Octave 3.8, jcobi/4

Percentage Accurate: 16.3% → 97.4%

Time: 24.5s

Alternatives: 11

Speedup: 53.0×

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: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \beta + \mathsf{fma}\left(i, 2, \alpha\right)\\ \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left(\frac{i}{\beta + \left(1 + i \cdot 2\right)} \cdot \frac{i + \beta}{\beta + \mathsf{fma}\left(i, 2, -1\right)}\right) \cdot \frac{i + \left(\alpha + \beta\right)}{t_0}}{\frac{t_0}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\beta + \left(-1 + \mathsf{fma}\left(i, 2, \alpha\right)\right)}}{1 + t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (fma i 2.0 alpha))))
   (if (<= alpha 2.3e+70)
     (/
      (*
       (*
        (/ i (+ beta (+ 1.0 (* i 2.0))))
        (/ (+ i beta) (+ beta (fma i 2.0 -1.0))))
       (/ (+ i (+ alpha beta)) t_0))
      (/ t_0 i))
     (/
      (* i (/ (+ alpha i) (+ beta (+ -1.0 (fma i 2.0 alpha)))))
      (+ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = beta + fma(i, 2.0, alpha);
	double tmp;
	if (alpha <= 2.3e+70) {
		tmp = (((i / (beta + (1.0 + (i * 2.0)))) * ((i + beta) / (beta + fma(i, 2.0, -1.0)))) * ((i + (alpha + beta)) / t_0)) / (t_0 / i);
	} else {
		tmp = (i * ((alpha + i) / (beta + (-1.0 + fma(i, 2.0, alpha))))) / (1.0 + t_0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(beta + fma(i, 2.0, alpha))
	tmp = 0.0
	if (alpha <= 2.3e+70)
		tmp = Float64(Float64(Float64(Float64(i / Float64(beta + Float64(1.0 + Float64(i * 2.0)))) * Float64(Float64(i + beta) / Float64(beta + fma(i, 2.0, -1.0)))) * Float64(Float64(i + Float64(alpha + beta)) / t_0)) / Float64(t_0 / i));
	else
		tmp = Float64(Float64(i * Float64(Float64(alpha + i) / Float64(beta + Float64(-1.0 + fma(i, 2.0, alpha))))) / Float64(1.0 + t_0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 2.3e+70], N[(N[(N[(N[(i / N[(beta + N[(1.0 + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / N[(beta + N[(i * 2.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(N[(alpha + i), $MachinePrecision] / N[(beta + N[(-1.0 + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.29999999999999994e70

