Octave 3.8, jcobi/2

Percentage Accurate: 63.5% → 96.8%

Time: 5.3s

Alternatives: 13

Speedup: 2.0×

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: 63.5% 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: 96.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 0.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 0.7

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 0.7

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

   3. Applied rewrites 0.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 0.7

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

   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 81.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 99.0%

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

Alternative 2: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := \left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)\\ t_2 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_3 := \frac{\frac{\frac{t\_1}{t\_2}}{t\_2 + 2} + 1}{2}\\ \mathbf{if}\;t\_3 \leq 0:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{elif}\;t\_3 \leq 0.9999999999999998:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(t\_0, 2, 4\right) \cdot t\_0} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{t\_0}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ alpha beta)))
        (t_1 (* (+ alpha beta) (- beta alpha)))
        (t_2 (+ (+ alpha beta) (* 2.0 i)))
        (t_3 (/ (+ (/ (/ t_1 t_2) (+ t_2 2.0)) 1.0) 2.0)))
   (if (<= t_3 0.0)
     (*
      0.5
      (/
       (- (+ beta (* -1.0 beta)) (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
       alpha))
     (if (<= t_3 0.9999999999999998)
       (+ (/ t_1 (* (fma t_0 2.0 4.0) t_0)) 0.5)
       (fma (/ (- beta alpha) t_0) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (alpha + beta));
	double t_1 = (alpha + beta) * (beta - alpha);
	double t_2 = (alpha + beta) + (2.0 * i);
	double t_3 = (((t_1 / t_2) / (t_2 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_3 <= 0.0) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else if (t_3 <= 0.9999999999999998) {
		tmp = (t_1 / (fma(t_0, 2.0, 4.0) * t_0)) + 0.5;
	} else {
		tmp = fma(((beta - alpha) / t_0), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(alpha + beta))
	t_1 = Float64(Float64(alpha + beta) * Float64(beta - alpha))
	t_2 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_3 = Float64(Float64(Float64(Float64(t_1 / t_2) / Float64(t_2 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_3 <= 0.0)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	elseif (t_3 <= 0.9999999999999998)
		tmp = Float64(Float64(t_1 / Float64(fma(t_0, 2.0, 4.0) * t_0)) + 0.5);
	else
		tmp = fma(Float64(Float64(beta - alpha) / t_0), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(t$95$1 / t$95$2), $MachinePrecision] / N[(t$95$2 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$3, 0.0], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9999999999999998], N[(N[(t$95$1 / N[(N[(t$95$0 * 2.0 + 4.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.5 + 0.5), $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. 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. associate-/l* N/A

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 0.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 0.7

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 0.7

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

   3. Applied rewrites 0.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 0.7

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

   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)) < 0.99999999999999978

   1. Initial program 98.6%

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

      1. lift-/.f64 N/A

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

      2. 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. div-add N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 98.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lift-fma.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 0.99999999999999978 < (/.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 33.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   5. Applied rewrites 50.7%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 98.7%

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

Alternative 3: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{t\_1 + 2} + 1}{2}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{elif}\;t\_2 \leq 0.9999999999999998:\\ \;\;\;\;\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(t\_0, 2, 4\right) \cdot t\_0} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{t\_0}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ alpha beta)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ t_1 2.0)) 1.0)
          2.0)))
   (if (<= t_2 0.0)
     (*
      0.5
      (/
       (- (+ beta (* -1.0 beta)) (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
       alpha))
     (if (<= t_2 0.9999999999999998)
       (+ (* (- beta alpha) (/ (+ alpha beta) (* (fma t_0 2.0 4.0) t_0))) 0.5)
       (fma (/ (- beta alpha) t_0) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (alpha + beta));
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else if (t_2 <= 0.9999999999999998) {
		tmp = ((beta - alpha) * ((alpha + beta) / (fma(t_0, 2.0, 4.0) * t_0))) + 0.5;
	} else {
		tmp = fma(((beta - alpha) / t_0), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(alpha + beta))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(t_1 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	elseif (t_2 <= 0.9999999999999998)
		tmp = Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / Float64(fma(t_0, 2.0, 4.0) * t_0))) + 0.5);
	else
		tmp = fma(Float64(Float64(beta - alpha) / t_0), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999999999999998], N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(N[(t$95$0 * 2.0 + 4.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.5 + 0.5), $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. 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. associate-/l* N/A

