Octave 3.8, jcobi/4

Percentage Accurate: 16.5% → 99.7%

Time: 7.1s

Alternatives: 15

Speedup: 75.4×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, 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) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

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

Initial Program: 16.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, 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) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + i\\ t_2 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \frac{\frac{t\_1}{2 + \frac{\beta + \alpha}{i}} \cdot \mathsf{fma}\left(\frac{t\_1}{t\_0 - -1}, \frac{i}{t\_0 - 1}, \frac{\beta}{t\_2 - -1} \cdot \frac{\alpha}{t\_2 - 1}\right)}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha)))
        (t_1 (+ (+ beta alpha) i))
        (t_2 (fma 2.0 i (+ alpha beta))))
   (/
    (*
     (/ t_1 (+ 2.0 (/ (+ beta alpha) i)))
     (fma
      (/ t_1 (- t_0 -1.0))
      (/ i (- t_0 1.0))
      (* (/ beta (- t_2 -1.0)) (/ alpha (- t_2 1.0)))))
    t_0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double t_1 = (beta + alpha) + i;
	double t_2 = fma(2.0, i, (alpha + beta));
	return ((t_1 / (2.0 + ((beta + alpha) / i))) * fma((t_1 / (t_0 - -1.0)), (i / (t_0 - 1.0)), ((beta / (t_2 - -1.0)) * (alpha / (t_2 - 1.0))))) / t_0;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + i)
	t_2 = fma(2.0, i, Float64(alpha + beta))
	return Float64(Float64(Float64(t_1 / Float64(2.0 + Float64(Float64(beta + alpha) / i))) * fma(Float64(t_1 / Float64(t_0 - -1.0)), Float64(i / Float64(t_0 - 1.0)), Float64(Float64(beta / Float64(t_2 - -1.0)) * Float64(alpha / Float64(t_2 - 1.0))))) / t_0)
end

Wolfram

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

Derivation

1. Initial program 16.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

3. Applied rewrites 24.9%

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

5. Applied rewrites 37.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. div-add N/A

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

   5. lift-fma.f64 N/A

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

   6. difference-of-sqr--1 N/A

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

   7. times-frac N/A

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

   8. lower-fma.f64 N/A

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

7. Applied rewrites 97.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-fma.f64 N/A

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

   5. difference-of-sqr--1 N/A

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

   6. metadata-eval N/A

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

   7. sub-flip N/A

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

   8. lift--.f64 N/A

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

   9. lift--.f64 N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-/.f64 99.7

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

9. Applied rewrites 99.7%

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

Alternative 2: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + i\\ t_2 := \frac{t\_1}{t\_0 - -1}\\ t_3 := \frac{t\_1}{2 + \frac{\beta + \alpha}{i}}\\ \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+148}:\\ \;\;\;\;\frac{t\_3 \cdot \mathsf{fma}\left(t\_2, \frac{1}{2 + \frac{\left(\alpha + \beta\right) - 1}{i}}, \beta \cdot \frac{\alpha}{\mathsf{fma}\left(t\_0, t\_0, -1\right)}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3 \cdot \mathsf{fma}\left(t\_2, \frac{i}{t\_0 - 1}, \frac{\alpha}{\beta}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha)))
        (t_1 (+ (+ beta alpha) i))
        (t_2 (/ t_1 (- t_0 -1.0)))
        (t_3 (/ t_1 (+ 2.0 (/ (+ beta alpha) i)))))
   (if (<= beta 1.4e+148)
     (/
      (*
       t_3
       (fma
        t_2
        (/ 1.0 (+ 2.0 (/ (- (+ alpha beta) 1.0) i)))
        (* beta (/ alpha (fma t_0 t_0 -1.0)))))
      t_0)
     (/ (* t_3 (fma t_2 (/ i (- t_0 1.0)) (/ alpha beta))) t_0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double t_1 = (beta + alpha) + i;
	double t_2 = t_1 / (t_0 - -1.0);
	double t_3 = t_1 / (2.0 + ((beta + alpha) / i));
	double tmp;
	if (beta <= 1.4e+148) {
		tmp = (t_3 * fma(t_2, (1.0 / (2.0 + (((alpha + beta) - 1.0) / i))), (beta * (alpha / fma(t_0, t_0, -1.0))))) / t_0;
	} else {
		tmp = (t_3 * fma(t_2, (i / (t_0 - 1.0)), (alpha / beta))) / t_0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + i)
	t_2 = Float64(t_1 / Float64(t_0 - -1.0))
	t_3 = Float64(t_1 / Float64(2.0 + Float64(Float64(beta + alpha) / i)))
	tmp = 0.0
	if (beta <= 1.4e+148)
		tmp = Float64(Float64(t_3 * fma(t_2, Float64(1.0 / Float64(2.0 + Float64(Float64(Float64(alpha + beta) - 1.0) / i))), Float64(beta * Float64(alpha / fma(t_0, t_0, -1.0))))) / t_0);
	else
		tmp = Float64(Float64(t_3 * fma(t_2, Float64(i / Float64(t_0 - 1.0)), Float64(alpha / beta))) / t_0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(2.0 + N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.4e+148], N[(N[(t$95$3 * N[(t$95$2 * N[(1.0 / N[(2.0 + N[(N[(N[(alpha + beta), $MachinePrecision] - 1.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(beta * N[(alpha / N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(t$95$3 * N[(t$95$2 * N[(i / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3999999999999999e148

