Octave 3.8, jcobi/4

Percentage Accurate: 16.6% → 99.7%

Time: 11.4s

Alternatives: 12

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: 16.6% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.6%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. lift--.f64 N/A

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

3. Applied rewrites 38.3%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. div-add N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. associate-*r/ N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. div-add-rev N/A

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

   12. lower-/.f64 N/A

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

5. Applied rewrites 94.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.7

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 99.7

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

7. Applied rewrites 99.7%

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

Alternative 2: 91.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.6%

   \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
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. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. lift--.f64 N/A

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

3. Applied rewrites 38.3%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. div-add N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. associate-*r/ N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. div-add-rev N/A

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

   12. lower-/.f64 N/A

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

5. Applied rewrites 94.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.7

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in alpha around 0

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

10. Applied rewrites 88.2%

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

11. Taylor expanded in alpha around 0

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

13. Applied rewrites 90.7%

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

14. Taylor expanded in alpha around 0

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

16. Applied rewrites 89.5%

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

17. Taylor expanded in alpha around 0

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

19. Applied rewrites 90.5%

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

20. Taylor expanded in alpha around 0

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

22. Applied rewrites 85.4%

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

23. Taylor expanded in alpha around 0

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

25. Applied rewrites 85.4%

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

Alternative 3: 85.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\alpha \leq 1.15 \cdot 10^{+108}:\\ \;\;\;\;\frac{\beta + i}{\mathsf{fma}\left(2, i, \beta\right) - 1} \cdot \frac{i \cdot \frac{\mathsf{fma}\left(\frac{\beta + i}{\mathsf{fma}\left(2, i, \beta\right)}, i, \frac{\beta \cdot \alpha}{\mathsf{fma}\left(2, i, \beta\right)}\right)}{\mathsf{fma}\left(2, i, \beta\right)}}{\mathsf{fma}\left(2, i, \beta\right) - -1}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{i \cdot \frac{\mathsf{fma}\left(\frac{\left(\beta + \alpha\right) + i}{t\_0}, i, \frac{\alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \beta\right)}{t\_0}}{t\_0 - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))))
   (if (<= alpha 1.15e+108)
     (*
      (/ (+ beta i) (- (fma 2.0 i beta) 1.0))
      (/
       (*
        i
        (/
         (fma
          (/ (+ beta i) (fma 2.0 i beta))
          i
          (/ (* beta alpha) (fma 2.0 i beta)))
         (fma 2.0 i beta)))
       (- (fma 2.0 i beta) -1.0)))
     (*
      1.0
      (/
       (*
        i
        (/
         (fma
          (/ (+ (+ beta alpha) i) t_0)
          i
          (* (/ alpha (fma 2.0 i (+ alpha beta))) beta))
         t_0))
       (- t_0 -1.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double tmp;
	if (alpha <= 1.15e+108) {
		tmp = ((beta + i) / (fma(2.0, i, beta) - 1.0)) * ((i * (fma(((beta + i) / fma(2.0, i, beta)), i, ((beta * alpha) / fma(2.0, i, beta))) / fma(2.0, i, beta))) / (fma(2.0, i, beta) - -1.0));
	} else {
		tmp = 1.0 * ((i * (fma((((beta + alpha) + i) / t_0), i, ((alpha / fma(2.0, i, (alpha + beta))) * beta)) / t_0)) / (t_0 - -1.0));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (alpha <= 1.15e+108)
		tmp = Float64(Float64(Float64(beta + i) / Float64(fma(2.0, i, beta) - 1.0)) * Float64(Float64(i * Float64(fma(Float64(Float64(beta + i) / fma(2.0, i, beta)), i, Float64(Float64(beta * alpha) / fma(2.0, i, beta))) / fma(2.0, i, beta))) / Float64(fma(2.0, i, beta) - -1.0)));
	else
		tmp = Float64(1.0 * Float64(Float64(i * Float64(fma(Float64(Float64(Float64(beta + alpha) + i) / t_0), i, Float64(Float64(alpha / fma(2.0, i, Float64(alpha + beta))) * beta)) / t_0)) / Float64(t_0 - -1.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 1.15e+108], N[(N[(N[(beta + i), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i * N[(N[(N[(N[(beta + i), $MachinePrecision] / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] * i + N[(N[(beta * alpha), $MachinePrecision] / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(i * N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] / t$95$0), $MachinePrecision] * i + N[(N[(alpha / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.1499999999999999e108

