Octave 3.8, jcobi/4

Percentage Accurate: 15.9% → 83.4%

Time: 17.5s

Alternatives: 10

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

real(8) function code(alpha, beta, i)
    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: 15.9% 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

real(8) function code(alpha, beta, i)
    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: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i + \left(\alpha + \beta\right)\\ t_1 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;i \leq 3 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{i \cdot t\_0}{t\_1}}{t\_1 + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, t\_0, \alpha \cdot \beta\right)}{t\_1}}{t\_1 + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{t\_0}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{\alpha - \left(-1 - \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ i (+ alpha beta))) (t_1 (fma i 2.0 (+ alpha beta))))
   (if (<= i 3e+140)
     (*
      (/ (/ (* i t_0) t_1) (+ t_1 1.0))
      (/ (/ (fma i t_0 (* alpha beta)) t_1) (+ t_1 -1.0)))
     (*
      (/
       (* i (/ t_0 (+ alpha (fma i 2.0 beta))))
       (- alpha (- -1.0 (fma i 2.0 beta))))
      0.25))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i + (alpha + beta);
	double t_1 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if (i <= 3e+140) {
		tmp = (((i * t_0) / t_1) / (t_1 + 1.0)) * ((fma(i, t_0, (alpha * beta)) / t_1) / (t_1 + -1.0));
	} else {
		tmp = ((i * (t_0 / (alpha + fma(i, 2.0, beta)))) / (alpha - (-1.0 - fma(i, 2.0, beta)))) * 0.25;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(i + Float64(alpha + beta))
	t_1 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (i <= 3e+140)
		tmp = Float64(Float64(Float64(Float64(i * t_0) / t_1) / Float64(t_1 + 1.0)) * Float64(Float64(fma(i, t_0, Float64(alpha * beta)) / t_1) / Float64(t_1 + -1.0)));
	else
		tmp = Float64(Float64(Float64(i * Float64(t_0 / Float64(alpha + fma(i, 2.0, beta)))) / Float64(alpha - Float64(-1.0 - fma(i, 2.0, beta)))) * 0.25);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 3e+140], N[(N[(N[(N[(i * t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i * t$95$0 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * N[(t$95$0 / N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha - N[(-1.0 - N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if i < 2.99999999999999997e140

   1. Initial program 39.4%

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

   4. Applied egg-rr 86.7%

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

   if 2.99999999999999997e140 < i

   1. Initial program 0.2%

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

   4. Applied egg-rr 8.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied egg-rr 8.0%

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

   7. Taylor expanded in i around -inf

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

   9. Simplified 83.1%

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

       1. lift-+.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. *-commutative N/A

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

   11. Applied egg-rr 83.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{\alpha - \left(-1 - \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t\_0 \cdot t\_0\\ t_2 := -1 + t\_1\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{t\_3 \cdot \left(t\_3 + \alpha \cdot \beta\right)}{t\_1}}{t\_2} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{i \cdot i}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* i 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (+ -1.0 t_1))
        (t_3 (* i (+ i (+ alpha beta)))))
   (if (<= (/ (/ (* t_3 (+ t_3 (* alpha beta))) t_1) t_2) 2e-18)
     (/ (* i i) t_2)
     0.0625)))

C

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = -1.0 + t_1;
	double t_3 = i * (i + (alpha + beta));
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= 2e-18) {
		tmp = (i * i) / t_2;
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    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) :: t_3
    real(8) :: tmp
    t_0 = (alpha + beta) + (i * 2.0d0)
    t_1 = t_0 * t_0
    t_2 = (-1.0d0) + t_1
    t_3 = i * (i + (alpha + beta))
    if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= 2d-18) then
        tmp = (i * i) / t_2
    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) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = -1.0 + t_1;
	double t_3 = i * (i + (alpha + beta));
	double tmp;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= 2e-18) {
		tmp = (i * i) / t_2;
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = -1.0 + t_1
	t_3 = i * (i + (alpha + beta))
	tmp = 0
	if (((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= 2e-18:
		tmp = (i * i) / t_2
	else:
		tmp = 0.0625
	return tmp

