Octave 3.8, jcobi/4

Percentage Accurate: 15.7% → 99.6%

Time: 9.7s

Alternatives: 19

Speedup: 115.0×

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: 15.7% 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: 99.6% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha)))
        (t_1 (fma 2.0 i (+ alpha beta)))
        (t_2 (/ i t_1))
        (t_3 (/ beta t_1)))
   (*
    (/ (fma t_2 beta (* t_2 (+ alpha i))) (- t_0 -1.0))
    (/ (fma t_2 (+ alpha i) (fma i t_3 (* alpha t_3))) (- t_0 1.0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double t_1 = fma(2.0, i, (alpha + beta));
	double t_2 = i / t_1;
	double t_3 = beta / t_1;
	return (fma(t_2, beta, (t_2 * (alpha + i))) / (t_0 - -1.0)) * (fma(t_2, (alpha + i), fma(i, t_3, (alpha * t_3))) / (t_0 - 1.0));
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	t_1 = fma(2.0, i, Float64(alpha + beta))
	t_2 = Float64(i / t_1)
	t_3 = Float64(beta / t_1)
	return Float64(Float64(fma(t_2, beta, Float64(t_2 * Float64(alpha + i))) / Float64(t_0 - -1.0)) * Float64(fma(t_2, Float64(alpha + i), fma(i, t_3, Float64(alpha * t_3))) / Float64(t_0 - 1.0)))
end

Wolfram

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

Derivation

1. Initial program 18.7%

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

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

   5. times-frac N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

4. Applied rewrites 41.8%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-+l+ N/A

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

   6. lift-+.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-+.f64 N/A

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

   12. lower-*.f64 99.7

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate-+l+ N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-lft-out N/A

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

   10. lift-*.f64 N/A

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

   11. +-commutative N/A

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

   12. associate-+l+ N/A

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

   13. lift-*.f64 N/A

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

10. Applied rewrites 99.7%

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

Alternative 2: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 - 1\\ t_3 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_4 := \left(\beta + \alpha\right) + i\\ t_5 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_6 := t\_4 \cdot \frac{i}{t\_5}\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(\beta \cdot \alpha + t\_3\right)}{t\_1}}{t\_2} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_4, i, \beta \cdot \alpha\right)}{t\_5} \cdot t\_6}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_6}{t\_5 - -1} \cdot \left(\left(0.25 + 0.25 \cdot \frac{\alpha + \beta}{i}\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right)\\ \end{array} \end{array} \]

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)))
        (t_4 (+ (+ beta alpha) i))
        (t_5 (fma 2.0 i (+ beta alpha)))
        (t_6 (* t_4 (/ i t_5))))
   (if (<= (/ (/ (* t_3 (+ (* beta alpha) t_3)) t_1) t_2) INFINITY)
     (/ (* (/ (fma t_4 i (* beta alpha)) t_5) t_6) t_2)
     (*
      (/ t_6 (- t_5 -1.0))
      (-
       (+ 0.25 (* 0.25 (/ (+ alpha beta) i)))
       (* 0.0625 (/ (* 2.0 (+ (+ alpha beta) (- (+ alpha beta) 1.0))) 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 t_4 = (beta + alpha) + i;
	double t_5 = fma(2.0, i, (beta + alpha));
	double t_6 = t_4 * (i / t_5);
	double tmp;
	if ((((t_3 * ((beta * alpha) + t_3)) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = ((fma(t_4, i, (beta * alpha)) / t_5) * t_6) / t_2;
	} else {
		tmp = (t_6 / (t_5 - -1.0)) * ((0.25 + (0.25 * ((alpha + beta) / i))) - (0.0625 * ((2.0 * ((alpha + beta) + ((alpha + beta) - 1.0))) / 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))
	t_4 = Float64(Float64(beta + alpha) + i)
	t_5 = fma(2.0, i, Float64(beta + alpha))
	t_6 = Float64(t_4 * Float64(i / t_5))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(Float64(beta * alpha) + t_3)) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(Float64(fma(t_4, i, Float64(beta * alpha)) / t_5) * t_6) / t_2);
	else
		tmp = Float64(Float64(t_6 / Float64(t_5 - -1.0)) * Float64(Float64(0.25 + Float64(0.25 * Float64(Float64(alpha + beta) / i))) - Float64(0.0625 * Float64(Float64(2.0 * Float64(Float64(alpha + beta) + Float64(Float64(alpha + beta) - 1.0))) / 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]}, Block[{t$95$4 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 * N[(i / t$95$5), $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], Infinity], N[(N[(N[(N[(t$95$4 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] * t$95$6), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(t$95$6 / N[(t$95$5 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 + N[(0.25 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(N[(2.0 * N[(N[(alpha + beta), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 56.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   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 0.0%