   1. Initial program 16.6%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift--.f64 N/A

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

   3. Applied rewrites 38.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in alpha around 0

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

   8. Applied rewrites 84.7%

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

   9. Taylor expanded in alpha around 0

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

   11. Applied rewrites 86.3%

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

   12. Taylor expanded in alpha around 0

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

   14. Applied rewrites 85.2%

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

   15. Taylor expanded in alpha around 0

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

   17. Applied rewrites 86.1%

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

   18. Taylor expanded in alpha around 0

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

   20. Applied rewrites 84.9%

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

   21. Taylor expanded in alpha around 0

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

   23. Applied rewrites 82.2%

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

   24. Taylor expanded in alpha around 0

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

   26. Applied rewrites 82.3%

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

   if 1.1499999999999999e108 < alpha

   1. Initial program 16.6%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift--.f64 N/A

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

   3. Applied rewrites 38.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 41.7%

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

Alternative 4: 80.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i\\ t_1 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_2 := t\_1 - -1\\ \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{t\_0}{t\_1 - 1} \cdot \frac{0.25 \cdot i}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{i \cdot \frac{\mathsf{fma}\left(\frac{t\_0}{t\_1}, i, \frac{\alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right)} \cdot \beta\right)}{t\_1}}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) i))
        (t_1 (fma 2.0 i (+ beta alpha)))
        (t_2 (- t_1 -1.0)))
   (if (<= beta 4.5e+125)
     (* (/ t_0 (- t_1 1.0)) (/ (* 0.25 i) t_2))
     (*
      1.0
      (/
       (*
        i
        (/
         (fma (/ t_0 t_1) i (* (/ alpha (fma 2.0 i (+ alpha beta))) beta))
         t_1))
       t_2)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (beta + alpha) + i;
	double t_1 = fma(2.0, i, (beta + alpha));
	double t_2 = t_1 - -1.0;
	double tmp;
	if (beta <= 4.5e+125) {
		tmp = (t_0 / (t_1 - 1.0)) * ((0.25 * i) / t_2);
	} else {
		tmp = 1.0 * ((i * (fma((t_0 / t_1), i, ((alpha / fma(2.0, i, (alpha + beta))) * beta)) / t_1)) / t_2);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(beta + alpha) + i)
	t_1 = fma(2.0, i, Float64(beta + alpha))
	t_2 = Float64(t_1 - -1.0)
	tmp = 0.0
	if (beta <= 4.5e+125)
		tmp = Float64(Float64(t_0 / Float64(t_1 - 1.0)) * Float64(Float64(0.25 * i) / t_2));
	else
		tmp = Float64(1.0 * Float64(Float64(i * Float64(fma(Float64(t_0 / t_1), i, Float64(Float64(alpha / fma(2.0, i, Float64(alpha + beta))) * beta)) / t_1)) / t_2));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - -1.0), $MachinePrecision]}, If[LessEqual[beta, 4.5e+125], N[(N[(t$95$0 / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 * i), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(i * N[(N[(N[(t$95$0 / t$95$1), $MachinePrecision] * i + N[(N[(alpha / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * beta), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.5e125

   1. Initial program 16.6%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift--.f64 N/A

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

   3. Applied rewrites 38.3%

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

   4. Taylor expanded in i around inf

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

      1. lower-*.f64 71.6

         \[\leadsto \frac{\left(\beta + \alpha\right) + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \frac{0.25 \cdot \color{blue}{i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \]

   6. Applied rewrites 71.6%

      \[\leadsto \frac{\left(\beta + \alpha\right) + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \frac{\color{blue}{0.25 \cdot i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \]

   if 4.5e125 < beta

   1. Initial program 16.6%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift--.f64 N/A

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

   3. Applied rewrites 38.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in alpha around inf

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

   10. Applied rewrites 41.7%

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

Alternative 5: 80.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] * N[(2.0 * i + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(beta + i), $MachinePrecision] * i), $MachinePrecision] / N[(N[(2.0 * i + beta), $MachinePrecision] * N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.125 * N[(beta + alpha), $MachinePrecision] + N[(0.0625 * i + N[(N[(beta + alpha), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $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 16.6%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