   1. Initial program 19.0%

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

   3. Simplified 37.7%

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

      1. *-commutative 37.7%

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

      2. fma-udef 37.7%

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

      3. difference-of-sqr--1 37.7%

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

      4. fma-udef 37.7%

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

      5. +-commutative 37.7%

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

      6. *-commutative 37.7%

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

      7. associate-+r+ 37.7%

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

      8. +-commutative 37.7%

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

      9. fma-udef 37.7%

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

      10. +-commutative 37.7%

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

      11. *-commutative 37.7%

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

      12. associate-+r+ 37.7%

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

      13. +-commutative 37.7%

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

   5. Applied egg-rr 42.8%

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

   6. Taylor expanded in alpha around 0 36.8%

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

      1. times-frac 98.8%

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

      2. *-commutative 98.8%

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

      3. associate--l+ 98.8%

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

      4. *-commutative 98.8%

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

   8. Simplified 98.8%

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

   10. Applied egg-rr 98.9%

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

   if 2.29999999999999994e70 < alpha

   1. Initial program 4.9%

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

   2. Taylor expanded in beta around inf 11.3%

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

      1. distribute-rgt-in 11.3%

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

      2. associate-+l+ 11.3%

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

      3. associate-+l+ 11.3%

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

   4. Applied egg-rr 11.3%

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

      1. distribute-rgt-in 11.3%

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

      2. associate-+r+ 11.3%

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

      3. difference-of-sqr-1 11.3%

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

      4. times-frac 16.1%

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

      5. +-commutative 16.1%

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

      6. +-commutative 16.1%

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

      7. associate-+r+ 16.1%

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

      8. *-commutative 16.1%

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

      9. fma-udef 16.1%

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

      10. +-commutative 16.1%

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

      11. sub-neg 16.1%

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

   6. Applied egg-rr 16.1%

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

      1. associate-*l/ 16.2%

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

      2. associate-+l+ 16.2%

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

   8. Simplified 16.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left(\frac{i}{\beta + \left(1 + i \cdot 2\right)} \cdot \frac{i + \beta}{\beta + \mathsf{fma}\left(i, 2, -1\right)}\right) \cdot \frac{i + \left(\alpha + \beta\right)}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}}{\frac{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\beta + \left(-1 + \mathsf{fma}\left(i, 2, \alpha\right)\right)}}{1 + \left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}\\ \end{array} \]

Alternative 2: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \beta + \mathsf{fma}\left(i, 2, \alpha\right)\\ t_1 := \beta + i \cdot 2\\ \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{i}{t_1} \cdot \frac{i + \left(\alpha + \beta\right)}{t_0}\right) \cdot \left(\frac{i}{1 + t_1} \cdot \frac{i + \beta}{\beta + \left(i \cdot 2 + -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{1 + t_0} \cdot \frac{i}{-1 + t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (fma i 2.0 alpha))) (t_1 (+ beta (* i 2.0))))
   (if (<= alpha 2.3e+70)
     (*
      (* (/ i t_1) (/ (+ i (+ alpha beta)) t_0))
      (* (/ i (+ 1.0 t_1)) (/ (+ i beta) (+ beta (+ (* i 2.0) -1.0)))))
     (* (/ (+ alpha i) (+ 1.0 t_0)) (/ i (+ -1.0 t_0))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = beta + fma(i, 2.0, alpha);
	double t_1 = beta + (i * 2.0);
	double tmp;
	if (alpha <= 2.3e+70) {
		tmp = ((i / t_1) * ((i + (alpha + beta)) / t_0)) * ((i / (1.0 + t_1)) * ((i + beta) / (beta + ((i * 2.0) + -1.0))));
	} else {
		tmp = ((alpha + i) / (1.0 + t_0)) * (i / (-1.0 + t_0));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(beta + fma(i, 2.0, alpha))
	t_1 = Float64(beta + Float64(i * 2.0))
	tmp = 0.0
	if (alpha <= 2.3e+70)
		tmp = Float64(Float64(Float64(i / t_1) * Float64(Float64(i + Float64(alpha + beta)) / t_0)) * Float64(Float64(i / Float64(1.0 + t_1)) * Float64(Float64(i + beta) / Float64(beta + Float64(Float64(i * 2.0) + -1.0)))));
	else
		tmp = Float64(Float64(Float64(alpha + i) / Float64(1.0 + t_0)) * Float64(i / Float64(-1.0 + t_0)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 2.3e+70], N[(N[(N[(i / t$95$1), $MachinePrecision] * N[(N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(i / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / N[(beta + N[(N[(i * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.29999999999999994e70

   1. Initial program 19.0%

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

   3. Simplified 37.7%

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

      1. *-commutative 37.7%

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

      2. fma-udef 37.7%

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

      3. difference-of-sqr--1 37.7%

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

      4. fma-udef 37.7%

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

      5. +-commutative 37.7%

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

      6. *-commutative 37.7%

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

      7. associate-+r+ 37.7%

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

      8. +-commutative 37.7%

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

      9. fma-udef 37.7%

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

      10. +-commutative 37.7%

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

      11. *-commutative 37.7%

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

      12. associate-+r+ 37.7%

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

      13. +-commutative 37.7%

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

   5. Applied egg-rr 42.8%

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

   6. Taylor expanded in alpha around 0 36.8%

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

      1. times-frac 98.8%

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

      2. *-commutative 98.8%

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

      3. associate--l+ 98.8%

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

      4. *-commutative 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in alpha around 0 98.8%