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   5. Applied rewrites 37.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. lift-fma.f64 N/A

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

      6. difference-of-sqr--1 N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 97.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. associate--l+ N/A

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

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

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 97.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 97.6

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

   9. Applied rewrites 97.6%

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

   if 1.3999999999999999e148 < beta

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   5. Applied rewrites 37.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. lift-fma.f64 N/A

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

      6. difference-of-sqr--1 N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 50.4

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

   10. Applied rewrites 50.4%

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

Alternative 3: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_1 := \frac{i}{t\_0 - 1}\\ t_2 := \left(\beta + \alpha\right) + i\\ t_3 := \frac{t\_2}{t\_0 - -1}\\ t_4 := \frac{t\_2}{2 + \frac{\beta + \alpha}{i}}\\ \mathbf{if}\;\beta \leq 10^{+110}:\\ \;\;\;\;\frac{t\_4 \cdot \mathsf{fma}\left(t\_3, t\_1, \beta \cdot \frac{\alpha}{\mathsf{fma}\left(t\_0, t\_0, -1\right)}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4 \cdot \mathsf{fma}\left(t\_3, t\_1, \frac{\alpha}{\beta}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha)))
        (t_1 (/ i (- t_0 1.0)))
        (t_2 (+ (+ beta alpha) i))
        (t_3 (/ t_2 (- t_0 -1.0)))
        (t_4 (/ t_2 (+ 2.0 (/ (+ beta alpha) i)))))
   (if (<= beta 1e+110)
     (/ (* t_4 (fma t_3 t_1 (* beta (/ alpha (fma t_0 t_0 -1.0))))) t_0)
     (/ (* t_4 (fma t_3 t_1 (/ alpha beta))) t_0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double t_1 = i / (t_0 - 1.0);
	double t_2 = (beta + alpha) + i;
	double t_3 = t_2 / (t_0 - -1.0);
	double t_4 = t_2 / (2.0 + ((beta + alpha) / i));
	double tmp;
	if (beta <= 1e+110) {
		tmp = (t_4 * fma(t_3, t_1, (beta * (alpha / fma(t_0, t_0, -1.0))))) / t_0;
	} else {
		tmp = (t_4 * fma(t_3, t_1, (alpha / beta))) / t_0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	t_1 = Float64(i / Float64(t_0 - 1.0))
	t_2 = Float64(Float64(beta + alpha) + i)
	t_3 = Float64(t_2 / Float64(t_0 - -1.0))
	t_4 = Float64(t_2 / Float64(2.0 + Float64(Float64(beta + alpha) / i)))
	tmp = 0.0
	if (beta <= 1e+110)
		tmp = Float64(Float64(t_4 * fma(t_3, t_1, Float64(beta * Float64(alpha / fma(t_0, t_0, -1.0))))) / t_0);
	else
		tmp = Float64(Float64(t_4 * fma(t_3, t_1, Float64(alpha / beta))) / t_0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / N[(2.0 + N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+110], N[(N[(t$95$4 * N[(t$95$3 * t$95$1 + N[(beta * N[(alpha / N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(t$95$4 * N[(t$95$3 * t$95$1 + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1e110

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   5. Applied rewrites 37.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. lift-fma.f64 N/A

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

      6. difference-of-sqr--1 N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 97.6%

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

   if 1e110 < beta

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   5. Applied rewrites 37.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. lift-fma.f64 N/A

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

      6. difference-of-sqr--1 N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 50.4