Julia

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(-1.0 + t_1)
	t_3 = Float64(i * Float64(i + Float64(alpha + beta)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(t_3 + Float64(alpha * beta))) / t_1) / t_2) <= 2e-18)
		tmp = Float64(Float64(i * i) / t_2);
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = -1.0 + t_1;
	t_3 = i * (i + (alpha + beta));
	tmp = 0.0;
	if ((((t_3 * (t_3 + (alpha * beta))) / t_1) / t_2) <= 2e-18)
		tmp = (i * i) / t_2;
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(t$95$3 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], 2e-18], N[(N[(i * i), $MachinePrecision] / t$95$2), $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))) < 2.0000000000000001e-18

   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 alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 23.3

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

   5. Simplified 23.3%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 25.9

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

   8. Simplified 25.9%

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

   9. Taylor expanded in i around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 93.7

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

   11. Simplified 93.7%

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

   if 2.0000000000000001e-18 < (/.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 15.9%

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

   4. Applied egg-rr 43.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied egg-rr 43.6%

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

   7. Taylor expanded in i around -inf

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

   9. Simplified 73.9%

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

   10. Taylor expanded in i around inf

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

   12. Simplified 73.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{-1 + \left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{i \cdot i}{-1 + \left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := \alpha - \left(-1 - \mathsf{fma}\left(i, 2, \beta\right)\right)\\ t_2 := i + \left(\alpha + \beta\right)\\ t_3 := \frac{t\_2}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\\ \mathbf{if}\;i \leq 3 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, t\_2, \alpha \cdot \beta\right)}{t\_0}}{t\_0 + -1} \cdot \left(i \cdot \frac{t\_3}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot t\_3}{t\_1} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ alpha beta)))
        (t_1 (- alpha (- -1.0 (fma i 2.0 beta))))
        (t_2 (+ i (+ alpha beta)))
        (t_3 (/ t_2 (+ alpha (fma i 2.0 beta)))))
   (if (<= i 3e+140)
     (* (/ (/ (fma i t_2 (* alpha beta)) t_0) (+ t_0 -1.0)) (* i (/ t_3 t_1)))
     (* (/ (* i t_3) t_1) 0.25))))

C

double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (alpha + beta));
	double t_1 = alpha - (-1.0 - fma(i, 2.0, beta));
	double t_2 = i + (alpha + beta);
	double t_3 = t_2 / (alpha + fma(i, 2.0, beta));
	double tmp;
	if (i <= 3e+140) {
		tmp = ((fma(i, t_2, (alpha * beta)) / t_0) / (t_0 + -1.0)) * (i * (t_3 / t_1));
	} else {
		tmp = ((i * t_3) / t_1) * 0.25;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(alpha + beta))
	t_1 = Float64(alpha - Float64(-1.0 - fma(i, 2.0, beta)))
	t_2 = Float64(i + Float64(alpha + beta))
	t_3 = Float64(t_2 / Float64(alpha + fma(i, 2.0, beta)))
	tmp = 0.0
	if (i <= 3e+140)
		tmp = Float64(Float64(Float64(fma(i, t_2, Float64(alpha * beta)) / t_0) / Float64(t_0 + -1.0)) * Float64(i * Float64(t_3 / t_1)));
	else
		tmp = Float64(Float64(Float64(i * t_3) / t_1) * 0.25);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha - N[(-1.0 - N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 3e+140], N[(N[(N[(N[(i * t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(i * N[(t$95$3 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision] * 0.25), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if i < 2.99999999999999997e140

   1. Initial program 39.4%

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

   4. Applied egg-rr 86.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied egg-rr 86.6%

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

   if 2.99999999999999997e140 < i

   1. Initial program 0.2%

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

   4. Applied egg-rr 8.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied egg-rr 8.0%

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

   7. Taylor expanded in i around -inf

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

   9. Simplified 83.1%

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

       1. lift-+.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. *-commutative N/A