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

   if 2.29999999999999994e70 < alpha

   1. Initial program 4.9%

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

   2. Taylor expanded in beta around inf 11.3%

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

      1. distribute-rgt-in 11.3%

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

      2. associate-+l+ 11.3%

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

      3. associate-+l+ 11.3%

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

   4. Applied egg-rr 11.3%

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

      1. *-commutative 11.3%

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

      2. distribute-rgt-in 11.3%

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

      3. associate-+r+ 11.3%

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

      4. difference-of-sqr-1 11.3%

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

      5. times-frac 16.1%

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

      6. +-commutative 16.1%

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

      7. +-commutative 16.1%

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

      8. +-commutative 16.1%

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

      9. associate-+r+ 16.1%

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

      10. *-commutative 16.1%

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

      11. fma-udef 16.1%

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

      12. sub-neg 16.1%

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

   6. Applied egg-rr 16.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{i + \left(\alpha + \beta\right)}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}\right) \cdot \left(\frac{i}{1 + \left(\beta + i \cdot 2\right)} \cdot \frac{i + \beta}{\beta + \left(i \cdot 2 + -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{1 + \left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)} \cdot \frac{i}{-1 + \left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}\\ \end{array} \]

Alternative 3: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \beta + \mathsf{fma}\left(i, 2, \alpha\right)\\ t_1 := \beta + i \cdot 2\\ \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{i}{t_1} \cdot \frac{i + \left(\alpha + \beta\right)}{t_0}\right) \cdot \left(\frac{i}{1 + t_1} \cdot \frac{i + \beta}{\beta + \left(i \cdot 2 + -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\beta + \left(-1 + \mathsf{fma}\left(i, 2, \alpha\right)\right)}}{1 + t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (fma i 2.0 alpha))) (t_1 (+ beta (* i 2.0))))
   (if (<= alpha 2.3e+70)
     (*
      (* (/ i t_1) (/ (+ i (+ alpha beta)) t_0))
      (* (/ i (+ 1.0 t_1)) (/ (+ i beta) (+ beta (+ (* i 2.0) -1.0)))))
     (/
      (* i (/ (+ alpha i) (+ beta (+ -1.0 (fma i 2.0 alpha)))))
      (+ 1.0 t_0)))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = beta + fma(i, 2.0, alpha);
	double t_1 = beta + (i * 2.0);
	double tmp;
	if (alpha <= 2.3e+70) {
		tmp = ((i / t_1) * ((i + (alpha + beta)) / t_0)) * ((i / (1.0 + t_1)) * ((i + beta) / (beta + ((i * 2.0) + -1.0))));
	} else {
		tmp = (i * ((alpha + i) / (beta + (-1.0 + fma(i, 2.0, alpha))))) / (1.0 + t_0);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(beta + fma(i, 2.0, alpha))
	t_1 = Float64(beta + Float64(i * 2.0))
	tmp = 0.0
	if (alpha <= 2.3e+70)
		tmp = Float64(Float64(Float64(i / t_1) * Float64(Float64(i + Float64(alpha + beta)) / t_0)) * Float64(Float64(i / Float64(1.0 + t_1)) * Float64(Float64(i + beta) / Float64(beta + Float64(Float64(i * 2.0) + -1.0)))));
	else
		tmp = Float64(Float64(i * Float64(Float64(alpha + i) / Float64(beta + Float64(-1.0 + fma(i, 2.0, alpha))))) / Float64(1.0 + t_0));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 2.3e+70], N[(N[(N[(i / t$95$1), $MachinePrecision] * N[(N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(i / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / N[(beta + N[(N[(i * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(N[(alpha + i), $MachinePrecision] / N[(beta + N[(-1.0 + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.29999999999999994e70

   1. Initial program 19.0%

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

   3. Simplified 37.7%

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

      1. *-commutative 37.7%

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

      2. fma-udef 37.7%

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

      3. difference-of-sqr--1 37.7%

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

      4. fma-udef 37.7%

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

      5. +-commutative 37.7%

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

      6. *-commutative 37.7%

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

      7. associate-+r+ 37.7%

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

      8. +-commutative 37.7%

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

      9. fma-udef 37.7%

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

      10. +-commutative 37.7%

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

      11. *-commutative 37.7%

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

      12. associate-+r+ 37.7%

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

      13. +-commutative 37.7%

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

   5. Applied egg-rr 42.8%

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

   6. Taylor expanded in alpha around 0 36.8%

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

      1. times-frac 98.8%

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

      2. *-commutative 98.8%

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

      3. associate--l+ 98.8%

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

      4. *-commutative 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in alpha around 0 98.8%

