Octave 3.8, jcobi/4

Percentage Accurate: 15.9% → 86.9%

Time: 14.7s

Alternatives: 8

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: 86.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+119}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + 1\right)} \cdot \frac{i}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.35e+119)
   (+ (/ 0.015625 (* i i)) 0.0625)
   (*
    (/ (+ i alpha) (+ alpha (+ (fma i 2.0 beta) 1.0)))
    (/ i (+ alpha (+ (fma i 2.0 beta) -1.0))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.35e+119) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = ((i + alpha) / (alpha + (fma(i, 2.0, beta) + 1.0))) * (i / (alpha + (fma(i, 2.0, beta) + -1.0)));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.35e+119)
		tmp = Float64(Float64(0.015625 / Float64(i * i)) + 0.0625);
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(alpha + Float64(fma(i, 2.0, beta) + 1.0))) * Float64(i / Float64(alpha + Float64(fma(i, 2.0, beta) + -1.0))));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.35e+119], N[(N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision] + 0.0625), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / N[(alpha + N[(N[(i * 2.0 + beta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / N[(alpha + N[(N[(i * 2.0 + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.3499999999999999e119

   1. Initial program 21.7%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Simplified 17.9%

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

   6. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\color{blue}{\frac{1}{16} \cdot {\beta}^{2}}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \color{blue}{\frac{-1}{256}} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \color{blue}{\frac{-1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \frac{-1}{256} \cdot \color{blue}{\frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

   8. Simplified 78.2%

      \[\leadsto \color{blue}{\left(0.0625 + \frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + -0.00390625 \cdot \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(\beta, \beta, -1\right), \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}} \]

   9. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

       3. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} + \frac{1}{16} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} + \frac{1}{16} \]

       5. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} + \frac{1}{16} \]

       6. unpow2 N/A

          \[\leadsto \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} + \frac{1}{16} \]

       7. lower-*.f64 78.3

          \[\leadsto \frac{0.015625}{\color{blue}{i \cdot i}} + 0.0625 \]

   11. Simplified 78.3%

       \[\leadsto \color{blue}{\frac{0.015625}{i \cdot i} + 0.0625} \]

   if 1.3499999999999999e119 < beta

   1. Initial program 1.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 31.1

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

   5. Simplified 31.1%

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

   7. Applied egg-rr 69.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+119}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + 1\right)} \cdot \frac{i}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+119}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{1 + \left(\beta + \alpha\right)} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.9e+119)
   (+ (/ 0.015625 (* i i)) 0.0625)
   (* (/ i (+ 1.0 (+ beta alpha))) (/ (+ i alpha) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.9e+119) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = (i / (1.0 + (beta + alpha))) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.9d+119) then
        tmp = (0.015625d0 / (i * i)) + 0.0625d0
    else
        tmp = (i / (1.0d0 + (beta + alpha))) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.9e+119) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = (i / (1.0 + (beta + alpha))) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.9e+119:
		tmp = (0.015625 / (i * i)) + 0.0625
	else:
		tmp = (i / (1.0 + (beta + alpha))) * ((i + alpha) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.9e+119)
		tmp = Float64(Float64(0.015625 / Float64(i * i)) + 0.0625);
	else
		tmp = Float64(Float64(i / Float64(1.0 + Float64(beta + alpha))) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.9e+119)
		tmp = (0.015625 / (i * i)) + 0.0625;
	else
		tmp = (i / (1.0 + (beta + alpha))) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.9e+119], N[(N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision] + 0.0625), $MachinePrecision], N[(N[(i / N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.90000000000000007e119

   1. Initial program 21.7%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Simplified 17.9%

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

   6. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\color{blue}{\frac{1}{16} \cdot {\beta}^{2}}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \color{blue}{\frac{-1}{256}} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \color{blue}{\frac{-1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \frac{-1}{256} \cdot \color{blue}{\frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

   8. Simplified 78.2%

      \[\leadsto \color{blue}{\left(0.0625 + \frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + -0.00390625 \cdot \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(\beta, \beta, -1\right), \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}} \]

   9. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

       3. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} + \frac{1}{16} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} + \frac{1}{16} \]

       5. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} + \frac{1}{16} \]

       6. unpow2 N/A

          \[\leadsto \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} + \frac{1}{16} \]

       7. lower-*.f64 78.3

          \[\leadsto \frac{0.015625}{\color{blue}{i \cdot i}} + 0.0625 \]