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

   10. Applied rewrites 50.4%

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

Alternative 4: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + i\\ \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{\beta + i}{2 + \frac{\beta}{i}} \cdot \mathsf{fma}\left(\frac{\beta + i}{\mathsf{fma}\left(2, i, \beta\right) - -1}, \frac{i}{\mathsf{fma}\left(2, i, \beta\right) - 1}, \beta \cdot \frac{\alpha}{\mathsf{fma}\left(\mathsf{fma}\left(2, i, \beta\right), \mathsf{fma}\left(2, i, \beta\right), -1\right)}\right)}{\mathsf{fma}\left(2, i, \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{2 + \frac{\beta + \alpha}{i}} \cdot \mathsf{fma}\left(\frac{t\_1}{t\_0 - -1}, \frac{i}{t\_0 - 1}, \frac{\alpha}{\beta}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))) (t_1 (+ (+ beta alpha) i)))
   (if (<= beta 1.2e+72)
     (/
      (*
       (/ (+ beta i) (+ 2.0 (/ beta i)))
       (fma
        (/ (+ beta i) (- (fma 2.0 i beta) -1.0))
        (/ i (- (fma 2.0 i beta) 1.0))
        (* beta (/ alpha (fma (fma 2.0 i beta) (fma 2.0 i beta) -1.0)))))
      (fma 2.0 i beta))
     (/
      (*
       (/ t_1 (+ 2.0 (/ (+ beta alpha) i)))
       (fma (/ t_1 (- t_0 -1.0)) (/ i (- t_0 1.0)) (/ alpha beta)))
      t_0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double t_1 = (beta + alpha) + i;
	double tmp;
	if (beta <= 1.2e+72) {
		tmp = (((beta + i) / (2.0 + (beta / i))) * fma(((beta + i) / (fma(2.0, i, beta) - -1.0)), (i / (fma(2.0, i, beta) - 1.0)), (beta * (alpha / fma(fma(2.0, i, beta), fma(2.0, i, beta), -1.0))))) / fma(2.0, i, beta);
	} else {
		tmp = ((t_1 / (2.0 + ((beta + alpha) / i))) * fma((t_1 / (t_0 - -1.0)), (i / (t_0 - 1.0)), (alpha / beta))) / t_0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + i)
	tmp = 0.0
	if (beta <= 1.2e+72)
		tmp = Float64(Float64(Float64(Float64(beta + i) / Float64(2.0 + Float64(beta / i))) * fma(Float64(Float64(beta + i) / Float64(fma(2.0, i, beta) - -1.0)), Float64(i / Float64(fma(2.0, i, beta) - 1.0)), Float64(beta * Float64(alpha / fma(fma(2.0, i, beta), fma(2.0, i, beta), -1.0))))) / fma(2.0, i, beta));
	else
		tmp = Float64(Float64(Float64(t_1 / Float64(2.0 + Float64(Float64(beta + alpha) / i))) * fma(Float64(t_1 / Float64(t_0 - -1.0)), Float64(i / Float64(t_0 - 1.0)), Float64(alpha / beta))) / t_0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, If[LessEqual[beta, 1.2e+72], N[(N[(N[(N[(beta + i), $MachinePrecision] / N[(2.0 + N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(beta + i), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(2.0 * i + beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + N[(beta * N[(alpha / N[(N[(2.0 * i + beta), $MachinePrecision] * N[(2.0 * i + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 / N[(2.0 + N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.20000000000000005e72

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   5. Applied rewrites 37.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. lift-fma.f64 N/A

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

      6. difference-of-sqr--1 N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in alpha around 0

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

   10. Applied rewrites 88.1%

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

   11. Taylor expanded in alpha around 0

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

   13. Applied rewrites 89.3%

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

   14. Taylor expanded in alpha around 0

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

   16. Applied rewrites 88.3%

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

   17. Taylor expanded in alpha around 0

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

   19. Applied rewrites 89.2%

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

   20. Taylor expanded in alpha around 0

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

   22. Applied rewrites 84.9%

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

   23. Taylor expanded in alpha around 0

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

   25. Applied rewrites 84.8%

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

   26. Taylor expanded in alpha around 0

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

   28. Applied rewrites 84.7%

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

   29. Taylor expanded in alpha around 0

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

   31. Applied rewrites 84.7%

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

   if 1.20000000000000005e72 < beta

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   5. Applied rewrites 37.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. lift-fma.f64 N/A

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

      6. difference-of-sqr--1 N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 50.4