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 0.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 0.7

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 0.7

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

   3. Applied rewrites 0.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 0.7

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

   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)) < 0.99999999999999978

   1. Initial program 98.6%

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

      1. lift-/.f64 N/A

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

      2. 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. div-add N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 98.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 98.5%

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

   if 0.99999999999999978 < (/.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 33.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   5. Applied rewrites 50.7%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 98.7%

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

Alternative 4: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{t\_1 + 2} + 1}{2}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{elif}\;t\_2 \leq 0.9999999999999998:\\ \;\;\;\;\mathsf{fma}\left(\beta - \alpha, \frac{\alpha + \beta}{\mathsf{fma}\left(t\_0, 2, 4\right) \cdot t\_0}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{t\_0}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ alpha beta)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ t_1 2.0)) 1.0)
          2.0)))
   (if (<= t_2 0.0)
     (*
      0.5
      (/
       (- (+ beta (* -1.0 beta)) (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
       alpha))
     (if (<= t_2 0.9999999999999998)
       (fma (- beta alpha) (/ (+ alpha beta) (* (fma t_0 2.0 4.0) t_0)) 0.5)
       (fma (/ (- beta alpha) t_0) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (alpha + beta));
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else if (t_2 <= 0.9999999999999998) {
		tmp = fma((beta - alpha), ((alpha + beta) / (fma(t_0, 2.0, 4.0) * t_0)), 0.5);
	} else {
		tmp = fma(((beta - alpha) / t_0), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(alpha + beta))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(t_1 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	elseif (t_2 <= 0.9999999999999998)
		tmp = fma(Float64(beta - alpha), Float64(Float64(alpha + beta) / Float64(fma(t_0, 2.0, 4.0) * t_0)), 0.5);
	else
		tmp = fma(Float64(Float64(beta - alpha) / t_0), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.9999999999999998], N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(N[(t$95$0 * 2.0 + 4.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.5 + 0.5), $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. 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. associate-/l* N/A

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 0.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 0.7

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 0.7

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

   3. Applied rewrites 0.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 0.7

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

   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)) < 0.99999999999999978

   1. Initial program 98.6%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 98.6%

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.5%

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

   if 0.99999999999999978 < (/.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 33.9%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   5. Applied rewrites 50.7%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 98.7%

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

Alternative 5: 95.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_1}}{t\_1 + 2} + 1}{2}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{elif}\;t\_2 \leq 0.52:\\ \;\;\;\;-0.5 \cdot \frac{\alpha \cdot \alpha}{\left(2 + t\_0\right) \cdot t\_0} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ alpha (* 2.0 i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ t_1 2.0)) 1.0)
          2.0)))
   (if (<= t_2 0.0)
     (*
      0.5
      (/
       (- (+ beta (* -1.0 beta)) (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
       alpha))
     (if (<= t_2 0.52)
       (+ (* -0.5 (/ (* alpha alpha) (* (+ 2.0 t_0) t_0))) 0.5)
       (fma (/ (- beta alpha) (fma i 2.0 (+ alpha beta))) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = alpha + (2.0 * i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = (((((alpha + beta) * (beta - alpha)) / t_1) / (t_1 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else if (t_2 <= 0.52) {
		tmp = (-0.5 * ((alpha * alpha) / ((2.0 + t_0) * t_0))) + 0.5;
	} else {
		tmp = fma(((beta - alpha) / fma(i, 2.0, (alpha + beta))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(alpha + Float64(2.0 * i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / Float64(t_1 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	elseif (t_2 <= 0.52)
		tmp = Float64(Float64(-0.5 * Float64(Float64(alpha * alpha) / Float64(Float64(2.0 + t_0) * t_0))) + 0.5);
	else
		tmp = fma(Float64(Float64(beta - alpha) / fma(i, 2.0, Float64(alpha + beta))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(alpha + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.52], N[(N[(-0.5 * N[(N[(alpha * alpha), $MachinePrecision] / N[(N[(2.0 + t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $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. 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. associate-/l* N/A

