Octave 3.8, jcobi/2

Percentage Accurate: 63.8% → 98.1%

Time: 5.9s

Alternatives: 14

Speedup: 0.9×

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: 63.8% 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: 98.1% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (- (- -2.0 (fma 2.0 i beta)) alpha))
        (t_1 (+ (+ alpha beta) (* 2.0 i))))
   (if (<=
        (/
         (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) (+ t_1 2.0)) 1.0)
         2.0)
        1e-16)
     (/
      (/ (- (fma -1.0 beta (* -1.0 (+ 2.0 (+ beta (* 2.0 i))))) (* 2.0 i)) t_0)
      2.0)
     (/
      (/
       (fma
        1.0
        t_0
        (* (+ alpha beta) (/ (- alpha beta) (fma 2.0 i (+ alpha beta)))))
       t_0)
      2.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (-2.0 - fma(2.0, i, 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) <= 1e-16) {
		tmp = ((fma(-1.0, beta, (-1.0 * (2.0 + (beta + (2.0 * i))))) - (2.0 * i)) / t_0) / 2.0;
	} else {
		tmp = (fma(1.0, t_0, ((alpha + beta) * ((alpha - beta) / fma(2.0, i, (alpha + beta))))) / t_0) / 2.0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(-2.0 - fma(2.0, i, 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) <= 1e-16)
		tmp = Float64(Float64(Float64(fma(-1.0, beta, Float64(-1.0 * Float64(2.0 + Float64(beta + Float64(2.0 * i))))) - Float64(2.0 * i)) / t_0) / 2.0);
	else
		tmp = Float64(Float64(fma(1.0, t_0, Float64(Float64(alpha + beta) * Float64(Float64(alpha - beta) / fma(2.0, i, Float64(alpha + beta))))) / t_0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(-2.0 - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] - alpha), $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], 1e-16], N[(N[(N[(N[(-1.0 * beta + N[(-1.0 * N[(2.0 + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 * t$95$0 + N[(N[(alpha + beta), $MachinePrecision] * N[(N[(alpha - beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$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)) < 9.9999999999999998e-17

   1. Initial program 63.8%

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

      1. lift-+.f64 N/A

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

      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 81.1%

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in alpha around inf

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 69.9

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

   8. Applied rewrites 69.9%

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

   if 9.9999999999999998e-17 < (/.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 63.8%

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

      1. lift-+.f64 N/A

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

      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 81.1%

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

   5. Applied rewrites 81.4%

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

Alternative 2: 98.0% accurate, 0.5× speedup

Math

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

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)
        1e-10)
     (/
      (/
       (- (fma -1.0 beta (* -1.0 (+ 2.0 (+ beta (* 2.0 i))))) (* 2.0 i))
       (- (- -2.0 (fma 2.0 i beta)) alpha))
      2.0)
     (/
      (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) <= 1e-10) {
		tmp = ((fma(-1.0, beta, (-1.0 * (2.0 + (beta + (2.0 * i))))) - (2.0 * i)) / ((-2.0 - fma(2.0, i, beta)) - alpha)) / 2.0;
	} 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) <= 1e-10)
		tmp = Float64(Float64(Float64(fma(-1.0, beta, Float64(-1.0 * Float64(2.0 + Float64(beta + Float64(2.0 * i))))) - Float64(2.0 * i)) / Float64(Float64(-2.0 - fma(2.0, i, beta)) - alpha)) / 2.0);
	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], 1e-10], N[(N[(N[(N[(-1.0 * beta + N[(-1.0 * N[(2.0 + N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 * i), $MachinePrecision]), $MachinePrecision] / N[(N[(-2.0 - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] - alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $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)) < 1.00000000000000004e-10

   1. Initial program 63.8%

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

      1. lift-+.f64 N/A

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

      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 81.1%

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in alpha around inf

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 69.9

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

   8. Applied rewrites 69.9%

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

   if 1.00000000000000004e-10 < (/.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 63.8%

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

      1. lift-+.f64 N/A

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

      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 81.1%

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

Alternative 3: 97.8% accurate, 0.5× speedup

Math

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

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)
        1e-10)
     (*
      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) <= 1e-10) {
		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) <= 1e-10)
		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], 1e-10], 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)) < 1.00000000000000004e-10

   1. Initial program 63.8%

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

   2. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}} \]
   3. 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 22.5