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

   10. Applied rewrites 50.4%

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

Alternative 5: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_1 := \left(\beta + \alpha\right) + i\\ \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+32}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_1}{2 + \frac{\beta + \alpha}{i}} \cdot \mathsf{fma}\left(\frac{t\_1}{t\_0 - -1}, \frac{i}{t\_0 - 1}, \frac{\alpha}{\beta}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))) (t_1 (+ (+ beta alpha) i)))
   (if (<= beta 6.5e+32)
     0.0625
     (/
      (*
       (/ t_1 (+ 2.0 (/ (+ beta alpha) i)))
       (fma (/ t_1 (- t_0 -1.0)) (/ i (- t_0 1.0)) (/ alpha beta)))
      t_0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double t_1 = (beta + alpha) + i;
	double tmp;
	if (beta <= 6.5e+32) {
		tmp = 0.0625;
	} else {
		tmp = ((t_1 / (2.0 + ((beta + alpha) / i))) * fma((t_1 / (t_0 - -1.0)), (i / (t_0 - 1.0)), (alpha / beta))) / t_0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	t_1 = Float64(Float64(beta + alpha) + i)
	tmp = 0.0
	if (beta <= 6.5e+32)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(t_1 / Float64(2.0 + Float64(Float64(beta + alpha) / i))) * fma(Float64(t_1 / Float64(t_0 - -1.0)), Float64(i / Float64(t_0 - 1.0)), Float64(alpha / beta))) / t_0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, If[LessEqual[beta, 6.5e+32], 0.0625, N[(N[(N[(t$95$1 / N[(2.0 + N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] + N[(alpha / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4999999999999994e32

   1. Initial program 16.5%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   3. Step-by-step derivation

   4. Applied rewrites 70.6%

      \[\leadsto \color{blue}{0.0625} \]

   if 6.4999999999999994e32 < beta

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   5. Applied rewrites 37.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. lift-fma.f64 N/A

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

      6. difference-of-sqr--1 N/A

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

      7. times-frac N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in beta around inf

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

      1. lower-/.f64 50.4

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

   10. Applied rewrites 50.4%

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

Alternative 6: 80.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ t_1 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_2 := \left(\beta + \alpha\right) + i\\ t_3 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_4 := t\_3 \cdot t\_3\\ t_5 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\frac{\frac{t\_1 \cdot \left(\beta \cdot \alpha + t\_1\right)}{t\_4}}{t\_4 - 1} \leq \infty:\\ \;\;\;\;\frac{\frac{t\_2}{2 + \frac{\beta + \alpha}{i}} \cdot \frac{\mathsf{fma}\left(t\_2, i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(t\_5, t\_5, -1\right)}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + t\_0\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.125 (/ beta i)))
        (t_1 (* i (+ (+ alpha beta) i)))
        (t_2 (+ (+ beta alpha) i))
        (t_3 (+ (+ alpha beta) (* 2.0 i)))
        (t_4 (* t_3 t_3))
        (t_5 (fma 2.0 i (+ beta alpha))))
   (if (<= (/ (/ (* t_1 (+ (* beta alpha) t_1)) t_4) (- t_4 1.0)) INFINITY)
     (/
      (*
       (/ t_2 (+ 2.0 (/ (+ beta alpha) i)))
       (/ (fma t_2 i (* beta alpha)) (fma t_5 t_5 -1.0)))
      t_5)
     (- (+ 0.0625 t_0) t_0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double t_1 = i * ((alpha + beta) + i);
	double t_2 = (beta + alpha) + i;
	double t_3 = (alpha + beta) + (2.0 * i);
	double t_4 = t_3 * t_3;
	double t_5 = fma(2.0, i, (beta + alpha));
	double tmp;
	if ((((t_1 * ((beta * alpha) + t_1)) / t_4) / (t_4 - 1.0)) <= ((double) INFINITY)) {
		tmp = ((t_2 / (2.0 + ((beta + alpha) / i))) * (fma(t_2, i, (beta * alpha)) / fma(t_5, t_5, -1.0))) / t_5;
	} else {
		tmp = (0.0625 + t_0) - t_0;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	t_1 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_2 = Float64(Float64(beta + alpha) + i)
	t_3 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_4 = Float64(t_3 * t_3)
	t_5 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (Float64(Float64(Float64(t_1 * Float64(Float64(beta * alpha) + t_1)) / t_4) / Float64(t_4 - 1.0)) <= Inf)
		tmp = Float64(Float64(Float64(t_2 / Float64(2.0 + Float64(Float64(beta + alpha) / i))) * Float64(fma(t_2, i, Float64(beta * alpha)) / fma(t_5, t_5, -1.0))) / t_5);
	else
		tmp = Float64(Float64(0.0625 + t_0) - t_0);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$3 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$1 * N[(N[(beta * alpha), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] / N[(t$95$4 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(t$95$2 / N[(2.0 + N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / N[(t$95$5 * t$95$5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], N[(N[(0.0625 + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64))) < +inf.0

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   5. Applied rewrites 37.0%

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

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i)) (+.f64 (+.f64 alpha beta) (*.f64 #s(literal 2 binary64) i))) #s(literal 1 binary64)))

