Octave 3.8, jcobi/4

Average Error: 53.6 → 10.5

Time: 1.4min

Precision: binary64

Cost: 7108

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t_0 \cdot t_0\\ t_2 := t_1 - 1\\ t_3 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_4 := i \cdot \left(\beta + \left(i + \alpha\right)\right)\\ \mathbf{if}\;\frac{\frac{t_3 \cdot \left(\beta \cdot \alpha + t_3\right)}{t_1}}{t_2} \leq \infty:\\ \;\;\;\;\frac{t_4 \cdot \frac{\beta \cdot \alpha + t_4}{\left(\alpha + \frac{\beta + \left(\beta + i \cdot 4\right)}{2}\right) \cdot \left(\alpha + \left(i + \left(\beta + i\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + \frac{\frac{\beta}{i}}{16}\right) + \frac{-0.125 \cdot \frac{\beta}{i}}{2}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (/
   (* (* 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)))

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (* t_0 t_0))
        (t_2 (- t_1 1.0))
        (t_3 (* i (+ (+ alpha beta) i)))
        (t_4 (* i (+ beta (+ i alpha)))))
   (if (<= (/ (/ (* t_3 (+ (* beta alpha) t_3)) t_1) t_2) INFINITY)
     (/
      (*
       t_4
       (/
        (+ (* beta alpha) t_4)
        (*
         (+ alpha (/ (+ beta (+ beta (* i 4.0))) 2.0))
         (+ alpha (+ i (+ beta i))))))
      t_2)
     (+ (+ 0.0625 (/ (/ beta i) 16.0)) (/ (* -0.125 (/ beta i)) 2.0)))))

C

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

↓

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 - 1.0;
	double t_3 = i * ((alpha + beta) + i);
	double t_4 = i * (beta + (i + alpha));
	double tmp;
	if ((((t_3 * ((beta * alpha) + t_3)) / t_1) / t_2) <= ((double) INFINITY)) {
		tmp = (t_4 * (((beta * alpha) + t_4) / ((alpha + ((beta + (beta + (i * 4.0))) / 2.0)) * (alpha + (i + (beta + i)))))) / t_2;
	} else {
		tmp = (0.0625 + ((beta / i) / 16.0)) + ((-0.125 * (beta / i)) / 2.0);
	}
	return tmp;
}

Java

public static double code(double alpha, double beta, double i) {
	return (((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);
}

↓

public static double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = t_0 * t_0;
	double t_2 = t_1 - 1.0;
	double t_3 = i * ((alpha + beta) + i);
	double t_4 = i * (beta + (i + alpha));
	double tmp;
	if ((((t_3 * ((beta * alpha) + t_3)) / t_1) / t_2) <= Double.POSITIVE_INFINITY) {
		tmp = (t_4 * (((beta * alpha) + t_4) / ((alpha + ((beta + (beta + (i * 4.0))) / 2.0)) * (alpha + (i + (beta + i)))))) / t_2;
	} else {
		tmp = (0.0625 + ((beta / i) / 16.0)) + ((-0.125 * (beta / i)) / 2.0);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	return (((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)

↓

def code(alpha, beta, i):
	t_0 = (alpha + beta) + (2.0 * i)
	t_1 = t_0 * t_0
	t_2 = t_1 - 1.0
	t_3 = i * ((alpha + beta) + i)
	t_4 = i * (beta + (i + alpha))
	tmp = 0
	if (((t_3 * ((beta * alpha) + t_3)) / t_1) / t_2) <= math.inf:
		tmp = (t_4 * (((beta * alpha) + t_4) / ((alpha + ((beta + (beta + (i * 4.0))) / 2.0)) * (alpha + (i + (beta + i)))))) / t_2
	else:
		tmp = (0.0625 + ((beta / i) / 16.0)) + ((-0.125 * (beta / i)) / 2.0)
	return tmp

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(i * Float64(Float64(alpha + beta) + i)) * Float64(Float64(beta * alpha) + Float64(i * Float64(Float64(alpha + beta) + i)))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(Float64(alpha + beta) + Float64(2.0 * i)))) / Float64(Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(Float64(alpha + beta) + Float64(2.0 * i))) - 1.0))
end