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

   if 2.29999999999999994e70 < alpha

   1. Initial program 4.9%

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

   2. Taylor expanded in beta around inf 11.3%

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

      1. distribute-rgt-in 11.3%

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

      2. associate-+l+ 11.3%

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

      3. associate-+l+ 11.3%

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

   4. Applied egg-rr 11.3%

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

      1. distribute-rgt-in 11.3%

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

      2. associate-+r+ 11.3%

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

      3. difference-of-sqr-1 11.3%

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

      4. times-frac 16.1%

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

      5. +-commutative 16.1%

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

      6. +-commutative 16.1%

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

      7. associate-+r+ 16.1%

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

      8. *-commutative 16.1%

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

      9. fma-udef 16.1%

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

      10. +-commutative 16.1%

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

      11. sub-neg 16.1%

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

   6. Applied egg-rr 16.1%

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

      1. associate-*l/ 16.2%

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

      2. associate-+l+ 16.2%

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

   8. Simplified 16.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.3 \cdot 10^{+70}:\\ \;\;\;\;\left(\frac{i}{\beta + i \cdot 2} \cdot \frac{i + \left(\alpha + \beta\right)}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)}\right) \cdot \left(\frac{i}{1 + \left(\beta + i \cdot 2\right)} \cdot \frac{i + \beta}{\beta + \left(i \cdot 2 + -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\beta + \left(-1 + \mathsf{fma}\left(i, 2, \alpha\right)\right)}}{1 + \left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}\\ \end{array} \]

Alternative 4: 96.9% accurate, 0.4× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* i 2.0))))
   (*
    (* (/ i t_0) (/ (+ i (+ alpha beta)) (+ beta (fma i 2.0 alpha))))
    (* (/ i (+ 1.0 t_0)) (/ (+ i beta) (+ beta (+ (* i 2.0) -1.0)))))))

C

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

Julia

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

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / N[(beta + N[(N[(i * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 15.3%