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

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 13.1%

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

   5. Taylor expanded in i around inf

      \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\color{blue}{\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)} - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \color{blue}{\frac{1}{4} \cdot \frac{\alpha + \beta}{i}}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \color{blue}{\frac{\alpha + \beta}{i}}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\color{blue}{\alpha + \beta}}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \color{blue}{\frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \color{blue}{\frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \color{blue}{\left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\alpha + \beta\right) + \color{blue}{\left(\left(\alpha + \beta\right) - 1\right)}\right)}{i}\right) \]

      13. lower-+.f64 82.0

          \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(0.25 + 0.25 \cdot \frac{\alpha + \beta}{i}\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\alpha + \beta\right) + \left(\color{blue}{\left(\alpha + \beta\right)} - 1\right)\right)}{i}\right) \]

   7. Applied rewrites 82.0%

      \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \color{blue}{\left(\left(0.25 + 0.25 \cdot \frac{\alpha + \beta}{i}\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := t\_1 - 1\\ t_3 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_4 := \left(\beta + \alpha\right) + i\\ t_5 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(\beta \cdot \alpha + t\_3\right)}{t\_1}}{t\_2} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_4, i, \beta \cdot \alpha\right)}{t\_5} \cdot \left(t\_4 \cdot \frac{i}{t\_5}\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + -0.125 \cdot \frac{\left(\alpha + \beta\right) \cdot -0.75}{i}\right) - 0.0078125 \cdot \frac{\mathsf{fma}\left(-2, -2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right), 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)\right)}{i}\\ \end{array} \end{array} \]

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)))
        (t_4 (+ (+ beta alpha) i))
        (t_5 (fma 2.0 i (+ beta alpha))))
   (if (<= (/ (/ (* t_3 (+ (* beta alpha) t_3)) t_1) t_2) INFINITY)
     (/ (* (/ (fma t_4 i (* beta alpha)) t_5) (* t_4 (/ i t_5))) t_2)
     (-
      (+ 0.0625 (* -0.125 (/ (* (+ alpha beta) -0.75) i)))
      (*
       0.0078125
       (/
        (fma
         -2.0
         (* -2.0 (+ (+ alpha beta) (- (+ alpha beta) 1.0)))
         (* 4.0 (+ 1.0 (+ alpha beta))))
        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 t_4 = (beta + alpha) + i;
	double t_5 = fma(2.0, i, (beta + alpha));
	double tmp;
	if ((((t_3 * ((beta * alpha) + t_3)) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = ((fma(t_4, i, (beta * alpha)) / t_5) * (t_4 * (i / t_5))) / t_2;
	} else {
		tmp = (0.0625 + (-0.125 * (((alpha + beta) * -0.75) / i))) - (0.0078125 * (fma(-2.0, (-2.0 * ((alpha + beta) + ((alpha + beta) - 1.0))), (4.0 * (1.0 + (alpha + beta)))) / 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))
	t_4 = Float64(Float64(beta + alpha) + i)
	t_5 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(Float64(beta * alpha) + t_3)) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(Float64(fma(t_4, i, Float64(beta * alpha)) / t_5) * Float64(t_4 * Float64(i / t_5))) / t_2);
	else
		tmp = Float64(Float64(0.0625 + Float64(-0.125 * Float64(Float64(Float64(alpha + beta) * -0.75) / i))) - Float64(0.0078125 * Float64(fma(-2.0, Float64(-2.0 * Float64(Float64(alpha + beta) + Float64(Float64(alpha + beta) - 1.0))), Float64(4.0 * Float64(1.0 + Float64(alpha + beta)))) / 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]}, Block[{t$95$4 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * i + N[(beta + alpha), $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], Infinity], N[(N[(N[(N[(t$95$4 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] * N[(t$95$4 * N[(i / t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(-0.125 * N[(N[(N[(alpha + beta), $MachinePrecision] * -0.75), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0078125 * N[(N[(-2.0 * N[(-2.0 * N[(N[(alpha + beta), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(1.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $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 56.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   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 0.0%

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

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 13.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in i around -inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)} - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{-1}{8} \cdot \frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \color{blue}{\frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