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

   11. Applied egg-rr 83.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \cdot \left(i \cdot \frac{\frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{\alpha - \left(-1 - \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{\alpha - \left(-1 - \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ t_1 := i + \left(\alpha + \beta\right)\\ t_2 := \alpha - \left(-1 - \mathsf{fma}\left(i, 2, \beta\right)\right)\\ \mathbf{if}\;i \leq 3 \cdot 10^{+140}:\\ \;\;\;\;\frac{1}{\left(\left(\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, -1\right)\right) \cdot \frac{t\_0}{\mathsf{fma}\left(i, t\_1, \alpha \cdot \beta\right)}\right) \cdot \left(t\_2 \cdot \frac{t\_0}{i \cdot t\_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{t\_1}{t\_0}}{t\_2} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ alpha (fma i 2.0 beta)))
        (t_1 (+ i (+ alpha beta)))
        (t_2 (- alpha (- -1.0 (fma i 2.0 beta)))))
   (if (<= i 3e+140)
     (/
      1.0
      (*
       (*
        (+ (+ alpha beta) (fma i 2.0 -1.0))
        (/ t_0 (fma i t_1 (* alpha beta))))
       (* t_2 (/ t_0 (* i t_1)))))
     (* (/ (* i (/ t_1 t_0)) t_2) 0.25))))

C

double code(double alpha, double beta, double i) {
	double t_0 = alpha + fma(i, 2.0, beta);
	double t_1 = i + (alpha + beta);
	double t_2 = alpha - (-1.0 - fma(i, 2.0, beta));
	double tmp;
	if (i <= 3e+140) {
		tmp = 1.0 / ((((alpha + beta) + fma(i, 2.0, -1.0)) * (t_0 / fma(i, t_1, (alpha * beta)))) * (t_2 * (t_0 / (i * t_1))));
	} else {
		tmp = ((i * (t_1 / t_0)) / t_2) * 0.25;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(alpha + fma(i, 2.0, beta))
	t_1 = Float64(i + Float64(alpha + beta))
	t_2 = Float64(alpha - Float64(-1.0 - fma(i, 2.0, beta)))
	tmp = 0.0
	if (i <= 3e+140)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(alpha + beta) + fma(i, 2.0, -1.0)) * Float64(t_0 / fma(i, t_1, Float64(alpha * beta)))) * Float64(t_2 * Float64(t_0 / Float64(i * t_1)))));
	else
		tmp = Float64(Float64(Float64(i * Float64(t_1 / t_0)) / t_2) * 0.25);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(alpha - N[(-1.0 - N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 3e+140], N[(1.0 / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(i * t$95$1 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * N[(t$95$0 / N[(i * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * 0.25), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if i < 2.99999999999999997e140

   1. Initial program 39.4%

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

   4. Applied egg-rr 86.7%

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

   6. Applied egg-rr 86.0%

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

   if 2.99999999999999997e140 < i

   1. Initial program 0.2%

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

   4. Applied egg-rr 8.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied egg-rr 8.0%

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

   7. Taylor expanded in i around -inf

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

   9. Simplified 83.1%

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

       1. lift-+.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. *-commutative N/A

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

   11. Applied egg-rr 83.1%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3 \cdot 10^{+140}:\\ \;\;\;\;\frac{1}{\left(\left(\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, -1\right)\right) \cdot \frac{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}\right) \cdot \left(\left(\alpha - \left(-1 - \mathsf{fma}\left(i, 2, \beta\right)\right)\right) \cdot \frac{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}{i \cdot \left(i + \left(\alpha + \beta\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{\alpha - \left(-1 - \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha - \left(-1 - \mathsf{fma}\left(i, 2, \beta\right)\right)\\ t_1 := i + \left(\alpha + \beta\right)\\ t_2 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \mathbf{if}\;i \leq 3 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, t\_1, \alpha \cdot \beta\right)}{t\_2}}{\left(\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, -1\right)\right) \cdot \left(t\_0 \cdot \frac{t\_2}{i \cdot t\_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{t\_1}{t\_2}}{t\_0} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (- alpha (- -1.0 (fma i 2.0 beta))))
        (t_1 (+ i (+ alpha beta)))
        (t_2 (+ alpha (fma i 2.0 beta))))
   (if (<= i 3e+140)
     (/
      (/ (fma i t_1 (* alpha beta)) t_2)
      (* (+ (+ alpha beta) (fma i 2.0 -1.0)) (* t_0 (/ t_2 (* i t_1)))))
     (* (/ (* i (/ t_1 t_2)) t_0) 0.25))))