   11. Simplified 78.3%

       \[\leadsto \color{blue}{\frac{0.015625}{i \cdot i} + 0.0625} \]

   if 2.90000000000000007e119 < beta

   1. Initial program 1.8%

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

   4. Applied egg-rr 30.8%

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

   5. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 31.2

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

   7. Simplified 31.2%

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

   8. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 66.5

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

   10. Simplified 66.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+119}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{1 + \left(\beta + \alpha\right)} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+119}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.9e+119)
   (+ (/ 0.015625 (* i i)) 0.0625)
   (* (/ (+ i alpha) beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.9e+119) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.9d+119) then
        tmp = (0.015625d0 / (i * i)) + 0.0625d0
    else
        tmp = ((i + alpha) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.9e+119) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.9e+119:
		tmp = (0.015625 / (i * i)) + 0.0625
	else:
		tmp = ((i + alpha) / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.9e+119)
		tmp = Float64(Float64(0.015625 / Float64(i * i)) + 0.0625);
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.9e+119)
		tmp = (0.015625 / (i * i)) + 0.0625;
	else
		tmp = ((i + alpha) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.9e+119], N[(N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision] + 0.0625), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.90000000000000007e119

   1. Initial program 21.7%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Simplified 17.9%

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

   6. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\color{blue}{\frac{1}{16} \cdot {\beta}^{2}}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \color{blue}{\frac{-1}{256}} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \color{blue}{\frac{-1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \frac{-1}{256} \cdot \color{blue}{\frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

   8. Simplified 78.2%

      \[\leadsto \color{blue}{\left(0.0625 + \frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + -0.00390625 \cdot \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(\beta, \beta, -1\right), \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}} \]

   9. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

       3. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} + \frac{1}{16} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} + \frac{1}{16} \]

       5. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} + \frac{1}{16} \]

       6. unpow2 N/A

          \[\leadsto \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} + \frac{1}{16} \]

       7. lower-*.f64 78.3

          \[\leadsto \frac{0.015625}{\color{blue}{i \cdot i}} + 0.0625 \]

   11. Simplified 78.3%

       \[\leadsto \color{blue}{\frac{0.015625}{i \cdot i} + 0.0625} \]

   if 2.90000000000000007e119 < beta

   1. Initial program 1.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 31.3

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

   5. Simplified 31.3%

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

      1. lift-+.f64 N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 66.4

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

   7. Applied egg-rr 66.4%

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

Alternative 4: 76.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+119}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.9e+119)
   (+ (/ 0.015625 (* i i)) 0.0625)
   (/ (/ (* i i) beta) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.9e+119) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = ((i * i) / beta) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.9d+119) then
        tmp = (0.015625d0 / (i * i)) + 0.0625d0
    else
        tmp = ((i * i) / beta) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.9e+119) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = ((i * i) / beta) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.9e+119:
		tmp = (0.015625 / (i * i)) + 0.0625
	else:
		tmp = ((i * i) / beta) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.9e+119)
		tmp = Float64(Float64(0.015625 / Float64(i * i)) + 0.0625);
	else
		tmp = Float64(Float64(Float64(i * i) / beta) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.9e+119)
		tmp = (0.015625 / (i * i)) + 0.0625;
	else
		tmp = ((i * i) / beta) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.9e+119], N[(N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision] + 0.0625), $MachinePrecision], N[(N[(N[(i * i), $MachinePrecision] / beta), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.90000000000000007e119

   1. Initial program 21.7%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Simplified 17.9%

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

   6. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\color{blue}{\frac{1}{16} \cdot {\beta}^{2}}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \color{blue}{\frac{-1}{256}} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \color{blue}{\frac{-1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \frac{-1}{256} \cdot \color{blue}{\frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

   8. Simplified 78.2%

      \[\leadsto \color{blue}{\left(0.0625 + \frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + -0.00390625 \cdot \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(\beta, \beta, -1\right), \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}} \]

   9. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

       3. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} + \frac{1}{16} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} + \frac{1}{16} \]

       5. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} + \frac{1}{16} \]

       6. unpow2 N/A

          \[\leadsto \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} + \frac{1}{16} \]

       7. lower-*.f64 78.3

          \[\leadsto \frac{0.015625}{\color{blue}{i \cdot i}} + 0.0625 \]

   11. Simplified 78.3%

       \[\leadsto \color{blue}{\frac{0.015625}{i \cdot i} + 0.0625} \]

   if 2.90000000000000007e119 < beta

   1. Initial program 1.8%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 31.3

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

   5. Simplified 31.3%

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

      1. lift-+.f64 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 66.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 66.4