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 0.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 0.7

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 0.7

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

   3. Applied rewrites 0.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 0.7

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

   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)) < 0.52000000000000002

   1. Initial program 98.6%

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

      1. lift-/.f64 N/A

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

      2. 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. div-add N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 98.6%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. flip3-+ N/A

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

      12. sum-cubes N/A

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-out-- N/A

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

      18. lift--.f64 N/A

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

      19. *-commutative N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in beta around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 97.6

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

   8. Applied rewrites 97.6%

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

   if 0.52000000000000002 < (/.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 37.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 96.8%

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

Alternative 6: 95.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 0.02:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.52:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1
         (/
          (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ t_0 2.0)) 1.0)
          2.0)))
   (if (<= t_1 0.02)
     (*
      0.5
      (/
       (- (+ beta (* -1.0 beta)) (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
       alpha))
     (if (<= t_1 0.52)
       0.5
       (fma (/ (- beta alpha) (fma i 2.0 (+ alpha beta))) 0.5 0.5)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = (((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0;
	double tmp;
	if (t_1 <= 0.02) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else if (t_1 <= 0.52) {
		tmp = 0.5;
	} else {
		tmp = fma(((beta - alpha) / fma(i, 2.0, (alpha + beta))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(t_0 + 2.0)) + 1.0) / 2.0)
	tmp = 0.0
	if (t_1 <= 0.02)
		tmp = Float64(0.5 * Float64(Float64(Float64(beta + Float64(-1.0 * beta)) - Float64(-1.0 * Float64(2.0 + fma(2.0, beta, Float64(4.0 * i))))) / alpha));
	elseif (t_1 <= 0.52)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(beta - alpha) / fma(i, 2.0, Float64(alpha + beta))), 0.5, 0.5);
	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[(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]}, If[LessEqual[t$95$1, 0.02], N[(0.5 * N[(N[(N[(beta + N[(-1.0 * beta), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(2.0 + N[(2.0 * beta + N[(4.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.52], 0.5, N[(N[(N[(beta - alpha), $MachinePrecision] / N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $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)) < 0.0200000000000000004

   1. Initial program 4.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l* N/A

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 2.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 2.9

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 2.9

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

   3. Applied rewrites 2.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 88.4

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

   8. Applied rewrites 88.4%

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

   if 0.0200000000000000004 < (/.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)) < 0.52000000000000002

   1. Initial program 100.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 67.6%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 98.3%

       \[\leadsto \color{blue}{0.5} \]

   if 0.52000000000000002 < (/.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 37.5%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   5. Applied rewrites 53.4%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 96.8%

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

Alternative 7: 96.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 0.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 0.7

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 0.7

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

   3. Applied rewrites 0.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 0.7

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

   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 81.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l* N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 99.0%

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

Alternative 8: 96.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 0.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 0.7

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 0.7

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

   3. Applied rewrites 0.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 0.7

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

   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 81.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 99.0%

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

   5. Applied rewrites 85.5%

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

   7. Applied rewrites 99.0%

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

Alternative 9: 96.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. difference-of-squares N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 0.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 0.7

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

      25. lower-+.f64 0.7

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

   3. Applied rewrites 0.7%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 0.7

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

   5. Applied rewrites 0.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

   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 81.0%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 99.0%

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

   5. Applied rewrites 85.5%

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

   7. Applied rewrites 99.0%

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

Alternative 10: 77.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0) <= 0.5d0) then
        tmp = 0.5d0
    else
        tmp = (0.5d0 * ((beta - alpha) / (2.0d0 + (alpha + beta)))) + 0.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 70.6%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 74.8%

