Octave 3.8, jcobi/4

Percentage Accurate: 16.5% → 84.3%

Time: 18.9s

Alternatives: 5

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: 16.5% 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: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;i \leq 8.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta + i \cdot 2}{i + \beta}} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \left(\beta + \alpha\right)}}}{t_0 \cdot t_0 + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha))))
   (if (<= i 8.5e+150)
     (/
      (*
       (/ i (/ (+ beta (* i 2.0)) (+ i beta)))
       (/ i (/ (fma i 2.0 (+ beta alpha)) (+ i (+ beta alpha)))))
      (+ (* t_0 t_0) -1.0))
     0.0625)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double tmp;
	if (i <= 8.5e+150) {
		tmp = ((i / ((beta + (i * 2.0)) / (i + beta))) * (i / (fma(i, 2.0, (beta + alpha)) / (i + (beta + alpha))))) / ((t_0 * t_0) + -1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	tmp = 0.0
	if (i <= 8.5e+150)
		tmp = Float64(Float64(Float64(i / Float64(Float64(beta + Float64(i * 2.0)) / Float64(i + beta))) * Float64(i / Float64(fma(i, 2.0, Float64(beta + alpha)) / Float64(i + Float64(beta + alpha))))) / Float64(Float64(t_0 * t_0) + -1.0));
	else
		tmp = 0.0625;
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i 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[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[i, 8.5e+150], N[(N[(N[(i / N[(N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision] / N[(i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 2 regimes

2. if i < 8.4999999999999999e150

   1. Initial program 39.5%

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

      1. times-frac 80.1%

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

      2. +-commutative 80.1%

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

      3. +-commutative 80.1%

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

      4. +-commutative 80.1%

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

      5. *-commutative 80.1%

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

      6. fma-udef 80.1%

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

      7. +-commutative 80.1%

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

      8. +-commutative 80.1%

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

      9. +-commutative 80.1%

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

      10. fma-def 80.1%

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

      11. +-commutative 80.1%

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

      12. +-commutative 80.1%

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

      13. *-commutative 80.1%

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

      14. fma-udef 80.1%

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

      15. +-commutative 80.1%

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

   3. Applied egg-rr 80.1%

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

      1. *-commutative 80.1%

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

      2. +-commutative 80.1%

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

      3. +-commutative 80.1%

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

      4. *-commutative 80.1%

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

      5. +-commutative 80.1%

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

      6. associate-/l* 80.3%

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

      7. +-commutative 80.3%

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

      8. +-commutative 80.3%

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

      9. +-commutative 80.3%

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

   5. Simplified 80.3%

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

   6. Taylor expanded in alpha around 0 77.5%

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

      1. associate-/l* 79.2%

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

      2. *-commutative 79.2%

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

   8. Simplified 79.2%

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

   if 8.4999999999999999e150 < i

   1. Initial program 0.1%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.1%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in i around inf 82.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 8.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta + i \cdot 2}{i + \beta}} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \left(\beta + \alpha\right)}}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 2: 81.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (i * 2.0d0) + (beta + alpha)
    t_1 = t_0 * t_0
    t_2 = i * (i + (beta + alpha))
    t_3 = ((t_2 * (t_2 + (beta * alpha))) / t_1) / (t_1 + (-1.0d0))
    if (t_3 <= 0.1d0) then
        tmp = t_3
    else
        tmp = (0.0625d0 + (0.0625d0 * ((beta * 2.0d0) / i))) + ((-0.125d0) * ((beta + alpha) / i))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 * N[(t$95$2 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 0.1], t$95$3, N[(N[(0.0625 + N[(0.0625 * N[(N[(beta * 2.0), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.125 * N[(N[(beta + alpha), $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 0.8%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 8.7%

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

   3. Simplified 8.7%

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

   4. Taylor expanded in i around inf 76.3%

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

      1. cancel-sign-sub-inv 76.3%

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

      2. distribute-lft-out 76.3%

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

      3. metadata-eval 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in alpha around 0 72.6%

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

      1. associate-*r/ 72.2%

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

      2. *-commutative 72.2%

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

   9. Simplified 72.2%

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

4. Final simplification 77.5%

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

Alternative 3: 77.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (0.0625d0 + (0.0625d0 * ((beta * 2.0d0) / i))) + ((-0.125d0) * (beta / i))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.1%

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

   1. associate-/l/ 16.9%

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

   2. associate-*l* 16.8%

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

   3. times-frac 26.4%

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

3. Simplified 26.4%

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

4. Taylor expanded in i around inf 76.3%

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

   1. cancel-sign-sub-inv 76.3%

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

   2. distribute-lft-out 76.3%

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

   3. metadata-eval 76.3%

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

6. Simplified 76.3%

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

7. Taylor expanded in alpha around 0 73.3%

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

   1. associate-*r/ 72.9%

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

   2. *-commutative 72.9%

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

9. Simplified 72.9%

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

10. Taylor expanded in alpha around 0 74.7%

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

11. Final simplification 74.7%

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

Alternative 4: 74.8% accurate, 10.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+245}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.8e+245) 0.0625 (/ 0.0 i)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.8e+245) {
		tmp = 0.0625;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 6.8d+245) then
        tmp = 0.0625d0
    else
        tmp = 0.0d0 / i
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.8e+245) {
		tmp = 0.0625;
	} else {
		tmp = 0.0 / i;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.8e+245:
		tmp = 0.0625
	else:
		tmp = 0.0 / i
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.8e+245)
		tmp = 0.0625;
	else
		tmp = Float64(0.0 / i);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6.8e+245)
		tmp = 0.0625;
	else
		tmp = 0.0 / i;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6.8e+245], 0.0625, N[(0.0 / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.79999999999999996e245

   1. Initial program 21.1%

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

      1. associate-/l/ 17.7%

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

      2. associate-*l* 17.7%

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

      3. times-frac 27.7%

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

   3. Simplified 27.7%

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

   4. Taylor expanded in i around inf 75.6%

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

   if 6.79999999999999996e245 < beta

   1. Initial program 0.0%

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

      1. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in i around inf 24.8%

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

      1. cancel-sign-sub-inv 24.8%

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

      2. distribute-lft-out 24.8%

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

      3. metadata-eval 24.8%

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

   6. Simplified 24.8%

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

      1. expm1-log1p-u 6.8%

         \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\alpha + \beta\right)}{i}\right) + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{i}\right)\right)} \]

   8. Applied egg-rr 6.8%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\alpha + \beta\right)}{i}\right) + -0.125 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\alpha + \beta}{i}\right)\right)} \]

   9. Taylor expanded in i around 0 29.0%

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

       1. distribute-rgt-out 29.0%

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

       2. metadata-eval 29.0%

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

       3. mul0-rgt 29.0%

          \[\leadsto \frac{\color{blue}{0}}{i} \]

   11. Simplified 29.0%

       \[\leadsto \color{blue}{\frac{0}{i}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.8 \cdot 10^{+245}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]

Alternative 5: 71.4% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.1%

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

   1. associate-/l/ 16.9%

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

   2. associate-*l* 16.8%

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

   3. times-frac 26.4%

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

3. Simplified 26.4%

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

4. Taylor expanded in i around inf 72.4%

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

5. Final simplification 72.4%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023333 
(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)))