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

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\color{blue}{\left(\alpha + \beta\right) \cdot \left(-1 - \frac{-1}{4}\right)}}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\color{blue}{\left(\alpha + \beta\right) \cdot \left(-1 - \frac{-1}{4}\right)}}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\color{blue}{\left(\alpha + \beta\right)} \cdot \left(-1 - \frac{-1}{4}\right)}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\left(\alpha + \beta\right) \cdot \color{blue}{\frac{-3}{4}}}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\left(\alpha + \beta\right) \cdot \frac{-3}{4}}{i}\right) - \color{blue}{\frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\left(\alpha + \beta\right) \cdot \frac{-3}{4}}{i}\right) - \frac{1}{128} \cdot \color{blue}{\frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i}} \]

   9. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\left(0.0625 + -0.125 \cdot \frac{\left(\alpha + \beta\right) \cdot -0.75}{i}\right) - 0.0078125 \cdot \frac{\mathsf{fma}\left(-2, -2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right), 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)\right)}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 80.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \beta + 2 \cdot i\\ t_3 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_4 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(\beta \cdot \alpha + t\_3\right)}{t\_1}}{t\_1 - 1} \leq \infty:\\ \;\;\;\;\frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{t\_4}}{t\_4 - -1} \cdot \frac{i \cdot \left(\beta + i\right)}{t\_2 \cdot \left(t\_2 - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + -0.125 \cdot \frac{\left(\alpha + \beta\right) \cdot -0.75}{i}\right) - 0.0078125 \cdot \frac{\mathsf{fma}\left(-2, -2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right), 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (+ beta (* 2.0 i)))
        (t_3 (* i (+ (+ alpha beta) i)))
        (t_4 (fma 2.0 i (+ beta alpha))))
   (if (<= (/ (/ (* t_3 (+ (* beta alpha) t_3)) t_1) (- t_1 1.0)) INFINITY)
     (*
      (/ (* (+ (+ beta alpha) i) (/ i t_4)) (- t_4 -1.0))
      (/ (* i (+ beta i)) (* t_2 (- t_2 1.0))))
     (-
      (+ 0.0625 (* -0.125 (/ (* (+ alpha beta) -0.75) i)))
      (*
       0.0078125
       (/
        (fma
         -2.0
         (* -2.0 (+ (+ alpha beta) (- (+ alpha beta) 1.0)))
         (* 4.0 (+ 1.0 (+ alpha beta))))
        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 = beta + (2.0 * i);
	double t_3 = i * ((alpha + beta) + i);
	double t_4 = fma(2.0, i, (beta + alpha));
	double tmp;
	if ((((t_3 * ((beta * alpha) + t_3)) / t_1) / (t_1 - 1.0)) <= ((double) INFINITY)) {
		tmp = ((((beta + alpha) + i) * (i / t_4)) / (t_4 - -1.0)) * ((i * (beta + i)) / (t_2 * (t_2 - 1.0)));
	} else {
		tmp = (0.0625 + (-0.125 * (((alpha + beta) * -0.75) / i))) - (0.0078125 * (fma(-2.0, (-2.0 * ((alpha + beta) + ((alpha + beta) - 1.0))), (4.0 * (1.0 + (alpha + beta)))) / 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(beta + Float64(2.0 * i))
	t_3 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_4 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(Float64(beta * alpha) + t_3)) / t_1) / Float64(t_1 - 1.0)) <= Inf)
		tmp = Float64(Float64(Float64(Float64(Float64(beta + alpha) + i) * Float64(i / t_4)) / Float64(t_4 - -1.0)) * Float64(Float64(i * Float64(beta + i)) / Float64(t_2 * Float64(t_2 - 1.0))));
	else
		tmp = Float64(Float64(0.0625 + Float64(-0.125 * Float64(Float64(Float64(alpha + beta) * -0.75) / i))) - Float64(0.0078125 * Float64(fma(-2.0, Float64(-2.0 * Float64(Float64(alpha + beta) + Float64(Float64(alpha + beta) - 1.0))), Float64(4.0 * Float64(1.0 + Float64(alpha + beta)))) / 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[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(N[(beta * alpha), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] * N[(i / t$95$4), $MachinePrecision]), $MachinePrecision] / N[(t$95$4 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + N[(-0.125 * N[(N[(N[(alpha + beta), $MachinePrecision] * -0.75), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0078125 * N[(N[(-2.0 * N[(-2.0 * N[(N[(alpha + beta), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(1.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $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 56.3%

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

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 13.5

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

   7. Applied rewrites 13.5%

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

   8. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 89.0

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

   10. Applied rewrites 89.0%

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

   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 0.0%

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

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 13.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in i around -inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)} - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{-1}{8} \cdot \frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \color{blue}{\frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