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

3. Simplified 35.2%

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

   1. *-commutative 35.2%

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

   2. fma-udef 35.2%

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

   3. difference-of-sqr--1 35.2%

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

   4. fma-udef 35.2%

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

   5. +-commutative 35.2%

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

   6. *-commutative 35.2%

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

   7. associate-+r+ 35.2%

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

   8. +-commutative 35.2%

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

   9. fma-udef 35.2%

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

   10. +-commutative 35.2%

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

   11. *-commutative 35.2%

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

   12. associate-+r+ 35.2%

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

   13. +-commutative 35.2%

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

5. Applied egg-rr 41.3%

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

6. Taylor expanded in alpha around 0 30.1%

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

   1. times-frac 83.6%

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

   2. *-commutative 83.6%

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

   3. associate--l+ 83.6%

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

   4. *-commutative 83.6%

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

8. Simplified 83.6%

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

9. Taylor expanded in alpha around 0 82.8%

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

10. Final simplification 82.8%

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

Alternative 5: 82.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := \beta + \mathsf{fma}\left(i, 2, \alpha\right)\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\left(\frac{i}{t_1} \cdot \frac{i + \beta}{t_0}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{1 + t_0} \cdot \frac{i + \beta}{\beta + \left(i \cdot 2 + -1\right)}\right) \cdot \left(\frac{i + \left(\alpha + \beta\right)}{t_1} \cdot \frac{i}{\beta}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* i 2.0))) (t_1 (+ beta (fma i 2.0 alpha))))
   (if (<= beta 2e+134)
     (* (* (/ i t_1) (/ (+ i beta) t_0)) 0.25)
     (*
      (* (/ i (+ 1.0 t_0)) (/ (+ i beta) (+ beta (+ (* i 2.0) -1.0))))
      (* (/ (+ i (+ alpha beta)) t_1) (/ i beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 2.0);
	double t_1 = beta + fma(i, 2.0, alpha);
	double tmp;
	if (beta <= 2e+134) {
		tmp = ((i / t_1) * ((i + beta) / t_0)) * 0.25;
	} else {
		tmp = ((i / (1.0 + t_0)) * ((i + beta) / (beta + ((i * 2.0) + -1.0)))) * (((i + (alpha + beta)) / t_1) * (i / beta));
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(i * 2.0))
	t_1 = Float64(beta + fma(i, 2.0, alpha))
	tmp = 0.0
	if (beta <= 2e+134)
		tmp = Float64(Float64(Float64(i / t_1) * Float64(Float64(i + beta) / t_0)) * 0.25);
	else
		tmp = Float64(Float64(Float64(i / Float64(1.0 + t_0)) * Float64(Float64(i + beta) / Float64(beta + Float64(Float64(i * 2.0) + -1.0)))) * Float64(Float64(Float64(i + Float64(alpha + beta)) / t_1) * Float64(i / beta)));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(beta + N[(i * 2.0 + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 2e+134], N[(N[(N[(i / t$95$1), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(N[(i / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / N[(beta + N[(N[(i * 2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.99999999999999984e134

   1. Initial program 19.9%

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

   3. Simplified 41.7%

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

   4. Taylor expanded in i around inf 78.4%

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

   5. Taylor expanded in alpha around 0 78.3%

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

      1. *-commutative 78.3%

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

   7. Simplified 78.3%

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

   if 1.99999999999999984e134 < beta

   1. Initial program 0.1%

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

   3. Simplified 14.0%

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

      1. *-commutative 14.0%

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

      2. fma-udef 14.0%

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

      3. difference-of-sqr--1 14.0%

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

      4. fma-udef 14.0%

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

      5. +-commutative 14.0%

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

      6. *-commutative 14.0%

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

      7. associate-+r+ 14.0%

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

      8. +-commutative 14.0%

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

      9. fma-udef 14.0%

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

      10. +-commutative 14.0%

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

      11. *-commutative 14.0%

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

      12. associate-+r+ 14.0%

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

      13. +-commutative 14.0%

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

   5. Applied egg-rr 26.6%

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

   6. Taylor expanded in alpha around 0 16.2%

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

      1. times-frac 88.7%

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

      2. *-commutative 88.7%

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

      3. associate--l+ 88.7%

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

      4. *-commutative 88.7%

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

   8. Simplified 88.7%

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

   9. Taylor expanded in beta around inf 68.9%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\left(\frac{i}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{1 + \left(\beta + i \cdot 2\right)} \cdot \frac{i + \beta}{\beta + \left(i \cdot 2 + -1\right)}\right) \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\beta + \mathsf{fma}\left(i, 2, \alpha\right)} \cdot \frac{i}{\beta}\right)\\ \end{array} \]

Alternative 6: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\alpha + \beta\right)\\ t_1 := t_0 \cdot t_0\\ t_2 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_3 := \frac{\frac{t_2 \cdot \left(t_2 + \alpha \cdot \beta\right)}{t_1}}{-1 + t_1}\\ \mathbf{if}\;t_3 \leq 0.1:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ alpha beta)))
        (t_1 (* t_0 t_0))
        (t_2 (* i (+ i (+ alpha beta))))
        (t_3 (/ (/ (* t_2 (+ t_2 (* alpha beta))) t_1) (+ -1.0 t_1))))
   (if (<= t_3 0.1)
     t_3
     (-
      (+ 0.0625 (* 0.0625 (/ (+ (* alpha 2.0) (* beta 2.0)) i)))
      (* 0.125 (/ (+ alpha beta) i))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (alpha + beta);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (i * 2.0d0) + (alpha + beta)
    t_1 = t_0 * t_0
    t_2 = i * (i + (alpha + beta))
    t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / ((-1.0d0) + t_1)
    if (t_3 <= 0.1d0) then
        tmp = t_3
    else
        tmp = (0.0625d0 + (0.0625d0 * (((alpha * 2.0d0) + (beta * 2.0d0)) / i))) - (0.125d0 * ((alpha + beta) / i))
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (alpha + beta);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1);
	double tmp;
	if (t_3 <= 0.1) {
		tmp = t_3;
	} else {
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i));
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	t_0 = (i * 2.0) + (alpha + beta)
	t_1 = t_0 * t_0
	t_2 = i * (i + (alpha + beta))
	t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1)
	tmp = 0
	if t_3 <= 0.1:
		tmp = t_3
	else:
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i))
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(alpha + beta))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(alpha + beta)))
	t_3 = Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(alpha * beta))) / t_1) / Float64(-1.0 + t_1))
	tmp = 0.0
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) - Float64(0.125 * Float64(Float64(alpha + beta) / i)));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = (i * 2.0) + (alpha + beta);
	t_1 = t_0 * t_0;
	t_2 = i * (i + (alpha + beta));
	t_3 = ((t_2 * (t_2 + (alpha * beta))) / t_1) / (-1.0 + t_1);
	tmp = 0.0;
	if (t_3 <= 0.1)
		tmp = t_3;
	else
		tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 * N[(t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.1], t$95$3, N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < 0.10000000000000001

