (-
       (* -1.0 (* 0.0 (/ beta alpha)))
       (*
        -1.0
        (/
         (* -1.0 (fma -1.0 (+ (fma 2.0 i beta) 2.0) (* -1.0 (fma 2.0 i beta))))
         alpha)))
      2.0)
     (/
      (+
       (/ (* (+ beta alpha) (/ (- beta alpha) (fma 2.0 i (+ beta alpha)))) t_1)
       1.0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 + 2.0;
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / t_1) + 1.0) / 2.0) <= 2e-16) {
		tmp = ((-1.0 * (0.0 * (beta / alpha))) - (-1.0 * ((-1.0 * fma(-1.0, (fma(2.0, i, beta) + 2.0), (-1.0 * fma(2.0, i, beta)))) / alpha))) / 2.0;
	} else {
		tmp = ((((beta + alpha) * ((beta - alpha) / fma(2.0, i, (beta + alpha)))) / t_1) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 + 2.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) + 1.0) / 2.0) <= 2e-16)
		tmp = Float64(Float64(Float64(-1.0 * Float64(0.0 * Float64(beta / alpha))) - Float64(-1.0 * Float64(Float64(-1.0 * fma(-1.0, Float64(fma(2.0, i, beta) + 2.0), Float64(-1.0 * fma(2.0, i, beta)))) / alpha))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta + alpha) * Float64(Float64(beta - alpha) / fma(2.0, i, Float64(beta + alpha)))) / t_1) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 2.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 2e-16], N[(N[(N[(-1.0 * N[(0.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(-1.0 * N[(-1.0 * N[(N[(2.0 * i + beta), $MachinePrecision] + 2.0), $MachinePrecision] + N[(-1.0 * N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2e-16

   1. Initial program 2.1%

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

   3. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. div-sub N/A

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

      3. div-add N/A

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

      4. associate-*r/ N/A

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

      5. +-commutative N/A

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

      6. lower--.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha} - \frac{\color{blue}{-1} \cdot \left(-1 \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right)}{\alpha}\right)}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha} - \frac{\color{blue}{-1 \cdot \left(-1 \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right)}}{\alpha}\right)}{2} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha} - \frac{-1 \cdot \color{blue}{\left(-1 \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right)}}{\alpha}\right)}{2} \]

   5. Applied rewrites 90.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha} - \frac{-1 \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(2, i, \beta\right) + 2, -1 \cdot \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}\right)}}{2} \]

