Octave 3.8, jcobi/4

Percentage Accurate: 16.3% → 97.3%

Time: 24.2s

Alternatives: 9

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: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{i}{\beta + i \cdot 2} \cdot \left(i + \beta\right)}{t_0 + 1} \cdot \frac{i \cdot \frac{\alpha + \left(i + \beta\right)}{t_0}}{t_0 + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma i 2.0 (+ alpha beta))))
   (if (<= alpha 4.3e+222)
     (*
      (/ (* (/ i (+ beta (* i 2.0))) (+ i beta)) (+ t_0 1.0))
      (/ (* i (/ (+ alpha (+ i beta)) t_0)) (+ t_0 -1.0)))
     (* (/ i beta) (/ (+ alpha i) beta)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if (alpha <= 4.3e+222) {
		tmp = (((i / (beta + (i * 2.0))) * (i + beta)) / (t_0 + 1.0)) * ((i * ((alpha + (i + beta)) / t_0)) / (t_0 + -1.0));
	} else {
		tmp = (i / beta) * ((alpha + i) / beta);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (alpha <= 4.3e+222)
		tmp = Float64(Float64(Float64(Float64(i / Float64(beta + Float64(i * 2.0))) * Float64(i + beta)) / Float64(t_0 + 1.0)) * Float64(Float64(i * Float64(Float64(alpha + Float64(i + beta)) / t_0)) / Float64(t_0 + -1.0)));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(alpha + i) / beta));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 4.3e+222], N[(N[(N[(N[(i / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i + beta), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(i * N[(N[(alpha + N[(i + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 4.2999999999999999e222

   1. Initial program 17.6%

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

      1. add-sqr-sqrt 17.6%

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

      2. +-commutative 17.6%

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

      3. +-commutative 17.6%

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

      4. +-commutative 17.6%

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

      5. fma-udef 17.6%

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

      6. *-commutative 17.6%

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

      7. times-frac 34.3%

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

   3. Applied egg-rr 34.3%

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

   4. Taylor expanded in alpha around 0 31.5%

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

      1. difference-of-sqr-1 31.5%

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

      2. times-frac 36.0%

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

      3. +-commutative 36.0%

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

      4. associate-/l* 93.5%

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

      5. associate-/r/ 93.5%

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

      6. *-commutative 93.5%

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

      7. +-commutative 93.5%

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

      8. *-commutative 93.5%

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

      9. fma-udef 93.5%

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

   6. Applied egg-rr 93.5%

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

   if 4.2999999999999999e222 < alpha

   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-/r* 0.0%

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

      2. times-frac 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in beta around inf 10.7%

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

   5. Taylor expanded in beta around inf 10.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 4.3 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{i}{\beta + i \cdot 2} \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i \cdot \frac{\alpha + \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternative 2: 86.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := t_0 + -1\\ t_2 := t_0 + 1\\ \mathbf{if}\;\beta \leq 9 \cdot 10^{+152}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot 0.25 + 0.5 \cdot \left(i + \left(\left(\alpha + \beta\right) \cdot -0.25\right) \cdot \frac{\alpha + \beta}{i}\right)}{t_2} \cdot \frac{\frac{i}{\beta + i \cdot 2} \cdot \left(i + \beta\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_2} \cdot \frac{i}{\frac{t_1}{\alpha + i}}\\ \end{array} \end{array} \]