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

   7. Applied egg-rr 66.4%

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

   8. Taylor expanded in i around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{{i}^{2}}{\beta}}}{\beta} \]

      2. unpow2 N/A

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

      3. lower-*.f64 50.9

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

   10. Simplified 50.9%

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

Alternative 5: 76.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+239}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 7.2e+239)
   (+ (/ 0.015625 (* i i)) 0.0625)
   (* (/ i beta) (/ alpha beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.2e+239) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 7.2d+239) then
        tmp = (0.015625d0 / (i * i)) + 0.0625d0
    else
        tmp = (i / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 7.2e+239) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 7.2e+239:
		tmp = (0.015625 / (i * i)) + 0.0625
	else:
		tmp = (i / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 7.2e+239)
		tmp = Float64(Float64(0.015625 / Float64(i * i)) + 0.0625);
	else
		tmp = Float64(Float64(i / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 7.2e+239)
		tmp = (0.015625 / (i * i)) + 0.0625;
	else
		tmp = (i / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 7.2e+239], N[(N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision] + 0.0625), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.2e239

   1. Initial program 18.5%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Simplified 14.9%

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

   6. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\color{blue}{\frac{1}{16} \cdot {\beta}^{2}}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \color{blue}{\frac{-1}{256}} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \color{blue}{\frac{-1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \frac{-1}{256} \cdot \color{blue}{\frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

   8. Simplified 66.4%

      \[\leadsto \color{blue}{\left(0.0625 + \frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + -0.00390625 \cdot \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(\beta, \beta, -1\right), \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}} \]

   9. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

       3. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} + \frac{1}{16} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} + \frac{1}{16} \]

       5. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} + \frac{1}{16} \]

       6. unpow2 N/A

          \[\leadsto \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} + \frac{1}{16} \]

       7. lower-*.f64 71.0

          \[\leadsto \frac{0.015625}{\color{blue}{i \cdot i}} + 0.0625 \]

   11. Simplified 71.0%

       \[\leadsto \color{blue}{\frac{0.015625}{i \cdot i} + 0.0625} \]

   if 7.2e239 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.4

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

   5. Simplified 62.4%

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

   6. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\alpha \cdot i}{{\beta}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\alpha \cdot i}}{{\beta}^{2}} \]

      3. unpow2 N/A

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

      4. lower-*.f64 63.1

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

   8. Simplified 63.1%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. lift-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 63.9

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

   10. Applied egg-rr 63.9%

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

Alternative 6: 75.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.16 \cdot 10^{+241}:\\ \;\;\;\;\frac{0.015625}{i \cdot i} + 0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{i}{\beta \cdot \beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.16e+241)
   (+ (/ 0.015625 (* i i)) 0.0625)
   (* i (/ i (* beta beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.16e+241) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = i * (i / (beta * beta));
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 1.16d+241) then
        tmp = (0.015625d0 / (i * i)) + 0.0625d0
    else
        tmp = i * (i / (beta * beta))
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.16e+241) {
		tmp = (0.015625 / (i * i)) + 0.0625;
	} else {
		tmp = i * (i / (beta * beta));
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.16e+241:
		tmp = (0.015625 / (i * i)) + 0.0625
	else:
		tmp = i * (i / (beta * beta))
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.16e+241)
		tmp = Float64(Float64(0.015625 / Float64(i * i)) + 0.0625);
	else
		tmp = Float64(i * Float64(i / Float64(beta * beta)));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.16e+241)
		tmp = (0.015625 / (i * i)) + 0.0625;
	else
		tmp = i * (i / (beta * beta));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.16e+241], N[(N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision] + 0.0625), $MachinePrecision], N[(i * N[(i / N[(beta * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.15999999999999999e241

   1. Initial program 18.5%

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

   3. Taylor expanded in alpha around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. sub-neg N/A

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

      19. unpow2 N/A

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

      20. metadata-eval N/A

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

      21. lower-fma.f64 N/A

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

   5. Simplified 14.9%

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

   6. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      4. associate-*r/ N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\color{blue}{\frac{1}{16} \cdot {\beta}^{2}}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \color{blue}{\frac{-1}{256}} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \color{blue}{\frac{-1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

      13. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \frac{-1}{256} \cdot \color{blue}{\frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

   8. Simplified 66.4%

      \[\leadsto \color{blue}{\left(0.0625 + \frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + -0.00390625 \cdot \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(\beta, \beta, -1\right), \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}} \]

   9. Taylor expanded in beta around 0

      \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

       2. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

       3. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} + \frac{1}{16} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} + \frac{1}{16} \]

       5. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} + \frac{1}{16} \]

       6. unpow2 N/A

          \[\leadsto \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} + \frac{1}{16} \]

       7. lower-*.f64 71.0

          \[\leadsto \frac{0.015625}{\color{blue}{i \cdot i}} + 0.0625 \]

   11. Simplified 71.0%

       \[\leadsto \color{blue}{\frac{0.015625}{i \cdot i} + 0.0625} \]

   if 1.15999999999999999e241 < beta

   1. Initial program 0.0%

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

   3. Taylor expanded in beta around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.4

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

   5. Simplified 62.4%

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

      1. lift-+.f64 N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.8

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

   7. Applied egg-rr 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 99.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-+.f64 99.7