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

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\color{blue}{\left(\alpha + \beta\right) \cdot \left(-1 - \frac{-1}{4}\right)}}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\color{blue}{\left(\alpha + \beta\right) \cdot \left(-1 - \frac{-1}{4}\right)}}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\color{blue}{\left(\alpha + \beta\right)} \cdot \left(-1 - \frac{-1}{4}\right)}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\left(\alpha + \beta\right) \cdot \color{blue}{\frac{-3}{4}}}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\left(\alpha + \beta\right) \cdot \frac{-3}{4}}{i}\right) - \color{blue}{\frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\left(\alpha + \beta\right) \cdot \frac{-3}{4}}{i}\right) - \frac{1}{128} \cdot \color{blue}{\frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i}} \]

   9. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\left(0.0625 + -0.125 \cdot \frac{\left(\alpha + \beta\right) \cdot -0.75}{i}\right) - 0.0078125 \cdot \frac{\mathsf{fma}\left(-2, -2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right), 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)\right)}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;i \cdot \frac{\alpha + i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

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-14)
     (* i (/ (+ alpha i) (* beta beta)))
     0.0625)))

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-14) {
		tmp = i * ((alpha + i) / (beta * beta));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = t_0 * t_0
    t_2 = i * ((alpha + beta) + i)
    if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0d0)) <= 2d-14) then
        tmp = i * ((alpha + i) / (beta * beta))
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double 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-14) {
		tmp = i * ((alpha + i) / (beta * beta));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = t_0 * t_0
	t_2 = i * ((alpha + beta) + i)
	tmp = 0
	if (((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 2e-14:
		tmp = i * ((alpha + i) / (beta * beta))
	else:
		tmp = 0.0625
	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-14)
		tmp = Float64(i * Float64(Float64(alpha + i) / Float64(beta * beta)));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = t_0 * t_0;
	t_2 = i * ((alpha + beta) + i);
	tmp = 0.0;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 2e-14)
		tmp = i * ((alpha + i) / (beta * beta));
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, 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-14], N[(i * N[(N[(alpha + i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0625]]]]

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))) < 2e-14

   1. Initial program 98.6%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 47.8

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in alpha around inf

      \[\leadsto \frac{\alpha \cdot i}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 5.1%

      \[\leadsto \alpha \cdot \color{blue}{\frac{i}{\beta \cdot \beta}} \]

   9. Taylor expanded in i around 0

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

   11. Applied rewrites 47.8%

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

   if 2e-14 < (/.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 14.4%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 77.6%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t\_0 \cdot t\_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ \mathbf{if}\;\frac{\frac{t\_2 \cdot \left(\beta \cdot \alpha + t\_2\right)}{t\_1}}{t\_1 - 1} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

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-14)
     (/ (* i i) (* beta beta))
     0.0625)))

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-14) {
		tmp = (i * i) / (beta * beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = t_0 * t_0
    t_2 = i * ((alpha + beta) + i)
    if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0d0)) <= 2d-14) then
        tmp = (i * i) / (beta * beta)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double 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-14) {
		tmp = (i * i) / (beta * beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = t_0 * t_0
	t_2 = i * ((alpha + beta) + i)
	tmp = 0
	if (((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 2e-14:
		tmp = (i * i) / (beta * beta)
	else:
		tmp = 0.0625
	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-14)
		tmp = Float64(Float64(i * i) / Float64(beta * beta));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = t_0 * t_0;
	t_2 = i * ((alpha + beta) + i);
	tmp = 0.0;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= 2e-14)
		tmp = (i * i) / (beta * beta);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, 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-14], N[(N[(i * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], 0.0625]]]]

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))) < 2e-14

   1. Initial program 98.6%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 47.8

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{{i}^{2}}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.8%

      \[\leadsto \frac{i \cdot i}{\color{blue}{\beta \cdot \beta}} \]

   if 2e-14 < (/.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 14.4%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 77.6%

      \[\leadsto \color{blue}{0.0625} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.6% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) i))
        (t_1 (fma 2.0 i (+ beta alpha)))
        (t_2 (/ i t_1)))
   (*
    (/ (* t_0 t_2) (- t_1 -1.0))
    (/ (fma t_2 t_0 (* beta (/ alpha 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));
	double t_2 = i / t_1;
	return ((t_0 * t_2) / (t_1 - -1.0)) * (fma(t_2, t_0, (beta * (alpha / 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))
	t_2 = Float64(i / t_1)
	return Float64(Float64(Float64(t_0 * t_2) / Float64(t_1 - -1.0)) * Float64(fma(t_2, t_0, Float64(beta * Float64(alpha / 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]}, Block[{t$95$2 = N[(i / t$95$1), $MachinePrecision]}, N[(N[(N[(t$95$0 * t$95$2), $MachinePrecision] / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * t$95$0 + N[(beta * N[(alpha / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 18.7%