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 (fma i 2.0 (+ alpha beta))) (t_1 (+ t_0 -1.0)) (t_2 (+ t_0 1.0)))
   (if (<= beta 9e+152)
     (*
      (/
       (+
        (* (+ alpha beta) 0.25)
        (* 0.5 (+ i (* (* (+ alpha beta) -0.25) (/ (+ alpha beta) i)))))
       t_2)
      (/ (* (/ i (+ beta (* i 2.0))) (+ i beta)) t_1))
     (* (/ 1.0 t_2) (/ i (/ t_1 (+ alpha i)))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (alpha + beta));
	double t_1 = t_0 + -1.0;
	double t_2 = t_0 + 1.0;
	double tmp;
	if (beta <= 9e+152) {
		tmp = ((((alpha + beta) * 0.25) + (0.5 * (i + (((alpha + beta) * -0.25) * ((alpha + beta) / i))))) / t_2) * (((i / (beta + (i * 2.0))) * (i + beta)) / t_1);
	} else {
		tmp = (1.0 / t_2) * (i / (t_1 / (alpha + i)));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(alpha + beta))
	t_1 = Float64(t_0 + -1.0)
	t_2 = Float64(t_0 + 1.0)
	tmp = 0.0
	if (beta <= 9e+152)
		tmp = Float64(Float64(Float64(Float64(Float64(alpha + beta) * 0.25) + Float64(0.5 * Float64(i + Float64(Float64(Float64(alpha + beta) * -0.25) * Float64(Float64(alpha + beta) / i))))) / t_2) * Float64(Float64(Float64(i / Float64(beta + Float64(i * 2.0))) * Float64(i + beta)) / t_1));
	else
		tmp = Float64(Float64(1.0 / t_2) * Float64(i / Float64(t_1 / Float64(alpha + i))));
	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[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + 1.0), $MachinePrecision]}, If[LessEqual[beta, 9e+152], N[(N[(N[(N[(N[(alpha + beta), $MachinePrecision] * 0.25), $MachinePrecision] + N[(0.5 * N[(i + N[(N[(N[(alpha + beta), $MachinePrecision] * -0.25), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(N[(i / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i + beta), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(i / N[(t$95$1 / N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.0000000000000002e152

   1. Initial program 19.7%

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

      1. add-sqr-sqrt 19.7%

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

      2. +-commutative 19.7%

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

      3. +-commutative 19.7%

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

      4. +-commutative 19.7%

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

      5. fma-udef 19.7%

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

      6. *-commutative 19.7%

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

      7. times-frac 35.7%

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

   3. Applied egg-rr 35.7%

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

   4. Taylor expanded in alpha around 0 34.5%

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

   5. Taylor expanded in i around -inf 29.2%

      \[\leadsto \frac{\frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i} \cdot \color{blue}{\left(-1 \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) - -0.25 \cdot \left(\alpha + \beta\right)\right) + \left(0.5 \cdot i + 0.5 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) - -0.25 \cdot \left(\alpha + \beta\right)\right)}{i}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   6. Step-by-step derivation

      1. mul-1-neg 29.2%

         \[\leadsto \frac{\frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i} \cdot \left(\color{blue}{\left(-\left(-0.5 \cdot \left(\alpha + \beta\right) - -0.25 \cdot \left(\alpha + \beta\right)\right)\right)} + \left(0.5 \cdot i + 0.5 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) - -0.25 \cdot \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. distribute-rgt-out-- 29.2%

         \[\leadsto \frac{\frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i} \cdot \left(\left(-\color{blue}{\left(\alpha + \beta\right) \cdot \left(-0.5 - -0.25\right)}\right) + \left(0.5 \cdot i + 0.5 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) - -0.25 \cdot \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      3. metadata-eval 29.2%

         \[\leadsto \frac{\frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i} \cdot \left(\left(-\left(\alpha + \beta\right) \cdot \color{blue}{-0.25}\right) + \left(0.5 \cdot i + 0.5 \cdot \frac{\left(\alpha + \beta\right) \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) - -0.25 \cdot \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      4. distribute-lft-out 29.2%

         \[\leadsto \frac{\frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i} \cdot \left(\left(-\left(\alpha + \beta\right) \cdot -0.25\right) + \color{blue}{0.5 \cdot \left(i + \frac{\left(\alpha + \beta\right) \cdot \left(-0.5 \cdot \left(\alpha + \beta\right) - -0.25 \cdot \left(\alpha + \beta\right)\right)}{i}\right)}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      5. associate-/l* 29.2%

         \[\leadsto \frac{\frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i} \cdot \left(\left(-\left(\alpha + \beta\right) \cdot -0.25\right) + 0.5 \cdot \left(i + \color{blue}{\frac{\alpha + \beta}{\frac{i}{-0.5 \cdot \left(\alpha + \beta\right) - -0.25 \cdot \left(\alpha + \beta\right)}}}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      6. distribute-rgt-out-- 29.2%

         \[\leadsto \frac{\frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i} \cdot \left(\left(-\left(\alpha + \beta\right) \cdot -0.25\right) + 0.5 \cdot \left(i + \frac{\alpha + \beta}{\frac{i}{\color{blue}{\left(\alpha + \beta\right) \cdot \left(-0.5 - -0.25\right)}}}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      7. metadata-eval 29.2%