   if 2e-16 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(2, i, \beta\right) + 2, -1 \cdot \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := {t\_0}^{2} \cdot \left(\beta + \alpha\right)\\ t_2 := 0.5 \cdot \left(\left(\frac{\beta}{t\_0} + 1\right) - \frac{\alpha}{t\_0}\right)\\ t_3 := \mathsf{fma}\left(2, t\_0, 2 \cdot \left(\beta + \alpha\right)\right)\\ t_4 := -1 \cdot \frac{\alpha}{2 + \alpha} - 1\\ t_5 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_6 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_5}}{t\_5 + 2} + 1}{2}\\ \mathbf{if}\;t\_6 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(2, i, \beta\right) + 2, -1 \cdot \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_6 \leq 0.5:\\ \;\;\;\;\frac{\frac{\frac{\alpha \cdot \alpha}{{\left(2 + \alpha\right)}^{2}}}{t\_4} - {t\_4}^{-1}}{2}\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{t\_3}^{2}}{{t\_0}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{t\_1}\right), t\_3 \cdot \frac{\beta - \alpha}{t\_1}\right), t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0))
        (t_1 (* (pow t_0 2.0) (+ beta alpha)))
        (t_2 (* 0.5 (- (+ (/ beta t_0) 1.0) (/ alpha t_0))))
        (t_3 (fma 2.0 t_0 (* 2.0 (+ beta alpha))))
        (t_4 (- (* -1.0 (/ alpha (+ 2.0 alpha))) 1.0))
        (t_5 (+ (+ alpha beta) (* 2.0 i)))
        (t_6
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_5) (+ t_5 2.0)) 1.0)
          2.0)))
   (if (<= t_6 2e-16)
     (/
      (-
       (* -1.0 (* 0.0 (/ beta alpha)))
       (*
        -1.0
        (/
         (* -1.0 (fma -1.0 (+ (fma 2.0 i beta) 2.0) (* -1.0 (fma 2.0 i beta))))
         alpha)))
      2.0)
     (if (<= t_6 0.5)
       (/
        (- (/ (/ (* alpha alpha) (pow (+ 2.0 alpha) 2.0)) t_4) (pow t_4 -1.0))
        2.0)
       (if (<= t_6 2.0)
         (fma
          i
          (*
           -0.5
           (fma
            i
            (fma
             -1.0
             (*
              (/ (pow t_3 2.0) (pow t_0 3.0))
              (/ (- beta alpha) (pow (+ beta alpha) 2.0)))
             (/ (* 4.0 (- beta alpha)) t_1))
            (* t_3 (/ (- beta alpha) t_1))))
          t_2)
         (fma
          i
          (*
           -0.5
           (/
            (-
             (fma
              -1.0
              (/ (fma 2.0 beta (* 2.0 (+ 2.0 beta))) alpha)
              (fma 4.0 (/ beta alpha) (* 12.0 (/ i alpha))))
             (+ 4.0 (* -4.0 (/ (+ 4.0 (+ beta (* 2.0 beta))) alpha))))
            alpha))
          t_2))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = pow(t_0, 2.0) * (beta + alpha);
	double t_2 = 0.5 * (((beta / t_0) + 1.0) - (alpha / t_0));
	double t_3 = fma(2.0, t_0, (2.0 * (beta + alpha)));
	double t_4 = (-1.0 * (alpha / (2.0 + alpha))) - 1.0;
	double t_5 = (alpha + beta) + (2.0 * i);
	double t_6 = (((((alpha + beta) * (beta - alpha)) / t_5) / (t_5 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_6 <= 2e-16) {
		tmp = ((-1.0 * (0.0 * (beta / alpha))) - (-1.0 * ((-1.0 * fma(-1.0, (fma(2.0, i, beta) + 2.0), (-1.0 * fma(2.0, i, beta)))) / alpha))) / 2.0;
	} else if (t_6 <= 0.5) {
		tmp = ((((alpha * alpha) / pow((2.0 + alpha), 2.0)) / t_4) - pow(t_4, -1.0)) / 2.0;
	} else if (t_6 <= 2.0) {
		tmp = fma(i, (-0.5 * fma(i, fma(-1.0, ((pow(t_3, 2.0) / pow(t_0, 3.0)) * ((beta - alpha) / pow((beta + alpha), 2.0))), ((4.0 * (beta - alpha)) / t_1)), (t_3 * ((beta - alpha) / t_1)))), t_2);
	} else {
		tmp = fma(i, (-0.5 * ((fma(-1.0, (fma(2.0, beta, (2.0 * (2.0 + beta))) / alpha), fma(4.0, (beta / alpha), (12.0 * (i / alpha)))) - (4.0 + (-4.0 * ((4.0 + (beta + (2.0 * beta))) / alpha)))) / alpha)), t_2);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64((t_0 ^ 2.0) * Float64(beta + alpha))
	t_2 = Float64(0.5 * Float64(Float64(Float64(beta / t_0) + 1.0) - Float64(alpha / t_0)))
	t_3 = fma(2.0, t_0, Float64(2.0 * Float64(beta + alpha)))
	t_4 = Float64(Float64(-1.0 * Float64(alpha / Float64(2.0 + alpha))) - 1.0)
	t_5 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_6 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_5) / Float64(t_5 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_6 <= 2e-16)
		tmp = Float64(Float64(Float64(-1.0 * Float64(0.0 * Float64(beta / alpha))) - Float64(-1.0 * Float64(Float64(-1.0 * fma(-1.0, Float64(fma(2.0, i, beta) + 2.0), Float64(-1.0 * fma(2.0, i, beta)))) / alpha))) / 2.0);
	elseif (t_6 <= 0.5)
		tmp = Float64(Float64(Float64(Float64(Float64(alpha * alpha) / (Float64(2.0 + alpha) ^ 2.0)) / t_4) - (t_4 ^ -1.0)) / 2.0);
	elseif (t_6 <= 2.0)
		tmp = fma(i, Float64(-0.5 * fma(i, fma(-1.0, Float64(Float64((t_3 ^ 2.0) / (t_0 ^ 3.0)) * Float64(Float64(beta - alpha) / (Float64(beta + alpha) ^ 2.0))), Float64(Float64(4.0 * Float64(beta - alpha)) / t_1)), Float64(t_3 * Float64(Float64(beta - alpha) / t_1)))), t_2);
	else
		tmp = fma(i, Float64(-0.5 * Float64(Float64(fma(-1.0, Float64(fma(2.0, beta, Float64(2.0 * Float64(2.0 + beta))) / alpha), fma(4.0, Float64(beta / alpha), Float64(12.0 * Float64(i / alpha)))) - Float64(4.0 + Float64(-4.0 * Float64(Float64(4.0 + Float64(beta + Float64(2.0 * beta))) / alpha)))) / alpha)), t_2);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(N[(N[(beta / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * t$95$0 + N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-1.0 * N[(alpha / N[(2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] / N[(t$95$5 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$6, 2e-16], N[(N[(N[(-1.0 * N[(0.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(-1.0 * N[(-1.0 * N[(N[(2.0 * i + beta), $MachinePrecision] + 2.0), $MachinePrecision] + N[(-1.0 * N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$6, 0.5], N[(N[(N[(N[(N[(alpha * alpha), $MachinePrecision] / N[Power[N[(2.0 + alpha), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] - N[Power[t$95$4, -1.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$6, 2.0], N[(i * N[(-0.5 * N[(i * N[(-1.0 * N[(N[(N[Power[t$95$3, 2.0], $MachinePrecision] / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(N[(beta - alpha), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(i * N[(-0.5 * N[(N[(N[(-1.0 * N[(N[(2.0 * beta + N[(2.0 * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + N[(4.0 * N[(beta / alpha), $MachinePrecision] + N[(12.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 + N[(-4.0 * N[(N[(4.0 + N[(beta + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2e-16

   1. Initial program 2.1%

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

   3. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. div-sub N/A

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

      3. div-add N/A

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

      4. associate-*r/ N/A

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

      5. +-commutative N/A

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

      6. lower--.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha} - \frac{\color{blue}{-1} \cdot \left(-1 \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right)}{\alpha}\right)}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha} - \frac{\color{blue}{-1 \cdot \left(-1 \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right)}}{\alpha}\right)}{2} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha} - \frac{-1 \cdot \color{blue}{\left(-1 \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right)}}{\alpha}\right)}{2} \]

   5. Applied rewrites 90.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha} - \frac{-1 \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(2, i, \beta\right) + 2, -1 \cdot \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}\right)}}{2} \]

   if 2e-16 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.5

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in beta around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{{\alpha}^{4}}{{\left(2 + \left(\alpha + 2 \cdot i\right)\right)}^{2} \cdot {\left(\alpha + 2 \cdot i\right)}^{2}} - 1}{\color{blue}{-1 \cdot \frac{{\alpha}^{2}}{\left(2 + \left(\alpha + 2 \cdot i\right)\right) \cdot \left(\alpha + 2 \cdot i\right)} - 1}}}{2} \]

   7. Applied rewrites 85.6%

      \[\leadsto \frac{\color{blue}{\frac{\frac{{\alpha}^{4}}{{\left(2 + \left(\alpha + 2 \cdot i\right)\right)}^{2} \cdot {\left(\alpha + 2 \cdot i\right)}^{2}} - 1}{-1 \cdot \frac{{\alpha}^{2}}{\left(2 + \left(\alpha + 2 \cdot i\right)\right) \cdot \left(\alpha + 2 \cdot i\right)} - 1}}}{2} \]

   8. Taylor expanded in i around 0

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

      1. div-sub N/A

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

      2. lower--.f64 N/A

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

   10. Applied rewrites 90.0%

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

   if 0.5 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2

   1. Initial program 99.8%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}\right) - \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right) + i \cdot \left(\frac{-1}{2} \cdot \left(i \cdot \left(-1 \cdot \frac{{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{3} \cdot {\left(\alpha + \beta\right)}^{2}} + 4 \cdot \frac{\beta - \alpha}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)\right) + \frac{-1}{2} \cdot \frac{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right) \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{\left(\mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right)\right)}^{2}}{{\left(\left(\beta + \alpha\right) + 2\right)}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), \mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right) \cdot \frac{\beta - \alpha}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right)} \]