   1. Initial program 16.5%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 77.4

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in alpha around 0

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      2. lower-/.f64 73.1

         \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied rewrites 73.1%

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\beta}{i} \]
   9. Step-by-step derivation

   10. Applied rewrites 74.3%

       \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \mathbf{if}\;\beta \leq 10^{+231}:\\ \;\;\;\;\left(0.0625 + t\_0\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.125 (/ beta i))))
   (if (<= beta 1e+231)
     (- (+ 0.0625 t_0) t_0)
     (/ (* (+ i alpha) (/ i beta)) (fma 2.0 i (+ beta alpha))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double tmp;
	if (beta <= 1e+231) {
		tmp = (0.0625 + t_0) - t_0;
	} else {
		tmp = ((i + alpha) * (i / beta)) / fma(2.0, i, (beta + alpha));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	tmp = 0.0
	if (beta <= 1e+231)
		tmp = Float64(Float64(0.0625 + t_0) - t_0);
	else
		tmp = Float64(Float64(Float64(i + alpha) * Float64(i / beta)) / fma(2.0, i, Float64(beta + alpha)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+231], N[(N[(0.0625 + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.0000000000000001e231

   1. Initial program 16.5%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 77.4

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in alpha around 0

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      2. lower-/.f64 73.1

         \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied rewrites 73.1%

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\beta}{i} \]
   9. Step-by-step derivation

   10. Applied rewrites 74.3%

       \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i} \]

   if 1.0000000000000001e231 < beta

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 12.8

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

   6. Applied rewrites 12.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 16.9

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

   8. Applied rewrites 16.9%

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

Alternative 8: 77.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \mathbf{if}\;\beta \leq 10^{+231}:\\ \;\;\;\;\left(0.0625 + t\_0\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \alpha}{\beta}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.125 (/ beta i))))
   (if (<= beta 1e+231)
     (- (+ 0.0625 t_0) t_0)
     (/ (* i (/ (+ i alpha) beta)) (fma 2.0 i (+ beta alpha))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	double tmp;
	if (beta <= 1e+231) {
		tmp = (0.0625 + t_0) - t_0;
	} else {
		tmp = (i * ((i + alpha) / beta)) / fma(2.0, i, (beta + alpha));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	tmp = 0.0
	if (beta <= 1e+231)
		tmp = Float64(Float64(0.0625 + t_0) - t_0);
	else
		tmp = Float64(Float64(i * Float64(Float64(i + alpha) / beta)) / fma(2.0, i, Float64(beta + alpha)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+231], N[(N[(0.0625 + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(i * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.0000000000000001e231

   1. Initial program 16.5%

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

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 77.4

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

   5. Taylor expanded in alpha around 0

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

      2. lower-/.f64 73.1

         \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied rewrites 73.1%

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   8. Taylor expanded in alpha around 0

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\beta}{i} \]
   9. Step-by-step derivation

   10. Applied rewrites 74.3%

       \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i} \]

   if 1.0000000000000001e231 < beta

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 12.8

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

   6. Applied rewrites 12.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 16.9

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 16.9

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

   8. Applied rewrites 16.9%

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

Alternative 9: 74.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \left(0.0625 + t\_0\right) - t\_0 \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.125 (/ beta i)))) (- (+ 0.0625 t_0) t_0)))

C

double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	return (0.0625 + t_0) - t_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 = 0.125d0 * (beta / i)
    code = (0.0625d0 + t_0) - t_0
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = 0.125 * (beta / i);
	return (0.0625 + t_0) - t_0;
}

Python

def code(alpha, beta, i):
	t_0 = 0.125 * (beta / i)
	return (0.0625 + t_0) - t_0

Julia

function code(alpha, beta, i)
	t_0 = Float64(0.125 * Float64(beta / i))
	return Float64(Float64(0.0625 + t_0) - t_0)
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = 0.125 * (beta / i);
	tmp = (0.0625 + t_0) - t_0;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, N[(N[(0.0625 + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision]]

Derivation

1. Initial program 16.5%

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

2. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8} \cdot \frac{\alpha + \beta}{i}} \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \color{blue}{\frac{1}{8}} \cdot \frac{\alpha + \beta}{i} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   4. lower-/.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 77.4

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

4. Applied rewrites 77.4%

   \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\mathsf{fma}\left(2, \alpha, 2 \cdot \beta\right)}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}} \]

5. Taylor expanded in alpha around 0

   \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\alpha + \beta}{i} \]

   2. lower-/.f64 73.1

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

7. Applied rewrites 73.1%

   \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \]

8. Taylor expanded in alpha around 0

   \[\leadsto \left(\frac{1}{16} + \frac{1}{8} \cdot \frac{\beta}{i}\right) - \frac{1}{8} \cdot \frac{\beta}{i} \]
9. Step-by-step derivation

