Octave 3.8, jcobi/4

Percentage Accurate: 16.3% → 85.4%

Time: 13.8s

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: 16.3% 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: 85.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+102}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 1\right)\right)} \cdot \frac{i + \alpha}{\alpha + \left(-1 + \mathsf{fma}\left(i, 2, \beta\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.2e+102)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (*
    (/ i (+ alpha (+ beta (fma i 2.0 1.0))))
    (/ (+ i alpha) (+ alpha (+ -1.0 (fma i 2.0 beta)))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.2e+102) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = (i / (alpha + (beta + fma(i, 2.0, 1.0)))) * ((i + alpha) / (alpha + (-1.0 + fma(i, 2.0, beta))));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.2e+102)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(i / Float64(alpha + Float64(beta + fma(i, 2.0, 1.0)))) * Float64(Float64(i + alpha) / Float64(alpha + Float64(-1.0 + fma(i, 2.0, beta)))));
	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, 2.2e+102], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / N[(alpha + N[(beta + N[(i * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / N[(alpha + N[(-1.0 + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.20000000000000007e102

   1. Initial program 22.2%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   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. Applied rewrites 19.4%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 95.0%

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

   9. Taylor expanded in beta around 0

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

   11. Applied rewrites 95.3%

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

   if 2.20000000000000007e102 < beta

   1. Initial program 3.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

   4. Applied rewrites 5.9%

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

   5. Taylor expanded in beta around inf

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

      1. lower-+.f64 24.7

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

   7. Applied rewrites 24.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-fma.f64 N/A

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

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

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

      6. times-frac N/A

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

   9. Applied rewrites 69.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+102}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\alpha + \left(\beta + \mathsf{fma}\left(i, 2, 1\right)\right)} \cdot \frac{i + \alpha}{\alpha + \left(-1 + \mathsf{fma}\left(i, 2, \beta\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+192}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.15e+192)
   (+ 0.0625 (/ 0.015625 (* i i)))
   (/ (* (+ i alpha) (/ i beta)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.15e+192) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i + alpha) * (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 <= 1.15d+192) then
        tmp = 0.0625d0 + (0.015625d0 / (i * i))
    else
        tmp = ((i + alpha) * (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 <= 1.15e+192) {
		tmp = 0.0625 + (0.015625 / (i * i));
	} else {
		tmp = ((i + alpha) * (i / beta)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.15e+192:
		tmp = 0.0625 + (0.015625 / (i * i))
	else:
		tmp = ((i + alpha) * (i / beta)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.15e+192)
		tmp = Float64(0.0625 + Float64(0.015625 / Float64(i * i)));
	else
		tmp = Float64(Float64(Float64(i + alpha) * Float64(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 <= 1.15e+192)
		tmp = 0.0625 + (0.015625 / (i * i));
	else
		tmp = ((i + alpha) * (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, 1.15e+192], N[(0.0625 + N[(0.015625 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.15e192

   1. Initial program 21.3%

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

   3. 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. Applied rewrites 18.5%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 82.0%

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

   9. Taylor expanded in beta around 0

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

   11. Applied rewrites 86.6%

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

   if 1.15e192 < 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 30.5

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

   5. Applied rewrites 30.5%

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

   7. Applied rewrites 79.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+192}:\\ \;\;\;\;0.0625 + \frac{0.015625}{i \cdot i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(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)))