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

   4. Applied rewrites 22.5%

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

   if 1.00000000000000004e-10 < (/.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 63.8%

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

      1. lift-+.f64 N/A

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

      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 81.1%

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

Alternative 4: 96.8% accurate, 0.6× speedup

Math

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

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)
        1e-10)
     (*
      0.5
      (/
       (- (+ beta (* -1.0 beta)) (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
       alpha))
     (/
      (fma
       beta
       (/ (/ (- beta alpha) (fma i 2.0 beta)) (- (fma i 2.0 beta) -2.0))
       1.0)
      2.0))))

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) <= 1e-10) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else {
		tmp = fma(beta, (((beta - alpha) / fma(i, 2.0, beta)) / (fma(i, 2.0, beta) - -2.0)), 1.0) / 2.0;
	}
	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) <= 1e-10)
		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(beta, Float64(Float64(Float64(beta - alpha) / fma(i, 2.0, beta)) / Float64(fma(i, 2.0, beta) - -2.0)), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, 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], 1e-10], 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 * N[(N[(N[(beta - alpha), $MachinePrecision] / N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(i * 2.0 + beta), $MachinePrecision] - -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)) < 1.00000000000000004e-10

   1. Initial program 63.8%

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

   2. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}} \]
   3. 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 22.5

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

   4. Applied rewrites 22.5%

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

   if 1.00000000000000004e-10 < (/.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 63.8%

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

      1. lift-+.f64 N/A

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

      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 81.1%

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

   4. Taylor expanded in alpha around 0

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

   6. Applied rewrites 79.7%

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

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 80.3%

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

   10. Taylor expanded in alpha around 0

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

   12. Applied rewrites 79.4%

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

Alternative 5: 96.6% accurate, 0.6× speedup

Math

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

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)
        1e-10)
     (*
      0.5
      (/
       (- (+ beta (* -1.0 beta)) (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
       alpha))
     (/
      (fma beta (/ (/ beta (+ beta (* 2.0 i))) (- (fma i 2.0 beta) -2.0)) 1.0)
      2.0))))

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) <= 1e-10) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else {
		tmp = fma(beta, ((beta / (beta + (2.0 * i))) / (fma(i, 2.0, beta) - -2.0)), 1.0) / 2.0;
	}
	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) <= 1e-10)
		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(beta, Float64(Float64(beta / Float64(beta + Float64(2.0 * i))) / Float64(fma(i, 2.0, beta) - -2.0)), 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, 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], 1e-10], 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 * N[(N[(beta / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(i * 2.0 + beta), $MachinePrecision] - -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)) < 1.00000000000000004e-10

   1. Initial program 63.8%

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

   2. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}} \]
   3. 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 22.5

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

   4. Applied rewrites 22.5%

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

   if 1.00000000000000004e-10 < (/.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 63.8%

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

      1. lift-+.f64 N/A

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

      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 81.1%

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

   4. Taylor expanded in alpha around 0

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

   6. Applied rewrites 79.7%

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

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 80.3%

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

   10. Taylor expanded in alpha around 0

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

   12. Applied rewrites 79.4%

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

   13. Taylor expanded in alpha around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 79.7

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

   15. Applied rewrites 79.7%

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

Alternative 6: 94.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 10^{-10}:\\ \;\;\;\;0.5 \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \mathsf{fma}\left(2, \beta, 4 \cdot i\right)\right)}{\alpha}\\ \mathbf{elif}\;t\_1 \leq 0.501:\\ \;\;\;\;\mathsf{fma}\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta}{\mathsf{fma}\left(2, i, \beta\right)}}{\left(i + i\right) - -2}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{\alpha - \beta}{\left(-2 - \alpha\right) - \beta} - -1}}\\ \end{array} \end{array} \]

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 1e-10)
     (*
      0.5
      (/
       (- (+ beta (* -1.0 beta)) (* -1.0 (+ 2.0 (fma 2.0 beta (* 4.0 i)))))
       alpha))
     (if (<= t_1 0.501)
       (fma
        (* (+ alpha beta) (/ (/ beta (fma 2.0 i beta)) (- (+ i i) -2.0)))
        0.5
        0.5)
       (/ 1.0 (/ 2.0 (- (/ (- alpha beta) (- (- -2.0 alpha) beta)) -1.0)))))))