C

double code(double alpha, double beta, double i) {
	double t_0 = alpha - (-1.0 - fma(i, 2.0, beta));
	double t_1 = i + (alpha + beta);
	double t_2 = alpha + fma(i, 2.0, beta);
	double tmp;
	if (i <= 3e+140) {
		tmp = (fma(i, t_1, (alpha * beta)) / t_2) / (((alpha + beta) + fma(i, 2.0, -1.0)) * (t_0 * (t_2 / (i * t_1))));
	} else {
		tmp = ((i * (t_1 / t_2)) / t_0) * 0.25;
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(alpha - Float64(-1.0 - fma(i, 2.0, beta)))
	t_1 = Float64(i + Float64(alpha + beta))
	t_2 = Float64(alpha + fma(i, 2.0, beta))
	tmp = 0.0
	if (i <= 3e+140)
		tmp = Float64(Float64(fma(i, t_1, Float64(alpha * beta)) / t_2) / Float64(Float64(Float64(alpha + beta) + fma(i, 2.0, -1.0)) * Float64(t_0 * Float64(t_2 / Float64(i * t_1)))));
	else
		tmp = Float64(Float64(Float64(i * Float64(t_1 / t_2)) / t_0) * 0.25);
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(alpha - N[(-1.0 - N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 3e+140], N[(N[(N[(i * t$95$1 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(t$95$2 / N[(i * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i * N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * 0.25), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if i < 2.99999999999999997e140

   1. Initial program 39.4%

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

   4. Applied egg-rr 86.7%

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

   6. Applied egg-rr 84.5%

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

   if 2.99999999999999997e140 < i

   1. Initial program 0.2%

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

   4. Applied egg-rr 8.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied egg-rr 8.0%

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

   7. Taylor expanded in i around -inf

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

   9. Simplified 83.1%

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

       1. lift-+.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. lift-+.f64 N/A

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

       8. lift-+.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. *-commutative N/A

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

   11. Applied egg-rr 83.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{\left(\left(\alpha + \beta\right) + \mathsf{fma}\left(i, 2, -1\right)\right) \cdot \left(\left(\alpha - \left(-1 - \mathsf{fma}\left(i, 2, \beta\right)\right)\right) \cdot \frac{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}{i \cdot \left(i + \left(\alpha + \beta\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{\alpha - \left(-1 - \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+153}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot \frac{\frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{\alpha - \left(-1 - \mathsf{fma}\left(i, 2, \beta\right)\right)}\right) \cdot \frac{i + 1}{\beta}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.45e+153)
   0.0625
   (*
    (*
     i
     (/
      (/ (+ i (+ alpha beta)) (+ alpha (fma i 2.0 beta)))
      (- alpha (- -1.0 (fma i 2.0 beta)))))
    (/ (+ i 1.0) beta))))

C

double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.45e+153) {
		tmp = 0.0625;
	} else {
		tmp = (i * (((i + (alpha + beta)) / (alpha + fma(i, 2.0, beta))) / (alpha - (-1.0 - fma(i, 2.0, beta))))) * ((i + 1.0) / beta);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.45e+153)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(Float64(Float64(i + Float64(alpha + beta)) / Float64(alpha + fma(i, 2.0, beta))) / Float64(alpha - Float64(-1.0 - fma(i, 2.0, beta))))) * Float64(Float64(i + 1.0) / beta));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := If[LessEqual[beta, 1.45e+153], 0.0625, N[(N[(i * N[(N[(N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha - N[(-1.0 - N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + 1.0), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.45000000000000001e153

   1. Initial program 22.6%

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

   4. Applied egg-rr 50.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied egg-rr 50.1%

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

   7. Taylor expanded in i around -inf

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

   9. Simplified 79.9%

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

   10. Taylor expanded in i around inf

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

   12. Simplified 80.2%

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

   if 1.45000000000000001e153 < 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

   4. Applied egg-rr 22.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied egg-rr 22.9%

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

   7. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 60.2

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

   9. Simplified 60.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+153}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot \frac{\frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{\alpha - \left(-1 - \mathsf{fma}\left(i, 2, \beta\right)\right)}\right) \cdot \frac{i + 1}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+154}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot i}{1 + t\_0}}{-1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ alpha (fma i 2.0 beta))))
   (if (<= beta 6.2e+154) 0.0625 (/ (/ (* i i) (+ 1.0 t_0)) (+ -1.0 t_0)))))