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

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

   5. times-frac N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

4. Applied rewrites 41.8%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 8: 85.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))) (t_1 (+ beta (* 2.0 i))))
   (*
    (* (/ i (+ 1.0 t_1)) (/ (+ beta i) t_1))
    (/
     (fma (/ i t_0) (+ (+ beta alpha) i) (* beta (/ alpha 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 t_1 = beta + (2.0 * i);
	return ((i / (1.0 + t_1)) * ((beta + i) / t_1)) * (fma((i / t_0), ((beta + alpha) + i), (beta * (alpha / t_0))) / (t_0 - 1.0));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 18.7%

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

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

   5. times-frac N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

4. Applied rewrites 41.8%

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

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in alpha around 0

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

   1. times-frac N/A

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

   2. div-add-rev N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-*.f64 88.4

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

9. Applied rewrites 88.4%

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

Alternative 9: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\beta + \alpha\right) + i\\ t_1 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ t_2 := \frac{t\_0 \cdot \frac{i}{t\_1}}{t\_1 - -1}\\ \mathbf{if}\;i \leq 2.1 \cdot 10^{+108}:\\ \;\;\;\;t\_2 \cdot \frac{\frac{\mathsf{fma}\left(t\_0, i, \beta \cdot \alpha\right)}{t\_1}}{t\_1 - 1}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \left(\left(0.25 + 0.25 \cdot \frac{\alpha + \beta}{i}\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ beta alpha) i))
        (t_1 (fma 2.0 i (+ beta alpha)))
        (t_2 (/ (* t_0 (/ i t_1)) (- t_1 -1.0))))
   (if (<= i 2.1e+108)
     (* t_2 (/ (/ (fma t_0 i (* beta alpha)) t_1) (- t_1 1.0)))
     (*
      t_2
      (-
       (+ 0.25 (* 0.25 (/ (+ alpha beta) i)))
       (* 0.0625 (/ (* 2.0 (+ (+ alpha beta) (- (+ alpha beta) 1.0))) i)))))))

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_0 * (i / t_1)) / (t_1 - -1.0);
	double tmp;
	if (i <= 2.1e+108) {
		tmp = t_2 * ((fma(t_0, i, (beta * alpha)) / t_1) / (t_1 - 1.0));
	} else {
		tmp = t_2 * ((0.25 + (0.25 * ((alpha + beta) / i))) - (0.0625 * ((2.0 * ((alpha + beta) + ((alpha + beta) - 1.0))) / i)));
	}
	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(Float64(t_0 * Float64(i / t_1)) / Float64(t_1 - -1.0))
	tmp = 0.0
	if (i <= 2.1e+108)
		tmp = Float64(t_2 * Float64(Float64(fma(t_0, i, Float64(beta * alpha)) / t_1) / Float64(t_1 - 1.0)));
	else
		tmp = Float64(t_2 * Float64(Float64(0.25 + Float64(0.25 * Float64(Float64(alpha + beta) / i))) - Float64(0.0625 * Float64(Float64(2.0 * Float64(Float64(alpha + beta) + Float64(Float64(alpha + beta) - 1.0))) / i))));
	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[(N[(t$95$0 * N[(i / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 2.1e+108], N[(t$95$2 * N[(N[(N[(t$95$0 * i + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(0.25 + N[(0.25 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(N[(2.0 * N[(N[(alpha + beta), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if i < 2.1000000000000001e108

   1. Initial program 52.4%

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

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 88.7%

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

   if 2.1000000000000001e108 < i

   1. Initial program 0.4%

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

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 16.4%

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

   5. Taylor expanded in i around inf

      \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)} \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \color{blue}{\left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\color{blue}{\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)} - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \color{blue}{\frac{1}{4} \cdot \frac{\alpha + \beta}{i}}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \color{blue}{\frac{\alpha + \beta}{i}}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\color{blue}{\alpha + \beta}}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \color{blue}{\frac{1}{16} \cdot \frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \color{blue}{\frac{2 \cdot \left(\alpha + \beta\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{\color{blue}{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \color{blue}{\left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}}{i}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\color{blue}{\left(\alpha + \beta\right)} + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right) - \frac{1}{16} \cdot \frac{2 \cdot \left(\left(\alpha + \beta\right) + \color{blue}{\left(\left(\alpha + \beta\right) - 1\right)}\right)}{i}\right) \]