   3. Applied rewrites 38.3%

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

   4. Taylor expanded in alpha around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 34.6

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

   6. Applied rewrites 34.6%

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

   7. Taylor expanded in alpha around 0

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

   9. Applied rewrites 35.0%

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

   10. Taylor expanded in alpha around 0

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

   12. Applied rewrites 34.8%

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

   13. Taylor expanded in alpha around 0

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

   15. Applied rewrites 36.6%

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

   16. Taylor expanded in alpha around 0

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

   18. Applied rewrites 35.0%

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

   19. Taylor expanded in alpha around 0

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

   21. Applied rewrites 33.5%

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

   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 16.6%

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

   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 77.0

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

   4. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. add-to-fraction N/A

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

      12. div-add-rev N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 77.0%

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

Alternative 6: 79.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_1 := t\_0 - -1\\ t_2 := \frac{\left(\beta + \alpha\right) + i}{t\_0 - 1}\\ \mathbf{if}\;\beta \leq 5.4 \cdot 10^{+187}:\\ \;\;\;\;t\_2 \cdot \frac{0.25 \cdot i}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \frac{i \cdot \frac{\alpha + i}{\beta}}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha)))
        (t_1 (- t_0 -1.0))
        (t_2 (/ (+ (+ beta alpha) i) (- t_0 1.0))))
   (if (<= beta 5.4e+187)
     (* t_2 (/ (* 0.25 i) t_1))
     (* t_2 (/ (* i (/ (+ alpha i) beta)) t_1)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double t_1 = t_0 - -1.0;
	double t_2 = ((beta + alpha) + i) / (t_0 - 1.0);
	double tmp;
	if (beta <= 5.4e+187) {
		tmp = t_2 * ((0.25 * i) / t_1);
	} else {
		tmp = t_2 * ((i * ((alpha + i) / beta)) / t_1);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	t_1 = Float64(t_0 - -1.0)
	t_2 = Float64(Float64(Float64(beta + alpha) + i) / Float64(t_0 - 1.0))
	tmp = 0.0
	if (beta <= 5.4e+187)
		tmp = Float64(t_2 * Float64(Float64(0.25 * i) / t_1));
	else
		tmp = Float64(t_2 * Float64(Float64(i * Float64(Float64(alpha + i) / beta)) / t_1));
	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[(t$95$0 - -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 5.4e+187], N[(t$95$2 * N[(N[(0.25 * i), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(i * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.40000000000000016e187

   1. Initial program 16.6%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift--.f64 N/A

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

   3. Applied rewrites 38.3%

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

   4. Taylor expanded in i around inf

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

      1. lower-*.f64 71.6

         \[\leadsto \frac{\left(\beta + \alpha\right) + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \frac{0.25 \cdot \color{blue}{i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \]

   6. Applied rewrites 71.6%

      \[\leadsto \frac{\left(\beta + \alpha\right) + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - 1} \cdot \frac{\color{blue}{0.25 \cdot i}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \]

   if 5.40000000000000016e187 < beta

   1. Initial program 16.6%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lift--.f64 N/A

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

   3. Applied rewrites 38.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. div-add-rev N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 16.7

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

   8. Applied rewrites 16.7%

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

Alternative 7: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 - 1\\ t_3 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(\beta \cdot \alpha + t\_3\right)}{t\_1}}{t\_2} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1 \cdot \left(i \cdot \left(-1 \cdot i\right)\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125, \beta + \alpha, \mathsf{fma}\left(0.0625, i, \left(\beta + \alpha\right) \cdot 0.125\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (- t_1 1.0))
        (t_3 (* i (+ (+ alpha beta) i))))
   (if (<= (/ (/ (* t_3 (+ (* beta alpha) t_3)) t_1) t_2) 2e-23)
     (/ (* -1.0 (* i (* -1.0 i))) t_2)
     (/
      (fma -0.125 (+ beta alpha) (fma 0.0625 i (* (+ beta alpha) 0.125)))
      i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 - 1.0;
	double t_3 = i * ((alpha + beta) + i);
	double tmp;
	if ((((t_3 * ((beta * alpha) + t_3)) / t_1) / t_2) <= 2e-23) {
		tmp = (-1.0 * (i * (-1.0 * i))) / t_2;
	} else {
		tmp = fma(-0.125, (beta + alpha), fma(0.0625, i, ((beta + alpha) * 0.125))) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(t_1 - 1.0)
	t_3 = Float64(i * Float64(Float64(alpha + beta) + i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(Float64(beta * alpha) + t_3)) / t_1) / t_2) <= 2e-23)
		tmp = Float64(Float64(-1.0 * Float64(i * Float64(-1.0 * i))) / t_2);
	else
		tmp = Float64(fma(-0.125, Float64(beta + alpha), fma(0.0625, i, Float64(Float64(beta + alpha) * 0.125))) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(N[(beta * alpha), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], 2e-23], N[(N[(-1.0 * N[(i * N[(-1.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(-0.125 * N[(beta + alpha), $MachinePrecision] + N[(0.0625 * i + N[(N[(beta + alpha), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $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))) < 1.99999999999999992e-23

   1. Initial program 16.6%

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

   2. Taylor expanded in alpha around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 13.6

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

   4. Applied rewrites 13.6%

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

   5. Taylor expanded in beta around 0

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

      1. lower-*.f64 17.8

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

   7. Applied rewrites 17.8%

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

   if 1.99999999999999992e-23 < (/.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 16.6%