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 <= 1e-10) {
		tmp = 0.5 * (((beta + (-1.0 * beta)) - (-1.0 * (2.0 + fma(2.0, beta, (4.0 * i))))) / alpha);
	} else if (t_1 <= 0.501) {
		tmp = fma(((alpha + beta) * ((beta / fma(2.0, i, beta)) / ((i + i) - -2.0))), 0.5, 0.5);
	} else {
		tmp = 1.0 / (2.0 / (((alpha - beta) / ((-2.0 - alpha) - beta)) - -1.0));
	}
	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 <= 1e-10)
		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 <= 0.501)
		tmp = fma(Float64(Float64(alpha + beta) * Float64(Float64(beta / fma(2.0, i, beta)) / Float64(Float64(i + i) - -2.0))), 0.5, 0.5);
	else
		tmp = Float64(1.0 / Float64(2.0 / Float64(Float64(Float64(alpha - beta) / Float64(Float64(-2.0 - alpha) - beta)) - -1.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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, 1e-10], 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, 0.501], N[(N[(N[(alpha + beta), $MachinePrecision] * N[(N[(beta / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(i + i), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], N[(1.0 / N[(2.0 / N[(N[(N[(alpha - beta), $MachinePrecision] / N[(N[(-2.0 - alpha), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $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)) < 1.00000000000000004e-10

   1. Initial program 63.8%

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

   2. Taylor expanded in alpha around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\left(\beta + -1 \cdot \beta\right) - -1 \cdot \left(2 + \left(2 \cdot \beta + 4 \cdot i\right)\right)}{\alpha}} \]
   3. 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 22.5

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

   4. Applied rewrites 22.5%

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

   if 1.00000000000000004e-10 < (/.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)) < 0.501000000000000001

   1. Initial program 63.8%

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

      1. lift-+.f64 N/A

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

      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 81.1%

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

   4. Taylor expanded in i around inf

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

      1. lower-*.f64 49.7

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in i around inf

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

      1. lower-*.f64 49.5

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

   9. Applied rewrites 49.5%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 59.1

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

   12. Applied rewrites 59.1%

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

       1. lift-/.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. div-add N/A

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

       5. mult-flip N/A

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

       6. metadata-eval N/A

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

       7. metadata-eval N/A

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

   14. Applied rewrites 59.1%

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

   if 0.501000000000000001 < (/.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 63.8%

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

   2. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 67.8

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

   4. Applied rewrites 67.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-to-mult N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 69.6

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

   6. Applied rewrites 69.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 69.6

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

   8. Applied rewrites 67.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. associate--r+ N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 67.8

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

   10. Applied rewrites 67.8%

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

Alternative 7: 79.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{-2 \cdot \beta}{\left(-2 - \mathsf{fma}\left(2, i, \beta\right)\right) - \alpha}}{2}\\ \mathbf{elif}\;t\_1 \leq 0.501:\\ \;\;\;\;\mathsf{fma}\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta}{\mathsf{fma}\left(2, i, \beta\right)}}{\left(i + i\right) - -2}, 0.5, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{\alpha - \beta}{\left(-2 - \alpha\right) - \beta} - -1}}\\ \end{array} \end{array} \]

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 0.0)
     (/ (/ (* -2.0 beta) (- (- -2.0 (fma 2.0 i beta)) alpha)) 2.0)
     (if (<= t_1 0.501)
       (fma
        (* (+ alpha beta) (/ (/ beta (fma 2.0 i beta)) (- (+ i i) -2.0)))
        0.5
        0.5)
       (/ 1.0 (/ 2.0 (- (/ (- alpha beta) (- (- -2.0 alpha) beta)) -1.0)))))))