C

double code(double alpha, double beta, double i) {
	double t_0 = alpha + fma(i, 2.0, beta);
	double tmp;
	if (beta <= 6.2e+154) {
		tmp = 0.0625;
	} else {
		tmp = ((i * i) / (1.0 + t_0)) / (-1.0 + t_0);
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(alpha + fma(i, 2.0, beta))
	tmp = 0.0
	if (beta <= 6.2e+154)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i * i) / Float64(1.0 + t_0)) / Float64(-1.0 + t_0));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 6.2e+154], 0.0625, N[(N[(N[(i * i), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.2000000000000003e154

   1. Initial program 22.6%

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

   4. Applied egg-rr 50.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied egg-rr 50.1%

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

   7. Taylor expanded in i around -inf

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

   9. Simplified 79.9%

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

   10. Taylor expanded in i around inf

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

   12. Simplified 80.2%

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

   if 6.2000000000000003e154 < 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 alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 6.2

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

   5. Simplified 6.2%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 8.9

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

   8. Simplified 8.9%

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

   10. Applied egg-rr 3.7%

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

   11. Taylor expanded in i around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 38.6

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

   13. Simplified 38.6%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+154}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot i}{1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}}{-1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+274}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{i + \beta}{\mathsf{fma}\left(t\_0, t\_0, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ alpha (fma i 2.0 beta))))
   (if (<= beta 2.8e+274) 0.0625 (* i (/ (+ i beta) (fma t_0 t_0 -1.0))))))

C

double code(double alpha, double beta, double i) {
	double t_0 = alpha + fma(i, 2.0, beta);
	double tmp;
	if (beta <= 2.8e+274) {
		tmp = 0.0625;
	} else {
		tmp = i * ((i + beta) / fma(t_0, t_0, -1.0));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	t_0 = Float64(alpha + fma(i, 2.0, beta))
	tmp = 0.0
	if (beta <= 2.8e+274)
		tmp = 0.0625;
	else
		tmp = Float64(i * Float64(Float64(i + beta) / fma(t_0, t_0, -1.0)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2.8e+274], 0.0625, N[(i * N[(N[(i + beta), $MachinePrecision] / N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.80000000000000009e274

   1. Initial program 19.3%

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

   4. Applied egg-rr 46.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied egg-rr 46.2%

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

   7. Taylor expanded in i around -inf

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

   9. Simplified 73.1%

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

   10. Taylor expanded in i around inf

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

   12. Simplified 73.2%

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

   if 2.80000000000000009e274 < 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 alpha around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 16.7

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

   5. Simplified 16.7%

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

   6. Taylor expanded in alpha around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 16.7

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

   8. Simplified 16.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   10. Applied egg-rr 68.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+274}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{i + \beta}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.8% 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

real(8) function code(alpha, beta, i)
    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 18.8%

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

4. Applied egg-rr 45.5%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

6. Applied egg-rr 45.5%

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

7. Taylor expanded in i around -inf

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

9. Simplified 71.5%

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

10. Taylor expanded in i around inf

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

12. Simplified 71.5%

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

Alternative 10: 11.9% accurate, 115.0× speedup

Math

\[\begin{array}{l} \\ 0.0009765625 \end{array} \]

FPCore

(FPCore (alpha beta i) :precision binary64 0.0009765625)

C

double code(double alpha, double beta, double i) {
	return 0.0009765625;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0009765625d0
end function

Java

public static double code(double alpha, double beta, double i) {
	return 0.0009765625;
}

Python

def code(alpha, beta, i):
	return 0.0009765625

Julia

function code(alpha, beta, i)
	return 0.0009765625
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = 0.0009765625;
end

Wolfram

code[alpha_, beta_, i_] := 0.0009765625

Derivation

1. Initial program 18.8%

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

4. Applied egg-rr 45.5%

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

5. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\frac{1}{1024}} \]
6. Step-by-step derivation

7. Simplified 12.0%

   \[\leadsto \color{blue}{0.0009765625} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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)))