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

   1. Initial program 3.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}\right) - \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right) + i \cdot \left(\frac{-1}{2} \cdot \left(i \cdot \left(-1 \cdot \frac{{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{3} \cdot {\left(\alpha + \beta\right)}^{2}} + 4 \cdot \frac{\beta - \alpha}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)\right) + \frac{-1}{2} \cdot \frac{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right) \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)} \]

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{\left(\mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right)\right)}^{2}}{{\left(\left(\beta + \alpha\right) + 2\right)}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), \mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right) \cdot \frac{\beta - \alpha}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right)} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{2 \cdot \beta + 2 \cdot \left(2 + \beta\right)}{\alpha} + \left(4 \cdot \frac{\beta}{\alpha} + 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\color{blue}{\alpha}}, \frac{1}{2} \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{2 \cdot \beta + 2 \cdot \left(2 + \beta\right)}{\alpha} + \left(4 \cdot \frac{\beta}{\alpha} + 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]

   8. Applied rewrites 28.3%

      \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\color{blue}{\alpha}}, 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(2, i, \beta\right) + 2, -1 \cdot \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 0.5:\\ \;\;\;\;\frac{\frac{\frac{\alpha \cdot \alpha}{{\left(2 + \alpha\right)}^{2}}}{-1 \cdot \frac{\alpha}{2 + \alpha} - 1} - {\left(-1 \cdot \frac{\alpha}{2 + \alpha} - 1\right)}^{-1}}{2}\\ \mathbf{elif}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{\left(\mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right)\right)}^{2}}{{\left(\left(\beta + \alpha\right) + 2\right)}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), \mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right) \cdot \frac{\beta - \alpha}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := {t\_0}^{2} \cdot \left(\beta + \alpha\right)\\ t_2 := 0.5 \cdot \left(\left(\frac{\beta}{t\_0} + 1\right) - \frac{\alpha}{t\_0}\right)\\ t_3 := \mathsf{fma}\left(2, t\_0, 2 \cdot \left(\beta + \alpha\right)\right)\\ t_4 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_5 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_4}}{t\_4 + 2} + 1}{2}\\ \mathbf{if}\;t\_5 \leq 0:\\ \;\;\;\;\frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(2, i, \beta\right) + 2, -1 \cdot \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{t\_3}^{2}}{{t\_0}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{t\_1}\right), t\_3 \cdot \frac{\beta - \alpha}{t\_1}\right), t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0))
        (t_1 (* (pow t_0 2.0) (+ beta alpha)))
        (t_2 (* 0.5 (- (+ (/ beta t_0) 1.0) (/ alpha t_0))))
        (t_3 (fma 2.0 t_0 (* 2.0 (+ beta alpha))))
        (t_4 (+ (+ alpha beta) (* 2.0 i)))
        (t_5
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_4) (+ t_4 2.0)) 1.0)
          2.0)))
   (if (<= t_5 0.0)
     (/
      (-
       (* -1.0 (* 0.0 (/ beta alpha)))
       (*
        -1.0
        (/
         (* -1.0 (fma -1.0 (+ (fma 2.0 i beta) 2.0) (* -1.0 (fma 2.0 i beta))))
         alpha)))
      2.0)
     (if (<= t_5 2.0)
       (fma
        i
        (*
         -0.5
         (fma
          i
          (fma
           -1.0
           (*
            (/ (pow t_3 2.0) (pow t_0 3.0))
            (/ (- beta alpha) (pow (+ beta alpha) 2.0)))
           (/ (* 4.0 (- beta alpha)) t_1))
          (* t_3 (/ (- beta alpha) t_1))))
        t_2)
       (fma
        i
        (*
         -0.5
         (/
          (-
           (fma
            -1.0
            (/ (fma 2.0 beta (* 2.0 (+ 2.0 beta))) alpha)
            (fma 4.0 (/ beta alpha) (* 12.0 (/ i alpha))))
           (+ 4.0 (* -4.0 (/ (+ 4.0 (+ beta (* 2.0 beta))) alpha))))
          alpha))
        t_2)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = pow(t_0, 2.0) * (beta + alpha);
	double t_2 = 0.5 * (((beta / t_0) + 1.0) - (alpha / t_0));
	double t_3 = fma(2.0, t_0, (2.0 * (beta + alpha)));
	double t_4 = (alpha + beta) + (2.0 * i);
	double t_5 = (((((alpha + beta) * (beta - alpha)) / t_4) / (t_4 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_5 <= 0.0) {
		tmp = ((-1.0 * (0.0 * (beta / alpha))) - (-1.0 * ((-1.0 * fma(-1.0, (fma(2.0, i, beta) + 2.0), (-1.0 * fma(2.0, i, beta)))) / alpha))) / 2.0;
	} else if (t_5 <= 2.0) {
		tmp = fma(i, (-0.5 * fma(i, fma(-1.0, ((pow(t_3, 2.0) / pow(t_0, 3.0)) * ((beta - alpha) / pow((beta + alpha), 2.0))), ((4.0 * (beta - alpha)) / t_1)), (t_3 * ((beta - alpha) / t_1)))), t_2);
	} else {
		tmp = fma(i, (-0.5 * ((fma(-1.0, (fma(2.0, beta, (2.0 * (2.0 + beta))) / alpha), fma(4.0, (beta / alpha), (12.0 * (i / alpha)))) - (4.0 + (-4.0 * ((4.0 + (beta + (2.0 * beta))) / alpha)))) / alpha)), t_2);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64((t_0 ^ 2.0) * Float64(beta + alpha))
	t_2 = Float64(0.5 * Float64(Float64(Float64(beta / t_0) + 1.0) - Float64(alpha / t_0)))
	t_3 = fma(2.0, t_0, Float64(2.0 * Float64(beta + alpha)))
	t_4 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_5 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_4) / Float64(t_4 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_5 <= 0.0)
		tmp = Float64(Float64(Float64(-1.0 * Float64(0.0 * Float64(beta / alpha))) - Float64(-1.0 * Float64(Float64(-1.0 * fma(-1.0, Float64(fma(2.0, i, beta) + 2.0), Float64(-1.0 * fma(2.0, i, beta)))) / alpha))) / 2.0);
	elseif (t_5 <= 2.0)
		tmp = fma(i, Float64(-0.5 * fma(i, fma(-1.0, Float64(Float64((t_3 ^ 2.0) / (t_0 ^ 3.0)) * Float64(Float64(beta - alpha) / (Float64(beta + alpha) ^ 2.0))), Float64(Float64(4.0 * Float64(beta - alpha)) / t_1)), Float64(t_3 * Float64(Float64(beta - alpha) / t_1)))), t_2);
	else
		tmp = fma(i, Float64(-0.5 * Float64(Float64(fma(-1.0, Float64(fma(2.0, beta, Float64(2.0 * Float64(2.0 + beta))) / alpha), fma(4.0, Float64(beta / alpha), Float64(12.0 * Float64(i / alpha)))) - Float64(4.0 + Float64(-4.0 * Float64(Float64(4.0 + Float64(beta + Float64(2.0 * beta))) / alpha)))) / alpha)), t_2);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(N[(N[(beta / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * t$95$0 + N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] / N[(t$95$4 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$5, 0.0], N[(N[(N[(-1.0 * N[(0.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(-1.0 * N[(-1.0 * N[(N[(2.0 * i + beta), $MachinePrecision] + 2.0), $MachinePrecision] + N[(-1.0 * N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(i * N[(-0.5 * N[(i * N[(-1.0 * N[(N[(N[Power[t$95$3, 2.0], $MachinePrecision] / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[(N[(beta - alpha), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(i * N[(-0.5 * N[(N[(N[(-1.0 * N[(N[(2.0 * beta + N[(2.0 * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] + N[(4.0 * N[(beta / alpha), $MachinePrecision] + N[(12.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 + N[(-4.0 * N[(N[(4.0 + N[(beta + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.0