   1. Initial program 99.7%

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

   if 0.10000000000000001 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

   1. Initial program 0.6%

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

   3. Simplified 24.0%

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

   4. Taylor expanded in i around inf 73.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}}{-1 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right)} \leq 0.1:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}}{-1 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i}\\ \end{array} \]

Alternative 7: 77.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \left(0.0625 + 0.0625 \cdot \frac{\alpha \cdot 2 + \beta \cdot 2}{i}\right) - 0.125 \cdot \frac{\alpha + \beta}{i} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (-
  (+ 0.0625 (* 0.0625 (/ (+ (* alpha 2.0) (* beta 2.0)) i)))
  (* 0.125 (/ (+ alpha beta) i))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i));
}

Fortran

NOTE: alpha and beta 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 + (0.0625d0 * (((alpha * 2.0d0) + (beta * 2.0d0)) / i))) - (0.125d0 * ((alpha + beta) / i))
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	return (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i));
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	return (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i))

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	return Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(alpha * 2.0) + Float64(beta * 2.0)) / i))) - Float64(0.125 * Float64(Float64(alpha + beta) / i)))
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta, i)
	tmp = (0.0625 + (0.0625 * (((alpha * 2.0) + (beta * 2.0)) / i))) - (0.125 * ((alpha + beta) / i));
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(alpha * 2.0), $MachinePrecision] + N[(beta * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.3%

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

3. Simplified 35.2%

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

4. Taylor expanded in i around inf 73.5%

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

5. Final simplification 73.5%

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

Alternative 8: 75.8% accurate, 3.1× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (+ (* 0.125 (/ (+ alpha beta) i)) (+ 0.0625 (/ -0.125 (/ i (+ alpha beta))))))

C

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

Fortran

NOTE: alpha and beta 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 = (0.125d0 * ((alpha + beta) / i)) + (0.0625d0 + ((-0.125d0) / (i / (alpha + beta))))
end function

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta, i)
	tmp = (0.125 * ((alpha + beta) / i)) + (0.0625 + (-0.125 / (i / (alpha + beta))));
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := N[(N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] + N[(0.0625 + N[(-0.125 / N[(i / N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.3%

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

3. Simplified 35.2%

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

4. Taylor expanded in i around inf 73.5%

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

   1. cancel-sign-sub-inv 73.5%

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

   2. +-commutative 73.5%

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

   3. associate-+l+ 73.5%

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

   4. associate-*r/ 73.5%

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

   5. distribute-lft-out 73.5%

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

   6. associate-*r* 73.5%

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

   7. metadata-eval 73.5%

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

   8. associate-*r/ 73.5%

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

   9. clear-num 70.3%

      \[\leadsto 0.125 \cdot \frac{\alpha + \beta}{i} + \left(0.0625 + \left(-0.125\right) \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}}\right) \]