      13. lower-+.f64 87.6

          \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \left(\left(0.25 + 0.25 \cdot \frac{\alpha + \beta}{i}\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\alpha + \beta\right) + \left(\color{blue}{\left(\alpha + \beta\right)} - 1\right)\right)}{i}\right) \]

   7. Applied rewrites 87.6%

      \[\leadsto \frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \color{blue}{\left(\left(0.25 + 0.25 \cdot \frac{\alpha + \beta}{i}\right) - 0.0625 \cdot \frac{2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right)}{i}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \beta + 2 \cdot i\\ t_1 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;i \leq 2.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{t\_1}}{t\_1 - -1} \cdot \frac{\frac{i \cdot \left(\beta + i\right)}{t\_0}}{t\_0 - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + -0.125 \cdot \frac{\left(\alpha + \beta\right) \cdot -0.75}{i}\right) - 0.0078125 \cdot \frac{\mathsf{fma}\left(-2, -2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right), 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)\right)}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* 2.0 i))) (t_1 (fma 2.0 i (+ beta alpha))))
   (if (<= i 2.1e+108)
     (*
      (/ (* (+ (+ beta alpha) i) (/ i t_1)) (- t_1 -1.0))
      (/ (/ (* i (+ beta i)) t_0) (- t_0 1.0)))
     (-
      (+ 0.0625 (* -0.125 (/ (* (+ alpha beta) -0.75) i)))
      (*
       0.0078125
       (/
        (fma
         -2.0
         (* -2.0 (+ (+ alpha beta) (- (+ alpha beta) 1.0)))
         (* 4.0 (+ 1.0 (+ alpha beta))))
        i))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = beta + (2.0 * i);
	double t_1 = fma(2.0, i, (beta + alpha));
	double tmp;
	if (i <= 2.1e+108) {
		tmp = ((((beta + alpha) + i) * (i / t_1)) / (t_1 - -1.0)) * (((i * (beta + i)) / t_0) / (t_0 - 1.0));
	} else {
		tmp = (0.0625 + (-0.125 * (((alpha + beta) * -0.75) / i))) - (0.0078125 * (fma(-2.0, (-2.0 * ((alpha + beta) + ((alpha + beta) - 1.0))), (4.0 * (1.0 + (alpha + beta)))) / i));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(2.0 * i))
	t_1 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (i <= 2.1e+108)
		tmp = Float64(Float64(Float64(Float64(Float64(beta + alpha) + i) * Float64(i / t_1)) / Float64(t_1 - -1.0)) * Float64(Float64(Float64(i * Float64(beta + i)) / t_0) / Float64(t_0 - 1.0)));
	else
		tmp = Float64(Float64(0.0625 + Float64(-0.125 * Float64(Float64(Float64(alpha + beta) * -0.75) / i))) - Float64(0.0078125 * Float64(fma(-2.0, Float64(-2.0 * Float64(Float64(alpha + beta) + Float64(Float64(alpha + beta) - 1.0))), Float64(4.0 * Float64(1.0 + Float64(alpha + beta)))) / i)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 2.1e+108], N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] * N[(i / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * N[(beta + i), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + N[(-0.125 * N[(N[(N[(alpha + beta), $MachinePrecision] * -0.75), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.0078125 * N[(N[(-2.0 * N[(-2.0 * N[(N[(alpha + beta), $MachinePrecision] + N[(N[(alpha + beta), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(1.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if i < 2.1000000000000001e108

   1. Initial program 52.4%

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

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in alpha around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 73.8

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

   7. Applied rewrites 73.8%

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

   if 2.1000000000000001e108 < i

   1. Initial program 0.4%

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

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 16.4%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in i around -inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i}} \]
   8. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)} - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{-1}{8} \cdot \frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \color{blue}{\frac{-1 \cdot \left(\alpha + \beta\right) - \frac{-1}{4} \cdot \left(\alpha + \beta\right)}{i}}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

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

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\color{blue}{\left(\alpha + \beta\right) \cdot \left(-1 - \frac{-1}{4}\right)}}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\color{blue}{\left(\alpha + \beta\right) \cdot \left(-1 - \frac{-1}{4}\right)}}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\color{blue}{\left(\alpha + \beta\right)} \cdot \left(-1 - \frac{-1}{4}\right)}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\left(\alpha + \beta\right) \cdot \color{blue}{\frac{-3}{4}}}{i}\right) - \frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\left(\alpha + \beta\right) \cdot \frac{-3}{4}}{i}\right) - \color{blue}{\frac{1}{128} \cdot \frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{-1}{8} \cdot \frac{\left(\alpha + \beta\right) \cdot \frac{-3}{4}}{i}\right) - \frac{1}{128} \cdot \color{blue}{\frac{-2 \cdot \left(-2 \cdot \left(\alpha + \beta\right) + -2 \cdot \left(\left(\alpha + \beta\right) - 1\right)\right) + 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)}{i}} \]