10. Applied rewrites 74.3%

    \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i} \]
11. Add Preprocessing

Alternative 10: 72.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+244}:\\ \;\;\;\;\frac{0.125 \cdot i}{\mathsf{fma}\left(i, 2, \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\alpha + i\right)}{\beta}}{\mathsf{fma}\left(2, i, \beta\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.35e+244)
   (/ (* 0.125 i) (fma i 2.0 alpha))
   (/ (/ (* i (+ alpha i)) beta) (fma 2.0 i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.35e+244) {
		tmp = (0.125 * i) / fma(i, 2.0, alpha);
	} else {
		tmp = ((i * (alpha + i)) / beta) / fma(2.0, i, beta);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.35e+244)
		tmp = Float64(Float64(0.125 * i) / fma(i, 2.0, alpha));
	else
		tmp = Float64(Float64(Float64(i * Float64(alpha + i)) / beta) / fma(2.0, i, beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.35e+244], N[(N[(0.125 * i), $MachinePrecision] / N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.34999999999999999e244

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   4. Taylor expanded in i around inf

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

      1. lower-*.f64 70.6

         \[\leadsto \frac{0.125 \cdot \color{blue}{i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \]

   6. Applied rewrites 70.6%

      \[\leadsto \frac{\color{blue}{0.125 \cdot i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \]

   7. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{1}{8} \cdot i}{\color{blue}{\alpha + 2 \cdot i}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + \color{blue}{2 \cdot i}} \]

      2. lower-*.f64 70.6

         \[\leadsto \frac{0.125 \cdot i}{\alpha + 2 \cdot \color{blue}{i}} \]

   9. Applied rewrites 70.6%

      \[\leadsto \frac{0.125 \cdot i}{\color{blue}{\alpha + 2 \cdot i}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + 2 \cdot \color{blue}{i}} \]

       2. lift-+.f64 N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + \color{blue}{2 \cdot i}} \]

       3. +-commutative N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{2 \cdot i + \color{blue}{\alpha}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{i \cdot 2 + \alpha} \]

       5. lower-fma.f64 70.6

          \[\leadsto \frac{0.125 \cdot i}{\mathsf{fma}\left(i, \color{blue}{2}, \alpha\right)} \]

   11. Applied rewrites 70.6%

       \[\leadsto \frac{0.125 \cdot i}{\mathsf{fma}\left(i, \color{blue}{2}, \alpha\right)} \]

   if 1.34999999999999999e244 < beta

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 12.8

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

   6. Applied rewrites 12.8%

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

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 12.8%

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

Alternative 11: 72.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+244}:\\ \;\;\;\;\frac{0.125 \cdot i}{\mathsf{fma}\left(i, 2, \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\beta}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.5e+244)
   (/ (* 0.125 i) (fma i 2.0 alpha))
   (/ (/ (* i i) beta) (fma 2.0 i (+ beta alpha)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.5e+244) {
		tmp = (0.125 * i) / fma(i, 2.0, alpha);
	} else {
		tmp = ((i * i) / beta) / fma(2.0, i, (beta + alpha));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.5e+244)
		tmp = Float64(Float64(0.125 * i) / fma(i, 2.0, alpha));
	else
		tmp = Float64(Float64(Float64(i * i) / beta) / fma(2.0, i, Float64(beta + alpha)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.5e+244], N[(N[(0.125 * i), $MachinePrecision] / N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * i), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.4999999999999999e244

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   4. Taylor expanded in i around inf

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

      1. lower-*.f64 70.6

         \[\leadsto \frac{0.125 \cdot \color{blue}{i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \]

   6. Applied rewrites 70.6%

      \[\leadsto \frac{\color{blue}{0.125 \cdot i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \]

   7. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{1}{8} \cdot i}{\color{blue}{\alpha + 2 \cdot i}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + \color{blue}{2 \cdot i}} \]

      2. lower-*.f64 70.6

         \[\leadsto \frac{0.125 \cdot i}{\alpha + 2 \cdot \color{blue}{i}} \]

   9. Applied rewrites 70.6%

      \[\leadsto \frac{0.125 \cdot i}{\color{blue}{\alpha + 2 \cdot i}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + 2 \cdot \color{blue}{i}} \]

       2. lift-+.f64 N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + \color{blue}{2 \cdot i}} \]

       3. +-commutative N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{2 \cdot i + \color{blue}{\alpha}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{i \cdot 2 + \alpha} \]

       5. lower-fma.f64 70.6

          \[\leadsto \frac{0.125 \cdot i}{\mathsf{fma}\left(i, \color{blue}{2}, \alpha\right)} \]