   1. Initial program 1.8%

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

   3. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. div-sub N/A

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

      3. div-add N/A

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

      4. associate-*r/ N/A

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

      5. +-commutative N/A

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

      6. lower--.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha} - \frac{\color{blue}{-1} \cdot \left(-1 \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right)}{\alpha}\right)}{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha} - \frac{\color{blue}{-1 \cdot \left(-1 \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right)}}{\alpha}\right)}{2} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha} - \frac{-1 \cdot \color{blue}{\left(-1 \cdot \left(2 + \left(\beta + 2 \cdot i\right)\right) + -1 \cdot \left(\beta + 2 \cdot i\right)\right)}}{\alpha}\right)}{2} \]

   5. Applied rewrites 90.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha} - \frac{-1 \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(2, i, \beta\right) + 2, -1 \cdot \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}\right)}}{2} \]

   if 0.0 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2

   1. Initial program 99.3%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}\right) - \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right) + i \cdot \left(\frac{-1}{2} \cdot \left(i \cdot \left(-1 \cdot \frac{{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{3} \cdot {\left(\alpha + \beta\right)}^{2}} + 4 \cdot \frac{\beta - \alpha}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)\right) + \frac{-1}{2} \cdot \frac{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right) \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)} \]

   5. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{\left(\mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right)\right)}^{2}}{{\left(\left(\beta + \alpha\right) + 2\right)}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), \mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right) \cdot \frac{\beta - \alpha}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right)} \]

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

   1. Initial program 3.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}\right) - \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right) + i \cdot \left(\frac{-1}{2} \cdot \left(i \cdot \left(-1 \cdot \frac{{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{3} \cdot {\left(\alpha + \beta\right)}^{2}} + 4 \cdot \frac{\beta - \alpha}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)\right) + \frac{-1}{2} \cdot \frac{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right) \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)} \]

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{\left(\mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right)\right)}^{2}}{{\left(\left(\beta + \alpha\right) + 2\right)}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), \mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right) \cdot \frac{\beta - \alpha}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right)} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{2 \cdot \beta + 2 \cdot \left(2 + \beta\right)}{\alpha} + \left(4 \cdot \frac{\beta}{\alpha} + 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\color{blue}{\alpha}}, \frac{1}{2} \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{2 \cdot \beta + 2 \cdot \left(2 + \beta\right)}{\alpha} + \left(4 \cdot \frac{\beta}{\alpha} + 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]