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

   5. Applied rewrites 74.2%

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

   7. Applied rewrites 47.5%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 73.1%

       \[\leadsto \color{blue}{0.5} \]

   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))

   1. Initial program 40.3%

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

      1. lift-/.f64 N/A

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

      2. 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. div-add N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 40.3%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. flip3-+ N/A

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

      12. sum-cubes N/A

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-out-- N/A

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

      18. lift--.f64 N/A

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

      19. *-commutative N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

   5. Applied rewrites 38.3%

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

   6. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-+.f64 92.7

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

   8. Applied rewrites 92.7%

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

Alternative 11: 76.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0d0)) + 1.0d0) / 2.0d0) <= 0.6d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.6) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	tmp = 0
	if ((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.6:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	tmp = 0.0;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.6)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

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)) < 0.599999999999999978

   1. Initial program 71.1%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 75.2%

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 48.2%

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

   8. Taylor expanded in i around inf

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

   10. Applied rewrites 72.9%

       \[\leadsto \color{blue}{0.5} \]

   if 0.599999999999999978 < (/.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 37.4%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 100.0%

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

   5. Applied rewrites 53.3%

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

   7. Applied rewrites 35.3%

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

   8. Taylor expanded in beta around inf

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

   10. Applied rewrites 90.7%

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

Alternative 12: 79.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq 400000:\\ \;\;\;\;0.5 \cdot \frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 400000.0)
   (+ (* 0.5 (/ (- beta alpha) (+ 2.0 (+ alpha beta)))) 0.5)
   (fma (/ (- beta alpha) (fma i 2.0 (+ alpha beta))) 0.5 0.5)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 400000.0) {
		tmp = (0.5 * ((beta - alpha) / (2.0 + (alpha + beta)))) + 0.5;
	} else {
		tmp = fma(((beta - alpha) / fma(i, 2.0, (alpha + beta))), 0.5, 0.5);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (i <= 400000.0)
		tmp = Float64(Float64(0.5 * Float64(Float64(beta - alpha) / Float64(2.0 + Float64(alpha + beta)))) + 0.5);
	else
		tmp = fma(Float64(Float64(beta - alpha) / fma(i, 2.0, Float64(alpha + beta))), 0.5, 0.5);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[i, 400000.0], N[(N[(0.5 * N[(N[(beta - alpha), $MachinePrecision] / N[(2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(beta - alpha), $MachinePrecision] / N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < 4e5

   1. Initial program 59.8%

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

      1. lift-/.f64 N/A

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

      2. 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. div-add N/A

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

      4. lower-+.f64 N/A

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

   3. Applied rewrites 59.8%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. flip3-+ N/A

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

      12. sum-cubes N/A

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-out-- N/A

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

      18. lift--.f64 N/A

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

      19. *-commutative N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f64 N/A

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

      22. lift-*.f64 N/A

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

      23. lift-*.f64 N/A

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in i around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-+.f64 74.3

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

   8. Applied rewrites 74.3%

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

   if 4e5 < i

   1. Initial program 67.3%

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
   2. 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. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 87.1%

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

   5. Applied rewrites 77.2%

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 85.0%

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

Alternative 13: 62.1% accurate, 73.0× speedup

Math

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

FPCore

(FPCore (alpha beta i) :precision binary64 0.5)

C

double code(double alpha, double beta, double i) {
	return 0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.5d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.5;
}

Python

def code(alpha, beta, i):
	return 0.5

Julia

function code(alpha, beta, i)
	return 0.5
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.5;
end

Wolfram

code[alpha_, beta_, i_] := 0.5

Derivation

1. Initial program 63.5%

   \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
2. 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. associate-/l/ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lower-fma.f64 N/A

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

3. Applied rewrites 80.7%

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

5. Applied rewrites 69.8%

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

7. Applied rewrites 45.3%

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

8. Taylor expanded in i around inf

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

10. Applied rewrites 62.1%

    \[\leadsto \color{blue}{0.5} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(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))