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in i around inf

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

       1. unpow2 N/A

          \[\leadsto \frac{\color{blue}{i \cdot i}}{{\beta}^{2}} \]

       2. associate-*r/ N/A

          \[\leadsto \color{blue}{i \cdot \frac{i}{{\beta}^{2}}} \]

       3. lower-*.f64 N/A

          \[\leadsto \color{blue}{i \cdot \frac{i}{{\beta}^{2}}} \]

       4. lower-/.f64 N/A

          \[\leadsto i \cdot \color{blue}{\frac{i}{{\beta}^{2}}} \]

       5. unpow2 N/A

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

       6. lower-*.f64 63.4

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

   12. Simplified 63.4%

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

Alternative 7: 71.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \frac{0.015625}{i \cdot i} + 0.0625 \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i) :precision binary64 (+ (/ 0.015625 (* i i)) 0.0625))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return (0.015625 / (i * i)) + 0.0625;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (0.015625d0 / (i * i)) + 0.0625d0
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	return (0.015625 / (i * i)) + 0.0625;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return (0.015625 / (i * i)) + 0.0625

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return Float64(Float64(0.015625 / Float64(i * i)) + 0.0625)
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = (0.015625 / (i * i)) + 0.0625;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := N[(N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision] + 0.0625), $MachinePrecision]

Derivation

1. Initial program 17.2%

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

3. Taylor expanded in alpha around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 N/A

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

   18. sub-neg N/A

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

   19. unpow2 N/A

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

   20. metadata-eval N/A

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

   21. lower-fma.f64 N/A

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

5. Simplified 13.9%

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

6. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) - \frac{1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]
7. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{{\beta}^{2}}{{i}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

   4. associate-*r/ N/A

      \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

   5. lower-/.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \color{blue}{\frac{\frac{1}{16} \cdot {\beta}^{2}}{{i}^{2}}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{\color{blue}{\frac{1}{16} \cdot {\beta}^{2}}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \color{blue}{\left(\beta \cdot \beta\right)}}{{i}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{\color{blue}{i \cdot i}}\right) + \left(\mathsf{neg}\left(\frac{1}{256}\right)\right) \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

   11. metadata-eval N/A

       \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \color{blue}{\frac{-1}{256}} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}} \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \color{blue}{\frac{-1}{256} \cdot \frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

   13. lower-/.f64 N/A

       \[\leadsto \left(\frac{1}{16} + \frac{\frac{1}{16} \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + \frac{-1}{256} \cdot \color{blue}{\frac{4 \cdot \left({\beta}^{2} - 1\right) + \left(4 \cdot {\beta}^{2} + 16 \cdot {\beta}^{2}\right)}{{i}^{2}}} \]

8. Simplified 61.7%

   \[\leadsto \color{blue}{\left(0.0625 + \frac{0.0625 \cdot \left(\beta \cdot \beta\right)}{i \cdot i}\right) + -0.00390625 \cdot \frac{\mathsf{fma}\left(4, \mathsf{fma}\left(\beta, \beta, -1\right), \left(\beta \cdot \beta\right) \cdot 20\right)}{i \cdot i}} \]

9. Taylor expanded in beta around 0

   \[\leadsto \color{blue}{\frac{1}{16} + \frac{1}{64} \cdot \frac{1}{{i}^{2}}} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

    2. lower-+.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{64} \cdot \frac{1}{{i}^{2}} + \frac{1}{16}} \]

    3. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{64} \cdot 1}{{i}^{2}}} + \frac{1}{16} \]

    4. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\frac{1}{64}}}{{i}^{2}} + \frac{1}{16} \]

    5. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{64}}{{i}^{2}}} + \frac{1}{16} \]

    6. unpow2 N/A

       \[\leadsto \frac{\frac{1}{64}}{\color{blue}{i \cdot i}} + \frac{1}{16} \]

    7. lower-*.f64 66.2

       \[\leadsto \frac{0.015625}{\color{blue}{i \cdot i}} + 0.0625 \]

11. Simplified 66.2%

    \[\leadsto \color{blue}{\frac{0.015625}{i \cdot i} + 0.0625} \]
12. Add Preprocessing

Alternative 8: 71.3% accurate, 115.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ 0.0625 \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i) :precision binary64 0.0625)

C

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

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
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

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

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return 0.0625

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return 0.0625
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = 0.0625;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 17.2%

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

3. Taylor expanded in i around inf

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

5. Simplified 65.9%

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

Reproduce

herbie shell --seed 2024207 
(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)))