   8. Applied rewrites 28.3%

      \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\color{blue}{\alpha}}, 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 0:\\ \;\;\;\;\frac{-1 \cdot \left(0 \cdot \frac{\beta}{\alpha}\right) - -1 \cdot \frac{-1 \cdot \mathsf{fma}\left(-1, \mathsf{fma}\left(2, i, \beta\right) + 2, -1 \cdot \mathsf{fma}\left(2, i, \beta\right)\right)}{\alpha}}{2}\\ \mathbf{elif}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{\left(\mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right)\right)}^{2}}{{\left(\left(\beta + \alpha\right) + 2\right)}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), \mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right) \cdot \frac{\beta - \alpha}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := {t\_0}^{2} \cdot \left(\beta + \alpha\right)\\ t_2 := 0.5 \cdot \left(\left(\frac{\beta}{t\_0} + 1\right) - \frac{\alpha}{t\_0}\right)\\ t_3 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_4 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_3}}{t\_3 + 2} + 1}{2}\\ t_5 := \mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)\\ t_6 := 4 + \left(\beta + 2 \cdot \beta\right)\\ t_7 := \mathsf{fma}\left(2, t\_0, 2 \cdot \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(i, -0.5 \cdot \frac{\frac{\mathsf{fma}\left(-4, \alpha, \mathsf{fma}\left(-1, t\_5, \mathsf{fma}\left(4, \beta, 12 \cdot i\right)\right)\right)}{\alpha} - \frac{-4 \cdot t\_6}{\alpha}}{\alpha}, 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right)\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{t\_7}^{2}}{{t\_0}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{t\_1}\right), t\_7 \cdot \frac{\beta - \alpha}{t\_1}\right), t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{t\_5}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{t\_6}{\alpha}\right)}{\alpha}, t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0))
        (t_1 (* (pow t_0 2.0) (+ beta alpha)))
        (t_2 (* 0.5 (- (+ (/ beta t_0) 1.0) (/ alpha t_0))))
        (t_3 (+ (+ alpha beta) (* 2.0 i)))
        (t_4
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_3) (+ t_3 2.0)) 1.0)
          2.0))
        (t_5 (fma 2.0 beta (* 2.0 (+ 2.0 beta))))
        (t_6 (+ 4.0 (+ beta (* 2.0 beta))))
        (t_7 (fma 2.0 t_0 (* 2.0 (+ beta alpha)))))
   (if (<= t_4 0.0)
     (fma
      i
      (*
       -0.5
       (/
        (-
         (/ (fma -4.0 alpha (fma -1.0 t_5 (fma 4.0 beta (* 12.0 i)))) alpha)
         (/ (* -4.0 t_6) alpha))
        alpha))
      (* 0.5 (/ (+ 2.0 (* 2.0 beta)) alpha)))
     (if (<= t_4 2.0)
       (fma
        i
        (*
         -0.5
         (fma
          i
          (fma
           -1.0
           (*
            (/ (pow t_7 2.0) (pow t_0 3.0))
            (/ (- beta alpha) (pow (+ beta alpha) 2.0)))
           (/ (* 4.0 (- beta alpha)) t_1))
          (* t_7 (/ (- beta alpha) t_1))))
        t_2)
       (fma
        i
        (*
         -0.5
         (/
          (-
           (fma
            -1.0
            (/ t_5 alpha)
            (fma 4.0 (/ beta alpha) (* 12.0 (/ i alpha))))
           (+ 4.0 (* -4.0 (/ t_6 alpha))))
          alpha))
        t_2)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = pow(t_0, 2.0) * (beta + alpha);
	double t_2 = 0.5 * (((beta / t_0) + 1.0) - (alpha / t_0));
	double t_3 = (alpha + beta) + (2.0 * i);
	double t_4 = (((((alpha + beta) * (beta - alpha)) / t_3) / (t_3 + 2.0)) + 1.0) / 2.0;
	double t_5 = fma(2.0, beta, (2.0 * (2.0 + beta)));
	double t_6 = 4.0 + (beta + (2.0 * beta));
	double t_7 = fma(2.0, t_0, (2.0 * (beta + alpha)));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = fma(i, (-0.5 * (((fma(-4.0, alpha, fma(-1.0, t_5, fma(4.0, beta, (12.0 * i)))) / alpha) - ((-4.0 * t_6) / alpha)) / alpha)), (0.5 * ((2.0 + (2.0 * beta)) / alpha)));
	} else if (t_4 <= 2.0) {
		tmp = fma(i, (-0.5 * fma(i, fma(-1.0, ((pow(t_7, 2.0) / pow(t_0, 3.0)) * ((beta - alpha) / pow((beta + alpha), 2.0))), ((4.0 * (beta - alpha)) / t_1)), (t_7 * ((beta - alpha) / t_1)))), t_2);
	} else {
		tmp = fma(i, (-0.5 * ((fma(-1.0, (t_5 / alpha), fma(4.0, (beta / alpha), (12.0 * (i / alpha)))) - (4.0 + (-4.0 * (t_6 / alpha)))) / alpha)), t_2);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64((t_0 ^ 2.0) * Float64(beta + alpha))
	t_2 = Float64(0.5 * Float64(Float64(Float64(beta / t_0) + 1.0) - Float64(alpha / t_0)))
	t_3 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_4 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_3) / Float64(t_3 + 2.0)) + 1.0) / 2.0)
	t_5 = fma(2.0, beta, Float64(2.0 * Float64(2.0 + beta)))
	t_6 = Float64(4.0 + Float64(beta + Float64(2.0 * beta)))
	t_7 = fma(2.0, t_0, Float64(2.0 * Float64(beta + alpha)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = fma(i, Float64(-0.5 * Float64(Float64(Float64(fma(-4.0, alpha, fma(-1.0, t_5, fma(4.0, beta, Float64(12.0 * i)))) / alpha) - Float64(Float64(-4.0 * t_6) / alpha)) / alpha)), Float64(0.5 * Float64(Float64(2.0 + Float64(2.0 * beta)) / alpha)));
	elseif (t_4 <= 2.0)
		tmp = fma(i, Float64(-0.5 * fma(i, fma(-1.0, Float64(Float64((t_7 ^ 2.0) / (t_0 ^ 3.0)) * Float64(Float64(beta - alpha) / (Float64(beta + alpha) ^ 2.0))), Float64(Float64(4.0 * Float64(beta - alpha)) / t_1)), Float64(t_7 * Float64(Float64(beta - alpha) / t_1)))), t_2);
	else
		tmp = fma(i, Float64(-0.5 * Float64(Float64(fma(-1.0, Float64(t_5 / alpha), fma(4.0, Float64(beta / alpha), Float64(12.0 * Float64(i / alpha)))) - Float64(4.0 + Float64(-4.0 * Float64(t_6 / alpha)))) / alpha)), t_2);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[(N[(N[(beta / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / N[(t$95$3 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * beta + N[(2.0 * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(4.0 + N[(beta + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(2.0 * t$95$0 + N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(i * N[(-0.5 * N[(N[(N[(N[(-4.0 * alpha + N[(-1.0 * t$95$5 + N[(4.0 * beta + N[(12.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(-4.0 * t$95$6), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(i * N[(-0.5 * N[(i * N[(-1.0 * N[(N[(N[Power[t$95$7, 2.0], $MachinePrecision] / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$7 * N[(N[(beta - alpha), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(i * N[(-0.5 * N[(N[(N[(-1.0 * N[(t$95$5 / alpha), $MachinePrecision] + N[(4.0 * N[(beta / alpha), $MachinePrecision] + N[(12.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 + N[(-4.0 * N[(t$95$6 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.0

   1. Initial program 1.8%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}\right) - \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right) + i \cdot \left(\frac{-1}{2} \cdot \left(i \cdot \left(-1 \cdot \frac{{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{3} \cdot {\left(\alpha + \beta\right)}^{2}} + 4 \cdot \frac{\beta - \alpha}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)\right) + \frac{-1}{2} \cdot \frac{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right) \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)} \]

   5. Applied rewrites 1.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{\left(\mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right)\right)}^{2}}{{\left(\left(\beta + \alpha\right) + 2\right)}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), \mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right) \cdot \frac{\beta - \alpha}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right)} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{2 \cdot \beta + 2 \cdot \left(2 + \beta\right)}{\alpha} + \left(4 \cdot \frac{\beta}{\alpha} + 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\color{blue}{\alpha}}, \frac{1}{2} \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{2 \cdot \beta + 2 \cdot \left(2 + \beta\right)}{\alpha} + \left(4 \cdot \frac{\beta}{\alpha} + 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]