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 <= 0.0) {
		tmp = ((-2.0 * beta) / ((-2.0 - fma(2.0, i, beta)) - alpha)) / 2.0;
	} else if (t_1 <= 0.501) {
		tmp = fma(((alpha + beta) * ((beta / fma(2.0, i, beta)) / ((i + i) - -2.0))), 0.5, 0.5);
	} else {
		tmp = 1.0 / (2.0 / (((alpha - beta) / ((-2.0 - alpha) - beta)) - -1.0));
	}
	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 <= 0.0)
		tmp = Float64(Float64(Float64(-2.0 * beta) / Float64(Float64(-2.0 - fma(2.0, i, beta)) - alpha)) / 2.0);
	elseif (t_1 <= 0.501)
		tmp = fma(Float64(Float64(alpha + beta) * Float64(Float64(beta / fma(2.0, i, beta)) / Float64(Float64(i + i) - -2.0))), 0.5, 0.5);
	else
		tmp = Float64(1.0 / Float64(2.0 / Float64(Float64(Float64(alpha - beta) / Float64(Float64(-2.0 - alpha) - beta)) - -1.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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, 0.0], N[(N[(N[(-2.0 * beta), $MachinePrecision] / N[(N[(-2.0 - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] - alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$1, 0.501], N[(N[(N[(alpha + beta), $MachinePrecision] * N[(N[(beta / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / N[(N[(i + i), $MachinePrecision] - -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], N[(1.0 / N[(2.0 / N[(N[(N[(alpha - beta), $MachinePrecision] / N[(N[(-2.0 - alpha), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $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 63.8%

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

      1. lift-+.f64 N/A

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

      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 81.1%

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in beta around inf

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

      1. lower-*.f64 27.5

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

   8. Applied rewrites 27.5%

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

   if 0.0 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.501000000000000001

   1. Initial program 63.8%

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

      1. lift-+.f64 N/A

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

      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 81.1%

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

   4. Taylor expanded in i around inf

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

      1. lower-*.f64 49.7

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in i around inf

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

      1. lower-*.f64 49.5

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

   9. Applied rewrites 49.5%

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

   10. Taylor expanded in alpha around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 59.1

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

   12. Applied rewrites 59.1%

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

       1. lift-/.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. div-add N/A

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

       5. mult-flip N/A

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

       6. metadata-eval N/A

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

       7. metadata-eval N/A

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

   14. Applied rewrites 59.1%

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

   if 0.501000000000000001 < (/.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 63.8%

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

   2. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 67.8

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

   4. Applied rewrites 67.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-to-mult N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 69.6

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

   6. Applied rewrites 69.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 69.6

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

   8. Applied rewrites 67.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. associate--r+ N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 67.8

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

   10. Applied rewrites 67.8%

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

Alternative 8: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 + 2\\ t_2 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_1} + 1}{2}\\ \mathbf{if}\;t\_2 \leq 10^{-16}:\\ \;\;\;\;\frac{\frac{-2 \cdot \beta}{\left(-2 - \mathsf{fma}\left(2, i, \beta\right)\right) - \alpha}}{2}\\ \mathbf{elif}\;t\_2 \leq 0.5000000005:\\ \;\;\;\;\frac{\frac{-1 \cdot \alpha}{t\_1} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{\alpha - \beta}{\left(-2 - \alpha\right) - \beta} - -1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (+ t_0 2.0))
        (t_2
         (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) 1.0) 2.0)))
   (if (<= t_2 1e-16)
     (/ (/ (* -2.0 beta) (- (- -2.0 (fma 2.0 i beta)) alpha)) 2.0)
     (if (<= t_2 0.5000000005)
       (/ (+ (/ (* -1.0 alpha) t_1) 1.0) 2.0)
       (/ 1.0 (/ 2.0 (- (/ (- alpha beta) (- (- -2.0 alpha) beta)) -1.0)))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 + 2.0;
	double t_2 = (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) + 1.0) / 2.0;
	double tmp;
	if (t_2 <= 1e-16) {
		tmp = ((-2.0 * beta) / ((-2.0 - fma(2.0, i, beta)) - alpha)) / 2.0;
	} else if (t_2 <= 0.5000000005) {
		tmp = (((-1.0 * alpha) / t_1) + 1.0) / 2.0;
	} else {
		tmp = 1.0 / (2.0 / (((alpha - beta) / ((-2.0 - alpha) - beta)) - -1.0));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 + 2.0)
	t_2 = Float64(Float64(Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) + 1.0) / 2.0)
	tmp = 0.0
	if (t_2 <= 1e-16)
		tmp = Float64(Float64(Float64(-2.0 * beta) / Float64(Float64(-2.0 - fma(2.0, i, beta)) - alpha)) / 2.0);
	elseif (t_2 <= 0.5000000005)
		tmp = Float64(Float64(Float64(Float64(-1.0 * alpha) / t_1) + 1.0) / 2.0);
	else
		tmp = Float64(1.0 / Float64(2.0 / Float64(Float64(Float64(alpha - beta) / Float64(Float64(-2.0 - alpha) - beta)) - -1.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-16], N[(N[(N[(-2.0 * beta), $MachinePrecision] / N[(N[(-2.0 - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] - alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$2, 0.5000000005], N[(N[(N[(N[(-1.0 * alpha), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(1.0 / N[(2.0 / N[(N[(N[(alpha - beta), $MachinePrecision] / N[(N[(-2.0 - alpha), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $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)) < 9.9999999999999998e-17