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

   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 77.0

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

   4. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. add-to-fraction N/A

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

      12. div-add-rev N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 77.0%

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

Alternative 8: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125, \beta + \alpha, \mathsf{fma}\left(0.0625, i, \left(\beta + \alpha\right) \cdot 0.125\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ (+ alpha beta) i))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) 2e-23)
     (/ (* i (+ alpha i)) (pow beta 2.0))
     (/
      (fma -0.125 (+ beta alpha) (fma 0.0625 i (* (+ beta alpha) 0.125)))
      i))))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = i * ((alpha + beta) + i);
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 2e-23) {
		tmp = (i * (alpha + i)) / pow(beta, 2.0);
	} else {
		tmp = fma(-0.125, (beta + alpha), fma(0.0625, i, ((beta + alpha) * 0.125))) / i;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(Float64(alpha + beta) + i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= 2e-23)
		tmp = Float64(Float64(i * Float64(alpha + i)) / (beta ^ 2.0));
	else
		tmp = Float64(fma(-0.125, Float64(beta + alpha), fma(0.0625, i, Float64(Float64(beta + alpha) * 0.125))) / i);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], 2e-23], N[(N[(i * N[(alpha + i), $MachinePrecision]), $MachinePrecision] / N[Power[beta, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(-0.125 * N[(beta + alpha), $MachinePrecision] + N[(0.0625 * i + N[(N[(beta + alpha), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $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))) < 1.99999999999999992e-23

   1. Initial program 16.6%

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

   2. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-pow.f64 9.2

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

   4. Applied rewrites 9.2%

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

   if 1.99999999999999992e-23 < (/.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 16.6%

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

   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 77.0

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

   4. Applied rewrites 77.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. add-to-fraction N/A

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

      12. div-add-rev N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 77.0%

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

Alternative 9: 77.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.125, \beta + \alpha, \mathsf{fma}\left(0.0625, i, \left(\beta + \alpha\right) \cdot 0.125\right)\right)}{i} \end{array} \]

FPCore

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

C

double code(double alpha, double beta, double i) {
	return fma(-0.125, (beta + alpha), fma(0.0625, i, ((beta + alpha) * 0.125))) / i;
}

Julia

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

Wolfram

code[alpha_, beta_, i_] := N[(N[(-0.125 * N[(beta + alpha), $MachinePrecision] + N[(0.0625 * i + N[(N[(beta + alpha), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]

Derivation

1. Initial program 16.6%

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

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 77.0

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

4. Applied rewrites 77.0%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lift-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. add-to-fraction N/A

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

   12. div-add-rev N/A

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

   13. lower-/.f64 N/A

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

6. Applied rewrites 77.0%

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

Alternative 10: 73.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \end{array} \]

FPCore

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

C

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

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 + (0.125d0 * (beta / i))) - (0.125d0 * ((alpha + beta) / i))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[alpha_, beta_, i_] := N[(N[(0.0625 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.6%

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

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 77.0

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

4. Applied rewrites 77.0%

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

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 73.3

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

7. Applied rewrites 73.3%

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

Alternative 11: 72.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+237}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\alpha + \beta}{i}, -0.125, 0.125 \cdot \frac{\beta}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.2e+237)
   0.0625
   (fma (/ (+ alpha beta) i) -0.125 (* 0.125 (/ beta i)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.2e+237) {
		tmp = 0.0625;
	} else {
		tmp = fma(((alpha + beta) / i), -0.125, (0.125 * (beta / i)));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.2e+237)
		tmp = 0.0625;
	else
		tmp = fma(Float64(Float64(alpha + beta) / i), -0.125, Float64(0.125 * Float64(beta / i)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 4.2e+237], 0.0625, N[(N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision] * -0.125 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.20000000000000029e237

   1. Initial program 16.6%

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

   2. Taylor expanded in i around inf

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

   4. Applied rewrites 70.7%

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

   if 4.20000000000000029e237 < beta

   1. Initial program 16.6%

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

   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 77.0

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

   4. Applied rewrites 77.0%

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

   5. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 5.9

         \[\leadsto 0.125 \cdot \frac{\alpha}{i} - 0.125 \cdot \frac{\alpha + \beta}{i} \]

   7. Applied rewrites 5.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval 5.9

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 5.9

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

   9. Applied rewrites 5.9%

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

   10. Taylor expanded in beta around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 6.5

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

   12. Applied rewrites 6.5%

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

Alternative 12: 70.7% 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 16.6%

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

2. Taylor expanded in i around inf

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

4. Applied rewrites 70.7%

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

Reproduce

herbie shell --seed 2025150 
(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)))