   8. Applied rewrites 26.7%

      \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\color{blue}{\alpha}}, 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]

   9. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

       2. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

       3. lift-*.f64 89.8

          \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

   11. Applied rewrites 89.8%

       \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

   12. Taylor expanded in alpha around 0

       \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\frac{\left(-4 \cdot \alpha + \left(-1 \cdot \left(2 \cdot \beta + 2 \cdot \left(2 + \beta\right)\right) + \left(4 \cdot \beta + 12 \cdot i\right)\right)\right) - -4 \cdot \left(4 + \left(\beta + 2 \cdot \beta\right)\right)}{\alpha}}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]
   13. Step-by-step derivation

       1. div-sub N/A

          \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\frac{-4 \cdot \alpha + \left(-1 \cdot \left(2 \cdot \beta + 2 \cdot \left(2 + \beta\right)\right) + \left(4 \cdot \beta + 12 \cdot i\right)\right)}{\alpha} - \frac{-4 \cdot \left(4 + \left(\beta + 2 \cdot \beta\right)\right)}{\alpha}}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

       2. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\frac{-4 \cdot \alpha + \left(-1 \cdot \left(2 \cdot \beta + 2 \cdot \left(2 + \beta\right)\right) + \left(4 \cdot \beta + 12 \cdot i\right)\right)}{\alpha} - \frac{-4 \cdot \left(4 + \left(\beta + 2 \cdot \beta\right)\right)}{\alpha}}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

   14. Applied rewrites 89.8%

       \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\frac{\mathsf{fma}\left(-4, \alpha, \mathsf{fma}\left(-1, \mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right), \mathsf{fma}\left(4, \beta, 12 \cdot i\right)\right)\right)}{\alpha} - \frac{-4 \cdot \left(4 + \left(\beta + 2 \cdot \beta\right)\right)}{\alpha}}{\alpha}, 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

   if 0.0 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 2

   1. Initial program 99.3%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}\right) - \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right) + i \cdot \left(\frac{-1}{2} \cdot \left(i \cdot \left(-1 \cdot \frac{{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{3} \cdot {\left(\alpha + \beta\right)}^{2}} + 4 \cdot \frac{\beta - \alpha}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)\right) + \frac{-1}{2} \cdot \frac{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right) \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)} \]

   5. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{\left(\mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right)\right)}^{2}}{{\left(\left(\beta + \alpha\right) + 2\right)}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), \mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right) \cdot \frac{\beta - \alpha}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right)} \]

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

   1. Initial program 3.1%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}\right) - \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right) + i \cdot \left(\frac{-1}{2} \cdot \left(i \cdot \left(-1 \cdot \frac{{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{3} \cdot {\left(\alpha + \beta\right)}^{2}} + 4 \cdot \frac{\beta - \alpha}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)\right) + \frac{-1}{2} \cdot \frac{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right) \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)} \]

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{\left(\mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right)\right)}^{2}}{{\left(\left(\beta + \alpha\right) + 2\right)}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), \mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right) \cdot \frac{\beta - \alpha}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right)} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{2 \cdot \beta + 2 \cdot \left(2 + \beta\right)}{\alpha} + \left(4 \cdot \frac{\beta}{\alpha} + 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\color{blue}{\alpha}}, \frac{1}{2} \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{2 \cdot \beta + 2 \cdot \left(2 + \beta\right)}{\alpha} + \left(4 \cdot \frac{\beta}{\alpha} + 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]

   8. Applied rewrites 28.3%

      \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\color{blue}{\alpha}}, 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 30.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + 2\\ t_1 := 4 + \left(\beta + 2 \cdot \beta\right)\\ t_2 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_3 := \mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_2}}{t\_2 + 2} + 1}{2} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(i, -0.5 \cdot \frac{\frac{\mathsf{fma}\left(-4, \alpha, \mathsf{fma}\left(-1, t\_3, \mathsf{fma}\left(4, \beta, 12 \cdot i\right)\right)\right)}{\alpha} - \frac{-4 \cdot t\_1}{\alpha}}{\alpha}, 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{t\_3}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{t\_1}{\alpha}\right)}{\alpha}, 0.5 \cdot \left(\left(\frac{\beta}{t\_0} + 1\right) - \frac{\alpha}{t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) 2.0))
        (t_1 (+ 4.0 (+ beta (* 2.0 beta))))
        (t_2 (+ (+ alpha beta) (* 2.0 i)))
        (t_3 (fma 2.0 beta (* 2.0 (+ 2.0 beta)))))
   (if (<=
        (/
         (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_2) (+ t_2 2.0)) 1.0)
         2.0)
        0.0)
     (fma
      i
      (*
       -0.5
       (/
        (-
         (/ (fma -4.0 alpha (fma -1.0 t_3 (fma 4.0 beta (* 12.0 i)))) alpha)
         (/ (* -4.0 t_1) alpha))
        alpha))
      (* 0.5 (/ (+ 2.0 (* 2.0 beta)) alpha)))
     (fma
      i
      (*
       -0.5
       (/
        (-
         (fma -1.0 (/ t_3 alpha) (fma 4.0 (/ beta alpha) (* 12.0 (/ i alpha))))
         (+ 4.0 (* -4.0 (/ t_1 alpha))))
        alpha))
      (* 0.5 (- (+ (/ beta t_0) 1.0) (/ alpha t_0)))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + 2.0;
	double t_1 = 4.0 + (beta + (2.0 * beta));
	double t_2 = (alpha + beta) + (2.0 * i);
	double t_3 = fma(2.0, beta, (2.0 * (2.0 + beta)));
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_2) / (t_2 + 2.0)) + 1.0) / 2.0) <= 0.0) {
		tmp = fma(i, (-0.5 * (((fma(-4.0, alpha, fma(-1.0, t_3, fma(4.0, beta, (12.0 * i)))) / alpha) - ((-4.0 * t_1) / alpha)) / alpha)), (0.5 * ((2.0 + (2.0 * beta)) / alpha)));
	} else {
		tmp = fma(i, (-0.5 * ((fma(-1.0, (t_3 / alpha), fma(4.0, (beta / alpha), (12.0 * (i / alpha)))) - (4.0 + (-4.0 * (t_1 / alpha)))) / alpha)), (0.5 * (((beta / t_0) + 1.0) - (alpha / t_0))));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + 2.0)
	t_1 = Float64(4.0 + Float64(beta + Float64(2.0 * beta)))
	t_2 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_3 = fma(2.0, beta, Float64(2.0 * Float64(2.0 + beta)))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_2) / Float64(t_2 + 2.0)) + 1.0) / 2.0) <= 0.0)
		tmp = fma(i, Float64(-0.5 * Float64(Float64(Float64(fma(-4.0, alpha, fma(-1.0, t_3, fma(4.0, beta, Float64(12.0 * i)))) / alpha) - Float64(Float64(-4.0 * t_1) / alpha)) / alpha)), Float64(0.5 * Float64(Float64(2.0 + Float64(2.0 * beta)) / alpha)));
	else
		tmp = fma(i, Float64(-0.5 * Float64(Float64(fma(-1.0, Float64(t_3 / alpha), fma(4.0, Float64(beta / alpha), Float64(12.0 * Float64(i / alpha)))) - Float64(4.0 + Float64(-4.0 * Float64(t_1 / alpha)))) / alpha)), Float64(0.5 * Float64(Float64(Float64(beta / t_0) + 1.0) - Float64(alpha / t_0))));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 + N[(beta + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * beta + N[(2.0 * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.0], N[(i * N[(-0.5 * N[(N[(N[(N[(-4.0 * alpha + N[(-1.0 * t$95$3 + N[(4.0 * beta + N[(12.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(-4.0 * t$95$1), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(i * N[(-0.5 * N[(N[(N[(-1.0 * N[(t$95$3 / alpha), $MachinePrecision] + N[(4.0 * N[(beta / alpha), $MachinePrecision] + N[(12.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 + N[(-4.0 * N[(t$95$1 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(N[(beta / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] - N[(alpha / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.0