   9. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\left(0.0625 + -0.125 \cdot \frac{\left(\alpha + \beta\right) \cdot -0.75}{i}\right) - 0.0078125 \cdot \frac{\mathsf{fma}\left(-2, -2 \cdot \left(\left(\alpha + \beta\right) + \left(\left(\alpha + \beta\right) - 1\right)\right), 4 \cdot \left(1 + \left(\alpha + \beta\right)\right)\right)}{i}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 77.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+92}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{t\_0}}{t\_0 - -1} \cdot \frac{\alpha + i}{t\_0 - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))))
   (if (<= beta 2.4e+92)
     0.0625
     (*
      (/ (* (+ (+ beta alpha) i) (/ i t_0)) (- t_0 -1.0))
      (/ (+ alpha i) (- 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 (beta <= 2.4e+92) {
		tmp = 0.0625;
	} else {
		tmp = ((((beta + alpha) + i) * (i / t_0)) / (t_0 - -1.0)) * ((alpha + i) / (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 (beta <= 2.4e+92)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta + alpha) + i) * Float64(i / t_0)) / Float64(t_0 - -1.0)) * Float64(Float64(alpha + i) / 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[beta, 2.4e+92], 0.0625, N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.40000000000000005e92

   1. Initial program 23.0%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 83.7%

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

   if 2.40000000000000005e92 < beta

   1. Initial program 2.0%

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

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 34.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 99.3

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in beta around inf

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

      1. lower-+.f64 67.5

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

   9. Applied rewrites 67.5%

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

Alternative 12: 77.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+174}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{t\_0}}{t\_0 - -1} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ beta alpha))))
   (if (<= beta 1.8e+174)
     0.0625
     (*
      (/ (* (+ (+ beta alpha) i) (/ i t_0)) (- t_0 -1.0))
      (/ (+ alpha i) beta)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (beta + alpha));
	double tmp;
	if (beta <= 1.8e+174) {
		tmp = 0.0625;
	} else {
		tmp = ((((beta + alpha) + i) * (i / t_0)) / (t_0 - -1.0)) * ((alpha + i) / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 1.8e+174)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta + alpha) + i) * Float64(i / t_0)) / Float64(t_0 - -1.0)) * Float64(Float64(alpha + i) / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.8e+174], 0.0625, N[(N[(N[(N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.8000000000000001e174

   1. Initial program 21.8%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 81.5%

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

   if 1.8000000000000001e174 < beta

   1. Initial program 0.0%

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

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 29.5%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 71.1

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

   7. Applied rewrites 71.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+174}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\beta + \alpha\right) + i\right) \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}{\mathsf{fma}\left(2, i, \beta + \alpha\right) - -1} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+174}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{1 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.9e+174)
   0.0625
   (* (/ i (+ 1.0 (+ alpha beta))) (/ (+ alpha i) beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.9e+174) {
		tmp = 0.0625;
	} else {
		tmp = (i / (1.0 + (alpha + beta))) * ((alpha + i) / beta);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.9d+174) then
        tmp = 0.0625d0
    else
        tmp = (i / (1.0d0 + (alpha + beta))) * ((alpha + i) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.9e+174) {
		tmp = 0.0625;
	} else {
		tmp = (i / (1.0 + (alpha + beta))) * ((alpha + i) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.9e+174:
		tmp = 0.0625
	else:
		tmp = (i / (1.0 + (alpha + beta))) * ((alpha + i) / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.9e+174)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / Float64(1.0 + Float64(alpha + beta))) * Float64(Float64(alpha + i) / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.9e+174)
		tmp = 0.0625;
	else
		tmp = (i / (1.0 + (alpha + beta))) * ((alpha + i) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.9e+174], 0.0625, N[(N[(i / N[(1.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9000000000000001e174

   1. Initial program 21.8%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 81.5%

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

   if 1.9000000000000001e174 < beta

   1. Initial program 0.0%

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

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

      5. times-frac N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

   4. Applied rewrites 29.5%

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

   5. Taylor expanded in beta around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 71.1

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 70.7

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

   10. Applied rewrites 70.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+174}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{1 + \left(\alpha + \beta\right)} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 77.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+174}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.9e+174) 0.0625 (* (/ (+ alpha i) beta) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.9e+174) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.9d+174) then
        tmp = 0.0625d0
    else
        tmp = ((alpha + i) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.9e+174) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.9e+174:
		tmp = 0.0625
	else:
		tmp = ((alpha + i) / beta) * (i / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.9e+174)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.9e+174)
		tmp = 0.0625;
	else
		tmp = ((alpha + i) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.9e+174], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9000000000000001e174