         \[\leadsto \frac{\frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i} \cdot \left(\left(-\left(\alpha + \beta\right) \cdot -0.25\right) + 0.5 \cdot \left(i + \frac{\alpha + \beta}{\frac{i}{\left(\alpha + \beta\right) \cdot \color{blue}{-0.25}}}\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   7. Simplified 29.2%

      \[\leadsto \frac{\frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i} \cdot \color{blue}{\left(\left(-\left(\alpha + \beta\right) \cdot -0.25\right) + 0.5 \cdot \left(i + \frac{\alpha + \beta}{\frac{i}{\left(\alpha + \beta\right) \cdot -0.25}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   9. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{\frac{\left(\beta + \alpha\right) \cdot 0.25 + 0.5 \cdot \left(i + \left(\left(\beta + \alpha\right) \cdot -0.25\right) \cdot \frac{\beta + \alpha}{i}\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\frac{i}{\beta + i \cdot 2} \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}} \]

   if 9.0000000000000002e152 < 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. Taylor expanded in beta around inf 28.7%

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

      1. *-un-lft-identity 28.7%

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

      2. difference-of-sqr-1 28.7%

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

      3. times-frac 47.0%

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

      4. +-commutative 47.0%

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

      5. *-commutative 47.0%

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

      6. fma-udef 47.0%

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

      7. +-commutative 47.0%

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

      8. +-commutative 47.0%

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

      9. *-commutative 47.0%

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

      10. fma-udef 47.0%

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

      11. sub-neg 47.0%

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

      12. metadata-eval 47.0%

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

   4. Applied egg-rr 47.0%

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

      1. associate-/l* 78.2%

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

      2. +-commutative 78.2%

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

   6. Simplified 78.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9 \cdot 10^{+152}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot 0.25 + 0.5 \cdot \left(i + \left(\left(\alpha + \beta\right) \cdot -0.25\right) \cdot \frac{\alpha + \beta}{i}\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{i}{\beta + i \cdot 2} \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}{\alpha + i}}\\ \end{array} \]

Alternative 3: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 3.45 \cdot 10^{+140}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{t_0} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 + 1} \cdot \frac{i}{\frac{t_0 + -1}{\alpha + i}}\\ \end{array} \end{array} \]

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 (fma i 2.0 (+ alpha beta))))
   (if (<= beta 3.45e+140)
     (* 0.25 (* (/ i t_0) (/ (+ i beta) (+ beta (* i 2.0)))))
     (* (/ 1.0 (+ t_0 1.0)) (/ i (/ (+ t_0 -1.0) (+ alpha i)))))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if (beta <= 3.45e+140) {
		tmp = 0.25 * ((i / t_0) * ((i + beta) / (beta + (i * 2.0))));
	} else {
		tmp = (1.0 / (t_0 + 1.0)) * (i / ((t_0 + -1.0) / (alpha + i)));
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 3.45e+140)
		tmp = Float64(0.25 * Float64(Float64(i / t_0) * Float64(Float64(i + beta) / Float64(beta + Float64(i * 2.0)))));
	else
		tmp = Float64(Float64(1.0 / Float64(t_0 + 1.0)) * Float64(i / Float64(Float64(t_0 + -1.0) / Float64(alpha + i))));
	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[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.45e+140], N[(0.25 * N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.4500000000000001e140

   1. Initial program 19.6%

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

      1. associate-/r* 17.8%

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

      2. times-frac 35.5%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in i around inf 82.5%

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

   5. Taylor expanded in alpha around 0 82.2%

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

   if 3.4500000000000001e140 < beta

   1. Initial program 1.9%

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

   2. Taylor expanded in beta around inf 28.2%

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

      1. *-un-lft-identity 28.2%

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

      2. difference-of-sqr-1 28.2%

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

      3. times-frac 45.0%

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

      4. +-commutative 45.0%

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

      5. *-commutative 45.0%

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

      6. fma-udef 45.0%

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

      7. +-commutative 45.0%

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

      8. +-commutative 45.0%

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

      9. *-commutative 45.0%

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

      10. fma-udef 45.0%

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

      11. sub-neg 45.0%

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

      12. metadata-eval 45.0%

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

   4. Applied egg-rr 45.0%

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

      1. associate-/l* 74.0%

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

      2. +-commutative 74.0%

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

   6. Simplified 74.0%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.45 \cdot 10^{+140}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}{\alpha + i}}\\ \end{array} \]

Alternative 4: 86.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+140}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{t_0} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{t_0 + 1}}{t_0 + -1}\\ \end{array} \end{array} \]