   10. un-div-inv 70.3%

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

   11. metadata-eval 70.3%

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

6. Applied egg-rr 70.3%

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

7. Final simplification 70.3%

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

Alternative 9: 75.8% accurate, 3.5× speedup

Math

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

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (+ (* 0.125 (/ (+ alpha beta) i)) (+ 0.0625 (/ -0.125 (/ i beta)))))

C

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

Fortran

NOTE: alpha and beta 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 = (0.125d0 * ((alpha + beta) / i)) + (0.0625d0 + ((-0.125d0) / (i / beta)))
end function

Java

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

Python

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

Julia

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

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp = code(alpha, beta, i)
	tmp = (0.125 * ((alpha + beta) / i)) + (0.0625 + (-0.125 / (i / beta)));
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := N[(N[(0.125 * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision] + N[(0.0625 + N[(-0.125 / N[(i / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.3%

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

3. Simplified 35.2%

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

4. Taylor expanded in i around inf 73.5%

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

   1. cancel-sign-sub-inv 73.5%

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

   2. +-commutative 73.5%

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

   3. associate-+l+ 73.5%

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

   4. associate-*r/ 73.5%

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

   5. distribute-lft-out 73.5%

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

   6. associate-*r* 73.5%

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

   7. metadata-eval 73.5%

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

   8. associate-*r/ 73.5%

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

   9. clear-num 70.3%

      \[\leadsto 0.125 \cdot \frac{\alpha + \beta}{i} + \left(0.0625 + \left(-0.125\right) \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}}\right) \]

   10. un-div-inv 70.3%

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

   11. metadata-eval 70.3%

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

6. Applied egg-rr 70.3%

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

7. Taylor expanded in alpha around 0 67.4%

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

8. Final simplification 67.4%

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

Alternative 10: 74.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+243}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot 0}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6e+243) 0.0625 (/ (* (+ alpha beta) 0.0) i)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6e+243) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + beta) * 0.0) / i;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 6d+243) then
        tmp = 0.0625d0
    else
        tmp = ((alpha + beta) * 0.0d0) / i
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6e+243) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + beta) * 0.0) / i;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6e+243:
		tmp = 0.0625
	else:
		tmp = ((alpha + beta) * 0.0) / i
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6e+243)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + beta) * 0.0) / i);
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6e+243)
		tmp = 0.0625;
	else
		tmp = ((alpha + beta) * 0.0) / i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6e+243], 0.0625, N[(N[(N[(alpha + beta), $MachinePrecision] * 0.0), $MachinePrecision] / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 5.99999999999999969e243

   1. Initial program 16.9%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in i around inf 70.3%

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

   if 5.99999999999999969e243 < beta

   1. Initial program 0.0%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in i around inf 49.4%

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

      1. cancel-sign-sub-inv 49.4%

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

      2. +-commutative 49.4%

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

      3. associate-+l+ 49.4%

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

      4. associate-*r/ 49.4%

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

      5. distribute-lft-out 49.4%

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

      6. associate-*r* 49.4%

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

      7. metadata-eval 49.4%

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

      8. associate-*r/ 49.4%

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

      9. clear-num 36.5%

         \[\leadsto 0.125 \cdot \frac{\alpha + \beta}{i} + \left(0.0625 + \left(-0.125\right) \cdot \color{blue}{\frac{1}{\frac{i}{\alpha + \beta}}}\right) \]

      10. un-div-inv 36.5%

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

      11. metadata-eval 36.5%

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

   6. Applied egg-rr 36.5%

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

   7. Taylor expanded in i around 0 41.1%

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

      1. distribute-rgt-out 41.1%

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

      2. +-commutative 41.1%

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

      3. metadata-eval 41.1%

         \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{0}}{i} \]

   9. Simplified 41.1%

      \[\leadsto \color{blue}{\frac{\left(\beta + \alpha\right) \cdot 0}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+243}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot 0}{i}\\ \end{array} \]

Alternative 11: 71.0% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta 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;
public static double code(double alpha, double beta, double i) {
	return 0.0625;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 15.3%

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

3. Simplified 35.2%

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

4. Taylor expanded in i around inf 64.8%

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

5. Final simplification 64.8%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023305 
(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)))