   1. Initial program 21.8%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 81.5%

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

   if 1.9000000000000001e174 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 70.6

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

   5. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 76.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+174}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.9e+174) 0.0625 (* (/ i beta) (/ i beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.9e+174) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.9d+174) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.9e+174) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.9e+174:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.9e+174)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.9e+174)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.9e+174], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9000000000000001e174

   1. Initial program 21.8%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 81.5%

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

   if 1.9000000000000001e174 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{i}{\beta} \cdot \frac{\color{blue}{i}}{\beta} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.2%

      \[\leadsto \frac{i}{\beta} \cdot \frac{\color{blue}{i}}{\beta} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 76.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+174}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta} \cdot i}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.9e+174) 0.0625 (/ (* (/ i beta) i) beta)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.9e+174) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * i) / beta;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.9d+174) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) * i) / beta
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.9e+174) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) * i) / beta;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.9e+174:
		tmp = 0.0625
	else:
		tmp = ((i / beta) * i) / beta
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.9e+174)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) * i) / beta);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.9e+174)
		tmp = 0.0625;
	else
		tmp = ((i / beta) * i) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.9e+174], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] * i), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9000000000000001e174

   1. Initial program 21.8%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 81.5%

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

   if 1.9000000000000001e174 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{{i}^{2}}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 19.4%

      \[\leadsto \frac{i \cdot i}{\color{blue}{\beta \cdot \beta}} \]
   9. Step-by-step derivation

   10. Applied rewrites 66.2%

       \[\leadsto \frac{\frac{i}{\beta} \cdot i}{\beta} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 73.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+174}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\beta} \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.9e+174) 0.0625 (* (/ (/ i beta) beta) i)))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.9e+174) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) / beta) * i;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.9d+174) then
        tmp = 0.0625d0
    else
        tmp = ((i / beta) / beta) * i
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.9e+174) {
		tmp = 0.0625;
	} else {
		tmp = ((i / beta) / beta) * i;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.9e+174:
		tmp = 0.0625
	else:
		tmp = ((i / beta) / beta) * i
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.9e+174)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i / beta) / beta) * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.9e+174)
		tmp = 0.0625;
	else
		tmp = ((i / beta) / beta) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.9e+174], 0.0625, N[(N[(N[(i / beta), $MachinePrecision] / beta), $MachinePrecision] * i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.9000000000000001e174

   1. Initial program 21.8%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 81.5%

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

   if 1.9000000000000001e174 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in alpha around 0

      \[\leadsto \frac{{i}^{2}}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 19.4%

      \[\leadsto \frac{i \cdot i}{\color{blue}{\beta \cdot \beta}} \]
   9. Step-by-step derivation

   10. Applied rewrites 45.1%

       \[\leadsto \frac{\frac{i}{\beta}}{\beta} \cdot i \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 71.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.16 \cdot 10^{+255}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.16e+255) 0.0625 (* alpha (/ i (* beta beta)))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.16e+255) {
		tmp = 0.0625;
	} else {
		tmp = alpha * (i / (beta * beta));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(alpha, beta, i)
use fmin_fmax_functions
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.16d+255) then
        tmp = 0.0625d0
    else
        tmp = alpha * (i / (beta * beta))
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.16e+255) {
		tmp = 0.0625;
	} else {
		tmp = alpha * (i / (beta * beta));
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.16e+255:
		tmp = 0.0625
	else:
		tmp = alpha * (i / (beta * beta))
	return tmp

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.16e+255)
		tmp = 0.0625;
	else
		tmp = Float64(alpha * Float64(i / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.16e+255)
		tmp = 0.0625;
	else
		tmp = alpha * (i / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.16e+255], 0.0625, N[(alpha * N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.16e255

   1. Initial program 19.6%

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

   3. Taylor expanded in i around inf

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

   5. Applied rewrites 77.3%

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

   if 1.16e255 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in alpha around inf

      \[\leadsto \frac{\alpha \cdot i}{\color{blue}{{\beta}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.3%

      \[\leadsto \alpha \cdot \color{blue}{\frac{i}{\beta \cdot \beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 70.1% accurate, 115.0× 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 18.7%

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

3. Taylor expanded in i around inf

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

5. Applied rewrites 73.9%

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

Reproduce

herbie shell --seed 2025009 
(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)))