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 (fma i 2.0 (+ alpha beta))))
   (if (<= beta 1.55e+140)
     (* 0.25 (* (/ i t_0) (/ (+ i beta) (+ beta (* i 2.0)))))
     (/ (* i (/ (+ alpha i) (+ t_0 1.0))) (+ t_0 -1.0)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if (beta <= 1.55e+140) {
		tmp = 0.25 * ((i / t_0) * ((i + beta) / (beta + (i * 2.0))));
	} else {
		tmp = (i * ((alpha + i) / (t_0 + 1.0))) / (t_0 + -1.0);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (beta <= 1.55e+140)
		tmp = Float64(0.25 * Float64(Float64(i / t_0) * Float64(Float64(i + beta) / Float64(beta + Float64(i * 2.0)))));
	else
		tmp = Float64(Float64(i * Float64(Float64(alpha + i) / Float64(t_0 + 1.0))) / Float64(t_0 + -1.0));
	end
	return tmp
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.55e+140], N[(0.25 * N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * N[(N[(alpha + i), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.55e140

   1. Initial program 19.6%

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

      1. associate-/r* 17.8%

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

      2. times-frac 35.5%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in i around inf 82.5%

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

   5. Taylor expanded in alpha around 0 82.2%

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

   if 1.55e140 < beta

   1. Initial program 1.9%

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

   2. Taylor expanded in beta around inf 28.2%

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

      1. *-commutative 28.2%

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

      2. difference-of-sqr-1 28.2%

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

      3. times-frac 74.0%

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

      4. +-commutative 74.0%

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

      5. *-commutative 74.0%

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

      6. fma-udef 74.0%

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

      7. +-commutative 74.0%

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

      8. +-commutative 74.0%

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

      9. *-commutative 74.0%

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

      10. fma-udef 74.0%

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

      11. sub-neg 74.0%

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

      12. metadata-eval 74.0%

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

   4. Applied egg-rr 74.0%

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

      1. associate-*r/ 74.0%

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

      2. +-commutative 74.0%

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

   6. Simplified 74.0%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.55 \cdot 10^{+140}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{\alpha + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}\\ \end{array} \]

Alternative 5: 85.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \beta}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta}}{\frac{t_0}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* i 2.0))))
   (if (<= beta 1.6e+154)
     (* 0.25 (* (/ i (fma i 2.0 (+ alpha beta))) (/ (+ i beta) t_0)))
     (/ (/ (+ alpha i) beta) (/ t_0 i)))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 2.0);
	double tmp;
	if (beta <= 1.6e+154) {
		tmp = 0.25 * ((i / fma(i, 2.0, (alpha + beta))) * ((i + beta) / t_0));
	} else {
		tmp = ((alpha + i) / beta) / (t_0 / i);
	}
	return tmp;
}

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(i * 2.0))
	tmp = 0.0
	if (beta <= 1.6e+154)
		tmp = Float64(0.25 * Float64(Float64(i / fma(i, 2.0, Float64(alpha + beta))) * Float64(Float64(i + beta) / t_0)));
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) / Float64(t_0 / i));
	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[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.6e+154], N[(0.25 * N[(N[(i / N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] / N[(t$95$0 / i), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.6e154

   1. Initial program 19.7%

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

      1. associate-/r* 17.4%

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

      2. times-frac 35.6%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in i around inf 82.3%

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

   5. Taylor expanded in alpha around 0 82.0%

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

   if 1.6e154 < 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-/r* 0.0%

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

      2. times-frac 11.3%

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

   3. Simplified 11.3%

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

   4. Taylor expanded in beta around inf 76.5%

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

   5. Taylor expanded in i around 0 76.5%

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

      1. *-rgt-identity 76.5%

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

      2. clear-num 76.5%

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

      3. associate-*l/ 76.5%

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

      4. *-un-lft-identity 76.5%

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

      5. +-commutative 76.5%

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

   7. Applied egg-rr 76.5%

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

   8. Taylor expanded in alpha around 0 76.4%

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

      1. *-commutative 76.4%

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

   10. Simplified 76.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \beta}{\beta + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta}}{\frac{\beta + i \cdot 2}{i}}\\ \end{array} \]