   11. Applied rewrites 70.6%

       \[\leadsto \frac{0.125 \cdot i}{\mathsf{fma}\left(i, \color{blue}{2}, \alpha\right)} \]

   if 1.4999999999999999e244 < beta

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 12.8

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

   6. Applied rewrites 12.8%

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

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 12.2%

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

Alternative 12: 72.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+245}:\\ \;\;\;\;\frac{0.125 \cdot i}{\mathsf{fma}\left(i, 2, \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha \cdot i}{\beta}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.55e+245)
   (/ (* 0.125 i) (fma i 2.0 alpha))
   (/ (/ (* alpha i) beta) (fma 2.0 i (+ beta alpha)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.55e+245) {
		tmp = (0.125 * i) / fma(i, 2.0, alpha);
	} else {
		tmp = ((alpha * i) / beta) / fma(2.0, i, (beta + alpha));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.55e+245)
		tmp = Float64(Float64(0.125 * i) / fma(i, 2.0, alpha));
	else
		tmp = Float64(Float64(Float64(alpha * i) / beta) / fma(2.0, i, Float64(beta + alpha)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.55e+245], N[(N[(0.125 * i), $MachinePrecision] / N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha * i), $MachinePrecision] / beta), $MachinePrecision] / N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5499999999999999e245

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   4. Taylor expanded in i around inf

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

      1. lower-*.f64 70.6

         \[\leadsto \frac{0.125 \cdot \color{blue}{i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \]

   6. Applied rewrites 70.6%

      \[\leadsto \frac{\color{blue}{0.125 \cdot i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \]

   7. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{1}{8} \cdot i}{\color{blue}{\alpha + 2 \cdot i}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + \color{blue}{2 \cdot i}} \]

      2. lower-*.f64 70.6

         \[\leadsto \frac{0.125 \cdot i}{\alpha + 2 \cdot \color{blue}{i}} \]

   9. Applied rewrites 70.6%

      \[\leadsto \frac{0.125 \cdot i}{\color{blue}{\alpha + 2 \cdot i}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + 2 \cdot \color{blue}{i}} \]

       2. lift-+.f64 N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + \color{blue}{2 \cdot i}} \]

       3. +-commutative N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{2 \cdot i + \color{blue}{\alpha}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{i \cdot 2 + \alpha} \]

       5. lower-fma.f64 70.6

          \[\leadsto \frac{0.125 \cdot i}{\mathsf{fma}\left(i, \color{blue}{2}, \alpha\right)} \]

   11. Applied rewrites 70.6%

       \[\leadsto \frac{0.125 \cdot i}{\mathsf{fma}\left(i, \color{blue}{2}, \alpha\right)} \]

   if 1.5499999999999999e245 < beta

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   4. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 12.8

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

   6. Applied rewrites 12.8%

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

   7. Taylor expanded in alpha around inf

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

      1. lower-*.f64 7.6

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

   9. Applied rewrites 7.6%

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

Alternative 13: 71.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+245}:\\ \;\;\;\;\frac{0.125 \cdot i}{\mathsf{fma}\left(i, 2, \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\beta \cdot \alpha\right) \cdot \frac{i}{\mathsf{fma}\left(\beta, \beta, -1\right) \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.55e+245)
   (/ (* 0.125 i) (fma i 2.0 alpha))
   (* (* beta alpha) (/ i (* (fma beta beta -1.0) beta)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.55e+245) {
		tmp = (0.125 * i) / fma(i, 2.0, alpha);
	} else {
		tmp = (beta * alpha) * (i / (fma(beta, beta, -1.0) * beta));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.55e+245)
		tmp = Float64(Float64(0.125 * i) / fma(i, 2.0, alpha));
	else
		tmp = Float64(Float64(beta * alpha) * Float64(i / Float64(fma(beta, beta, -1.0) * beta)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.55e+245], N[(N[(0.125 * i), $MachinePrecision] / N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision], N[(N[(beta * alpha), $MachinePrecision] * N[(i / N[(N[(beta * beta + -1.0), $MachinePrecision] * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.5499999999999999e245

   1. Initial program 16.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

   3. Applied rewrites 24.9%

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

   4. Taylor expanded in i around inf

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

      1. lower-*.f64 70.6

         \[\leadsto \frac{0.125 \cdot \color{blue}{i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \]

   6. Applied rewrites 70.6%

      \[\leadsto \frac{\color{blue}{0.125 \cdot i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \]

   7. Taylor expanded in beta around 0

      \[\leadsto \frac{\frac{1}{8} \cdot i}{\color{blue}{\alpha + 2 \cdot i}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + \color{blue}{2 \cdot i}} \]