   1. Initial program 1.8%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}\right) - \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right) + i \cdot \left(\frac{-1}{2} \cdot \left(i \cdot \left(-1 \cdot \frac{{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{3} \cdot {\left(\alpha + \beta\right)}^{2}} + 4 \cdot \frac{\beta - \alpha}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)\right) + \frac{-1}{2} \cdot \frac{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right) \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)} \]

   5. Applied rewrites 1.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{\left(\mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right)\right)}^{2}}{{\left(\left(\beta + \alpha\right) + 2\right)}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), \mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right) \cdot \frac{\beta - \alpha}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right)} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{2 \cdot \beta + 2 \cdot \left(2 + \beta\right)}{\alpha} + \left(4 \cdot \frac{\beta}{\alpha} + 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\color{blue}{\alpha}}, \frac{1}{2} \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{2 \cdot \beta + 2 \cdot \left(2 + \beta\right)}{\alpha} + \left(4 \cdot \frac{\beta}{\alpha} + 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]

   8. Applied rewrites 26.7%

      \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\color{blue}{\alpha}}, 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]

   9. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

       2. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

       3. lift-*.f64 89.8

          \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

   11. Applied rewrites 89.8%

       \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

   12. Taylor expanded in alpha around 0

       \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\frac{\left(-4 \cdot \alpha + \left(-1 \cdot \left(2 \cdot \beta + 2 \cdot \left(2 + \beta\right)\right) + \left(4 \cdot \beta + 12 \cdot i\right)\right)\right) - -4 \cdot \left(4 + \left(\beta + 2 \cdot \beta\right)\right)}{\alpha}}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]
   13. Step-by-step derivation

       1. div-sub N/A

          \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\frac{-4 \cdot \alpha + \left(-1 \cdot \left(2 \cdot \beta + 2 \cdot \left(2 + \beta\right)\right) + \left(4 \cdot \beta + 12 \cdot i\right)\right)}{\alpha} - \frac{-4 \cdot \left(4 + \left(\beta + 2 \cdot \beta\right)\right)}{\alpha}}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

       2. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\frac{-4 \cdot \alpha + \left(-1 \cdot \left(2 \cdot \beta + 2 \cdot \left(2 + \beta\right)\right) + \left(4 \cdot \beta + 12 \cdot i\right)\right)}{\alpha} - \frac{-4 \cdot \left(4 + \left(\beta + 2 \cdot \beta\right)\right)}{\alpha}}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

   14. Applied rewrites 89.8%

       \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\frac{\mathsf{fma}\left(-4, \alpha, \mathsf{fma}\left(-1, \mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right), \mathsf{fma}\left(4, \beta, 12 \cdot i\right)\right)\right)}{\alpha} - \frac{-4 \cdot \left(4 + \left(\beta + 2 \cdot \beta\right)\right)}{\alpha}}{\alpha}, 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

   if 0.0 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64))

   1. Initial program 79.3%

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

   3. Taylor expanded in i around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}\right) - \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right) + i \cdot \left(\frac{-1}{2} \cdot \left(i \cdot \left(-1 \cdot \frac{{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{3} \cdot {\left(\alpha + \beta\right)}^{2}} + 4 \cdot \frac{\beta - \alpha}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)\right) + \frac{-1}{2} \cdot \frac{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right) \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)} \]

   5. Applied rewrites 29.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{\left(\mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right)\right)}^{2}}{{\left(\left(\beta + \alpha\right) + 2\right)}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), \mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right) \cdot \frac{\beta - \alpha}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right)} \]

   6. Taylor expanded in alpha around inf

      \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{2 \cdot \beta + 2 \cdot \left(2 + \beta\right)}{\alpha} + \left(4 \cdot \frac{\beta}{\alpha} + 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\color{blue}{\alpha}}, \frac{1}{2} \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{2 \cdot \beta + 2 \cdot \left(2 + \beta\right)}{\alpha} + \left(4 \cdot \frac{\beta}{\alpha} + 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]

   8. Applied rewrites 13.7%

      \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\color{blue}{\alpha}}, 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 22.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(i, -0.5 \cdot \frac{\frac{\mathsf{fma}\left(-4, \alpha, \mathsf{fma}\left(-1, \mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right), \mathsf{fma}\left(4, \beta, 12 \cdot i\right)\right)\right)}{\alpha} - \frac{-4 \cdot \left(4 + \left(\beta + 2 \cdot \beta\right)\right)}{\alpha}}{\alpha}, 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (fma
  i
  (*
   -0.5
   (/
    (-
     (/
      (fma
       -4.0
       alpha
       (fma
        -1.0
        (fma 2.0 beta (* 2.0 (+ 2.0 beta)))
        (fma 4.0 beta (* 12.0 i))))
      alpha)
     (/ (* -4.0 (+ 4.0 (+ beta (* 2.0 beta)))) alpha))
    alpha))
  (* 0.5 (/ (+ 2.0 (* 2.0 beta)) alpha))))