Alternative 6: 85.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta}}{\frac{\beta + i \cdot 2}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.5e+154)
   0.0625
   (/ (/ (+ alpha i) beta) (/ (+ beta (* i 2.0)) i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.5e+154) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) / ((beta + (i * 2.0)) / 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 <= 1.5d+154) then
        tmp = 0.0625d0
    else
        tmp = ((alpha + i) / beta) / ((beta + (i * 2.0d0)) / 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 <= 1.5e+154) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) / ((beta + (i * 2.0)) / i);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.5e+154:
		tmp = 0.0625
	else:
		tmp = ((alpha + i) / beta) / ((beta + (i * 2.0)) / i)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.5e+154)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) / Float64(Float64(beta + Float64(i * 2.0)) / 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 <= 1.5e+154)
		tmp = 0.0625;
	else
		tmp = ((alpha + i) / beta) / ((beta + (i * 2.0)) / 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, 1.5e+154], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] / N[(N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.50000000000000013e154

   1. Initial program 19.7%

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

      1. associate-/r* 17.4%

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

      2. times-frac 35.6%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in i around inf 81.9%

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

   if 1.50000000000000013e154 < 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-/r* 0.0%

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

      2. times-frac 11.3%

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

   3. Simplified 11.3%

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

   4. Taylor expanded in beta around inf 76.5%

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

   5. Taylor expanded in i around 0 76.5%

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

      1. *-rgt-identity 76.5%

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

      2. clear-num 76.5%

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

      3. associate-*l/ 76.5%

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

      4. *-un-lft-identity 76.5%

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

      5. +-commutative 76.5%

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

   7. Applied egg-rr 76.5%

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

   8. Taylor expanded in alpha around 0 76.4%

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

      1. *-commutative 76.4%

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

   10. Simplified 76.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta}}{\frac{\beta + i \cdot 2}{i}}\\ \end{array} \]

Alternative 7: 85.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+154}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 4.8e+154) 0.0625 (* (/ i beta) (/ (+ alpha i) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.8e+154) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((alpha + i) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 4.8d+154) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * ((alpha + i) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 4.8e+154) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * ((alpha + i) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 4.8e+154:
		tmp = 0.0625
	else:
		tmp = (i / beta) * ((alpha + i) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 4.8e+154)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(alpha + i) / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 4.8e+154)
		tmp = 0.0625;
	else
		tmp = (i / beta) * ((alpha + i) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 4.8e+154], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.8000000000000003e154

   1. Initial program 19.7%

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

      1. associate-/r* 17.4%

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

      2. times-frac 35.6%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in i around inf 81.9%

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

   if 4.8000000000000003e154 < 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-/r* 0.0%

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

      2. times-frac 11.3%

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

   3. Simplified 11.3%

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

   4. Taylor expanded in beta around inf 76.5%

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

   5. Taylor expanded in beta around inf 76.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+154}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternative 8: 85.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+151}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.5e+151) 0.0625 (/ (/ (+ alpha i) beta) (/ beta i))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.5e+151) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) / (beta / 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 <= 6.5d+151) then
        tmp = 0.0625d0
    else
        tmp = ((alpha + i) / beta) / (beta / 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 <= 6.5e+151) {
		tmp = 0.0625;
	} else {
		tmp = ((alpha + i) / beta) / (beta / i);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.5e+151:
		tmp = 0.0625
	else:
		tmp = ((alpha + i) / beta) / (beta / i)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.5e+151)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) / Float64(beta / 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 <= 6.5e+151)
		tmp = 0.0625;
	else
		tmp = ((alpha + i) / beta) / (beta / 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, 6.5e+151], 0.0625, N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.5000000000000002e151

   1. Initial program 19.7%

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

      1. associate-/r* 17.4%

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

      2. times-frac 35.6%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in i around inf 81.9%

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

   if 6.5000000000000002e151 < 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-/r* 0.0%

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

      2. times-frac 11.3%

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

   3. Simplified 11.3%

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

   4. Taylor expanded in beta around inf 76.5%

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

   5. Taylor expanded in i around 0 76.5%

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

      1. *-rgt-identity 76.5%

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

      2. clear-num 76.5%

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

      3. associate-*l/ 76.5%

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

      4. *-un-lft-identity 76.5%

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

      5. +-commutative 76.5%

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

   7. Applied egg-rr 76.5%

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

   8. Taylor expanded in beta around inf 76.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+151}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\alpha + i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 9: 70.5% 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 16.0%

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

   1. associate-/r* 14.1%

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

   2. times-frac 31.1%

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

3. Simplified 31.1%

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

4. Taylor expanded in i around inf 70.1%

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

5. Final simplification 70.1%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023336 
(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)))