      2. lower-*.f64 70.6

         \[\leadsto \frac{0.125 \cdot i}{\alpha + 2 \cdot \color{blue}{i}} \]

   9. Applied rewrites 70.6%

      \[\leadsto \frac{0.125 \cdot i}{\color{blue}{\alpha + 2 \cdot i}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + 2 \cdot \color{blue}{i}} \]

       2. lift-+.f64 N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + \color{blue}{2 \cdot i}} \]

       3. +-commutative N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{2 \cdot i + \color{blue}{\alpha}} \]

       4. *-commutative N/A

          \[\leadsto \frac{\frac{1}{8} \cdot i}{i \cdot 2 + \alpha} \]

       5. lower-fma.f64 70.6

          \[\leadsto \frac{0.125 \cdot i}{\mathsf{fma}\left(i, \color{blue}{2}, \alpha\right)} \]

   11. Applied rewrites 70.6%

       \[\leadsto \frac{0.125 \cdot i}{\mathsf{fma}\left(i, \color{blue}{2}, \alpha\right)} \]

   if 1.5499999999999999e245 < beta

   1. Initial program 16.5%

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

   2. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-+.f64 7.3

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

   4. Applied rewrites 7.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 9.2

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 9.2

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

      13. lift--.f64 N/A

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

      14. sub-flip N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 9.2

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-+.f64 9.2

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lift-+.f64 9.2

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

      25. lift-+.f64 N/A

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

      26. +-commutative N/A

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

      27. lift-+.f64 9.2

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

   6. Applied rewrites 9.2%

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

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 8.5%

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

   10. Taylor expanded in alpha around 0

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

   12. Applied rewrites 6.3%

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

   13. Taylor expanded in alpha around 0

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

   15. Applied rewrites 5.3%

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

Alternative 14: 70.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot i}{\mathsf{fma}\left(i, 2, \alpha\right)} \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 (/ (* 0.125 i) (fma i 2.0 alpha)))

C

double code(double alpha, double beta, double i) {
	return (0.125 * i) / fma(i, 2.0, alpha);
}

Julia

function code(alpha, beta, i)
	return Float64(Float64(0.125 * i) / fma(i, 2.0, alpha))
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(0.125 * i), $MachinePrecision] / N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

3. Applied rewrites 24.9%

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

4. Taylor expanded in i around inf

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

   1. lower-*.f64 70.6

      \[\leadsto \frac{0.125 \cdot \color{blue}{i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \]

6. Applied rewrites 70.6%

   \[\leadsto \frac{\color{blue}{0.125 \cdot i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right)} \]

7. Taylor expanded in beta around 0

   \[\leadsto \frac{\frac{1}{8} \cdot i}{\color{blue}{\alpha + 2 \cdot i}} \]
8. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + \color{blue}{2 \cdot i}} \]

   2. lower-*.f64 70.6

      \[\leadsto \frac{0.125 \cdot i}{\alpha + 2 \cdot \color{blue}{i}} \]

9. Applied rewrites 70.6%

   \[\leadsto \frac{0.125 \cdot i}{\color{blue}{\alpha + 2 \cdot i}} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + 2 \cdot \color{blue}{i}} \]

    2. lift-+.f64 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot i}{\alpha + \color{blue}{2 \cdot i}} \]

    3. +-commutative N/A

       \[\leadsto \frac{\frac{1}{8} \cdot i}{2 \cdot i + \color{blue}{\alpha}} \]

    4. *-commutative N/A

       \[\leadsto \frac{\frac{1}{8} \cdot i}{i \cdot 2 + \alpha} \]

    5. lower-fma.f64 70.6

       \[\leadsto \frac{0.125 \cdot i}{\mathsf{fma}\left(i, \color{blue}{2}, \alpha\right)} \]

11. Applied rewrites 70.6%

    \[\leadsto \frac{0.125 \cdot i}{\mathsf{fma}\left(i, \color{blue}{2}, \alpha\right)} \]
12. Add Preprocessing

Alternative 15: 70.6% accurate, 75.4× speedup

Math

\[\begin{array}{l} \\ 0.0625 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.0625)

C

double code(double alpha, double beta, double i) {
	return 0.0625;
}

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 = 0.0625d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.0625;
}

Python

def code(alpha, beta, i):
	return 0.0625

Julia

function code(alpha, beta, i)
	return 0.0625
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.0625;
end

Wolfram

code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 16.5%

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

2. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\frac{1}{16}} \]
3. Step-by-step derivation

4. Applied rewrites 70.6%

   \[\leadsto \color{blue}{0.0625} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))