C

double code(double alpha, double beta, double i) {
	return fma(i, (-0.5 * (((fma(-4.0, alpha, fma(-1.0, fma(2.0, beta, (2.0 * (2.0 + beta))), fma(4.0, beta, (12.0 * i)))) / alpha) - ((-4.0 * (4.0 + (beta + (2.0 * beta)))) / alpha)) / alpha)), (0.5 * ((2.0 + (2.0 * beta)) / alpha)));
}

Julia

function code(alpha, beta, i)
	return fma(i, Float64(-0.5 * Float64(Float64(Float64(fma(-4.0, alpha, fma(-1.0, fma(2.0, beta, Float64(2.0 * Float64(2.0 + beta))), fma(4.0, beta, Float64(12.0 * i)))) / alpha) - Float64(Float64(-4.0 * Float64(4.0 + Float64(beta + Float64(2.0 * beta)))) / alpha)) / alpha)), Float64(0.5 * Float64(Float64(2.0 + Float64(2.0 * beta)) / alpha)))
end

Wolfram

code[alpha_, beta_, i_] := N[(i * N[(-0.5 * N[(N[(N[(N[(-4.0 * alpha + N[(-1.0 * N[(2.0 * beta + N[(2.0 * N[(2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * beta + N[(12.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(-4.0 * N[(4.0 + N[(beta + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(2.0 + N[(2.0 * beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in i around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{\beta}{2 + \left(\alpha + \beta\right)}\right) - \frac{\alpha}{2 + \left(\alpha + \beta\right)}\right) + i \cdot \left(\frac{-1}{2} \cdot \left(i \cdot \left(-1 \cdot \frac{{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{3} \cdot {\left(\alpha + \beta\right)}^{2}} + 4 \cdot \frac{\beta - \alpha}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)\right) + \frac{-1}{2} \cdot \frac{\left(2 \cdot \left(2 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\alpha + \beta\right)\right) \cdot \left(\beta - \alpha\right)}{{\left(2 + \left(\alpha + \beta\right)\right)}^{2} \cdot \left(\alpha + \beta\right)}\right)} \]

5. Applied rewrites 22.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(i, -0.5 \cdot \mathsf{fma}\left(i, \mathsf{fma}\left(-1, \frac{{\left(\mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right)\right)}^{2}}{{\left(\left(\beta + \alpha\right) + 2\right)}^{3}} \cdot \frac{\beta - \alpha}{{\left(\beta + \alpha\right)}^{2}}, \frac{4 \cdot \left(\beta - \alpha\right)}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), \mathsf{fma}\left(2, \left(\beta + \alpha\right) + 2, 2 \cdot \left(\beta + \alpha\right)\right) \cdot \frac{\beta - \alpha}{{\left(\left(\beta + \alpha\right) + 2\right)}^{2} \cdot \left(\beta + \alpha\right)}\right), 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right)} \]

6. Taylor expanded in alpha around inf

   \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{2 \cdot \beta + 2 \cdot \left(2 + \beta\right)}{\alpha} + \left(4 \cdot \frac{\beta}{\alpha} + 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\color{blue}{\alpha}}, \frac{1}{2} \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\left(-1 \cdot \frac{2 \cdot \beta + 2 \cdot \left(2 + \beta\right)}{\alpha} + \left(4 \cdot \frac{\beta}{\alpha} + 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]

8. Applied rewrites 16.7%

   \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\color{blue}{\alpha}}, 0.5 \cdot \left(\left(\frac{\beta}{\left(\beta + \alpha\right) + 2} + 1\right) - \frac{\alpha}{\left(\beta + \alpha\right) + 2}\right)\right) \]

9. Taylor expanded in alpha around inf

   \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

    2. lower-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

    3. lift-*.f64 22.5

       \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

11. Applied rewrites 22.5%

    \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right)}{\alpha}, \mathsf{fma}\left(4, \frac{\beta}{\alpha}, 12 \cdot \frac{i}{\alpha}\right)\right) - \left(4 + -4 \cdot \frac{4 + \left(\beta + 2 \cdot \beta\right)}{\alpha}\right)}{\alpha}, 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

12. Taylor expanded in alpha around 0

    \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\frac{\left(-4 \cdot \alpha + \left(-1 \cdot \left(2 \cdot \beta + 2 \cdot \left(2 + \beta\right)\right) + \left(4 \cdot \beta + 12 \cdot i\right)\right)\right) - -4 \cdot \left(4 + \left(\beta + 2 \cdot \beta\right)\right)}{\alpha}}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]
13. Step-by-step derivation

    1. div-sub N/A

       \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\frac{-4 \cdot \alpha + \left(-1 \cdot \left(2 \cdot \beta + 2 \cdot \left(2 + \beta\right)\right) + \left(4 \cdot \beta + 12 \cdot i\right)\right)}{\alpha} - \frac{-4 \cdot \left(4 + \left(\beta + 2 \cdot \beta\right)\right)}{\alpha}}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

    2. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(i, \frac{-1}{2} \cdot \frac{\frac{-4 \cdot \alpha + \left(-1 \cdot \left(2 \cdot \beta + 2 \cdot \left(2 + \beta\right)\right) + \left(4 \cdot \beta + 12 \cdot i\right)\right)}{\alpha} - \frac{-4 \cdot \left(4 + \left(\beta + 2 \cdot \beta\right)\right)}{\alpha}}{\alpha}, \frac{1}{2} \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]

14. Applied rewrites 22.6%

    \[\leadsto \mathsf{fma}\left(i, -0.5 \cdot \frac{\frac{\mathsf{fma}\left(-4, \alpha, \mathsf{fma}\left(-1, \mathsf{fma}\left(2, \beta, 2 \cdot \left(2 + \beta\right)\right), \mathsf{fma}\left(4, \beta, 12 \cdot i\right)\right)\right)}{\alpha} - \frac{-4 \cdot \left(4 + \left(\beta + 2 \cdot \beta\right)\right)}{\alpha}}{\alpha}, 0.5 \cdot \frac{2 + 2 \cdot \beta}{\alpha}\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))