   1. Initial program 63.8%

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

      1. lift-+.f64 N/A

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

      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 81.1%

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in beta around inf

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

      1. lower-*.f64 27.5

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

   8. Applied rewrites 27.5%

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

   if 9.9999999999999998e-17 < (/.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)) < 0.50000000050000004

   1. Initial program 63.8%

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

   2. Taylor expanded in alpha around inf

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

      1. lower-*.f64 62.6

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

   4. Applied rewrites 62.6%

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

   if 0.50000000050000004 < (/.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 63.8%

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

   2. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 67.8

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

   4. Applied rewrites 67.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-to-mult N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 69.6

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

   6. Applied rewrites 69.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 69.6

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

   8. Applied rewrites 67.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. associate--r+ N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 67.8

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

   10. Applied rewrites 67.8%

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

Alternative 9: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{-2 \cdot \beta}{\left(-2 - \mathsf{fma}\left(2, i, \beta\right)\right) - \alpha}}{2}\\ \mathbf{elif}\;t\_1 \leq 0.5000000005:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{\frac{\alpha - \beta}{\left(-2 - \alpha\right) - \beta} - -1}}\\ \end{array} \end{array} \]

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 0.0)
     (/ (/ (* -2.0 beta) (- (- -2.0 (fma 2.0 i beta)) alpha)) 2.0)
     (if (<= t_1 0.5000000005)
       0.5
       (/ 1.0 (/ 2.0 (- (/ (- alpha beta) (- (- -2.0 alpha) beta)) -1.0)))))))

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 <= 0.0) {
		tmp = ((-2.0 * beta) / ((-2.0 - fma(2.0, i, beta)) - alpha)) / 2.0;
	} else if (t_1 <= 0.5000000005) {
		tmp = 0.5;
	} else {
		tmp = 1.0 / (2.0 / (((alpha - beta) / ((-2.0 - alpha) - beta)) - -1.0));
	}
	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 <= 0.0)
		tmp = Float64(Float64(Float64(-2.0 * beta) / Float64(Float64(-2.0 - fma(2.0, i, beta)) - alpha)) / 2.0);
	elseif (t_1 <= 0.5000000005)
		tmp = 0.5;
	else
		tmp = Float64(1.0 / Float64(2.0 / Float64(Float64(Float64(alpha - beta) / Float64(Float64(-2.0 - alpha) - beta)) - -1.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(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, 0.0], N[(N[(N[(-2.0 * beta), $MachinePrecision] / N[(N[(-2.0 - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] - alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$1, 0.5000000005], 0.5, N[(1.0 / N[(2.0 / N[(N[(N[(alpha - beta), $MachinePrecision] / N[(N[(-2.0 - alpha), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $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 63.8%

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

      1. lift-+.f64 N/A

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

      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 81.1%

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in beta around inf

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

      1. lower-*.f64 27.5

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

   8. Applied rewrites 27.5%

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

   if 0.0 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.50000000050000004

   1. Initial program 63.8%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.9%

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

   if 0.50000000050000004 < (/.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 63.8%

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

   2. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 67.8

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

   4. Applied rewrites 67.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-to-mult N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 69.6

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

   6. Applied rewrites 69.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 69.6

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

      5. lift-+.f64 N/A

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

      6. add-flip N/A

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

   8. Applied rewrites 67.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. associate--r+ N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 67.8

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

   10. Applied rewrites 67.8%

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

Alternative 10: 78.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{-2 \cdot \beta}{\left(-2 - \mathsf{fma}\left(2, i, \beta\right)\right) - \alpha}}{2}\\ \mathbf{elif}\;t\_1 \leq 0.5000000005:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha - \beta}{\left(-2 - \alpha\right) - \beta}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

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 0.0)
     (/ (/ (* -2.0 beta) (- (- -2.0 (fma 2.0 i beta)) alpha)) 2.0)
     (if (<= t_1 0.5000000005)
       0.5
       (fma (/ (- alpha beta) (- (- -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 <= 0.0) {
		tmp = ((-2.0 * beta) / ((-2.0 - fma(2.0, i, beta)) - alpha)) / 2.0;
	} else if (t_1 <= 0.5000000005) {
		tmp = 0.5;
	} else {
		tmp = fma(((alpha - beta) / ((-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 <= 0.0)
		tmp = Float64(Float64(Float64(-2.0 * beta) / Float64(Float64(-2.0 - fma(2.0, i, beta)) - alpha)) / 2.0);
	elseif (t_1 <= 0.5000000005)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(alpha - beta) / Float64(Float64(-2.0 - 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, 0.0], N[(N[(N[(-2.0 * beta), $MachinePrecision] / N[(N[(-2.0 - N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] - alpha), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[t$95$1, 0.5000000005], 0.5, N[(N[(N[(alpha - beta), $MachinePrecision] / N[(N[(-2.0 - alpha), $MachinePrecision] - beta), $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 63.8%

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

      1. lift-+.f64 N/A

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

      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 81.1%

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in beta around inf

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

      1. lower-*.f64 27.5

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

   8. Applied rewrites 27.5%

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

   if 0.0 < (/.f64 (+.f64 (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) #s(literal 2 binary64))) #s(literal 1 binary64)) #s(literal 2 binary64)) < 0.50000000050000004

   1. Initial program 63.8%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.9%

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

   if 0.50000000050000004 < (/.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 63.8%

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

   2. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 67.8

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

   4. Applied rewrites 67.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 67.8

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

   6. Applied rewrites 67.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. associate--r+ N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 67.8

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

   8. Applied rewrites 67.8%

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

Alternative 11: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \leq 0.5000000005:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha - \beta}{\left(-2 - \alpha\right) - \beta}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

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)
        0.5000000005)
     0.5
     (fma (/ (- alpha beta) (- (- -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 tmp;
	if (((((((alpha + beta) * (beta - alpha)) / t_0) / (t_0 + 2.0)) + 1.0) / 2.0) <= 0.5000000005) {
		tmp = 0.5;
	} else {
		tmp = fma(((alpha - beta) / ((-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))
	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) <= 0.5000000005)
		tmp = 0.5;
	else
		tmp = fma(Float64(Float64(alpha - beta) / Float64(Float64(-2.0 - 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]}, 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], 0.5000000005], 0.5, N[(N[(N[(alpha - beta), $MachinePrecision] / N[(N[(-2.0 - alpha), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 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)) < 0.50000000050000004

   1. Initial program 63.8%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.9%

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

   if 0.50000000050000004 < (/.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 63.8%

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

   2. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 67.8

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

   4. Applied rewrites 67.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 67.8

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

   6. Applied rewrites 67.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. associate--r+ N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 67.8

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

   8. Applied rewrites 67.8%

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

Alternative 12: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t\_0}}{t\_0 + 2} + 1}{2} \leq 0.5000000005:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\beta}{2 + \beta}, 0.5, 0.5\right)\\ \end{array} \end{array} \]

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)
        0.5000000005)
     0.5
     (fma (/ beta (+ 2.0 beta)) 0.5 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) <= 0.5000000005) {
		tmp = 0.5;
	} else {
		tmp = fma((beta / (2.0 + beta)), 0.5, 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) <= 0.5000000005)
		tmp = 0.5;
	else
		tmp = fma(Float64(beta / Float64(2.0 + 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]}, 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], 0.5000000005], 0.5, N[(N[(beta / N[(2.0 + beta), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (/.f64 (/.f64 (*.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)) < 0.50000000050000004

   1. Initial program 63.8%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.9%

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

   if 0.50000000050000004 < (/.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 63.8%

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

   2. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 67.8

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

   4. Applied rewrites 67.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 67.8

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

   6. Applied rewrites 67.8%

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

   7. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 72.6

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

   9. Applied rewrites 72.6%

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

Alternative 13: 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 63.8%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 61.9%

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

   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 63.8%

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

   2. Taylor expanded in beta around inf

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

   4. Applied rewrites 32.8%

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

Alternative 14: 61.9% accurate, 41.7× 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 63.8%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 61.9%

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

Reproduce

herbie shell --seed 2025149 
(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))