↓

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(t_1 - 1.0)
	t_3 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_4 = Float64(i * Float64(beta + Float64(i + alpha)))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 * Float64(Float64(beta * alpha) + t_3)) / t_1) / t_2) <= Inf)
		tmp = Float64(Float64(t_4 * Float64(Float64(Float64(beta * alpha) + t_4) / Float64(Float64(alpha + Float64(Float64(beta + Float64(beta + Float64(i * 4.0))) / 2.0)) * Float64(alpha + Float64(i + Float64(beta + i)))))) / t_2);
	else
		tmp = Float64(Float64(0.0625 + Float64(Float64(beta / i) / 16.0)) + Float64(Float64(-0.125 * Float64(beta / i)) / 2.0));
	end
	return tmp
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (((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);
end

↓

function tmp_2 = code(alpha, beta, i)
	t_0 = (alpha + beta) + (2.0 * i);
	t_1 = t_0 * t_0;
	t_2 = t_1 - 1.0;
	t_3 = i * ((alpha + beta) + i);
	t_4 = i * (beta + (i + alpha));
	tmp = 0.0;
	if ((((t_3 * ((beta * alpha) + t_3)) / t_1) / t_2) <= Inf)
		tmp = (t_4 * (((beta * alpha) + t_4) / ((alpha + ((beta + (beta + (i * 4.0))) / 2.0)) * (alpha + (i + (beta + i)))))) / t_2;
	else
		tmp = (0.0625 + ((beta / i) / 16.0)) + ((-0.125 * (beta / i)) / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] * N[(N[(beta * alpha), $MachinePrecision] + N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(i * N[(beta + N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 * N[(N[(beta * alpha), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], Infinity], N[(N[(t$95$4 * N[(N[(N[(beta * alpha), $MachinePrecision] + t$95$4), $MachinePrecision] / N[(N[(alpha + N[(N[(beta + N[(beta + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(i + N[(beta + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(N[(beta / i), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < +inf.0

   1. Initial program 34.3

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

   2. Applied egg-rr 0.2

      \[\leadsto \frac{\color{blue}{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\alpha + \left(\beta + \left(i + i\right)\right)}}{\alpha + \left(\beta + \left(i + i\right)\right)} + 0}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   3. Simplified 0.2

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

      [Start] 0.2	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\alpha + \left(\beta + \left(i + i\right)\right)}}{\alpha + \left(\beta + \left(i + i\right)\right)} + 0}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-3 [=>] 0.2	\[ \frac{\color{blue}{0 + \left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\alpha + \left(\beta + \left(i + i\right)\right)}}{\alpha + \left(\beta + \left(i + i\right)\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-6 [=>] 0.2	\[ \frac{\color{blue}{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\alpha + \left(\beta + \left(i + i\right)\right)}}{\alpha + \left(\beta + \left(i + i\right)\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-47 [=>] 0.2	\[ \frac{\left(i \cdot \color{blue}{\left(\beta + \left(\alpha + i\right)\right)}\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\alpha + \left(\beta + \left(i + i\right)\right)}}{\alpha + \left(\beta + \left(i + i\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-3 [<=] 0.2	\[ \frac{\left(i \cdot \left(\beta + \color{blue}{\left(i + \alpha\right)}\right)\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\alpha + \left(\beta + \left(i + i\right)\right)}}{\alpha + \left(\beta + \left(i + i\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-53 [=>] 0.2	\[ \frac{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \color{blue}{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\left(\alpha + \left(\beta + \left(i + i\right)\right)\right) \cdot \left(\alpha + \left(\beta + \left(i + i\right)\right)\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-1 [=>] 0.2	\[ \frac{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \color{blue}{\beta \cdot \alpha}}{\left(\alpha + \left(\beta + \left(i + i\right)\right)\right) \cdot \left(\alpha + \left(\beta + \left(i + i\right)\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-3 [=>] 0.2	\[ \frac{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\color{blue}{\beta \cdot \alpha + i \cdot \left(i + \left(\alpha + \beta\right)\right)}}{\left(\alpha + \left(\beta + \left(i + i\right)\right)\right) \cdot \left(\alpha + \left(\beta + \left(i + i\right)\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-47 [=>] 0.2	\[ \frac{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\beta \cdot \alpha + i \cdot \color{blue}{\left(\beta + \left(\alpha + i\right)\right)}}{\left(\alpha + \left(\beta + \left(i + i\right)\right)\right) \cdot \left(\alpha + \left(\beta + \left(i + i\right)\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-3 [<=] 0.2	\[ \frac{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\beta \cdot \alpha + i \cdot \left(\beta + \color{blue}{\left(i + \alpha\right)}\right)}{\left(\alpha + \left(\beta + \left(i + i\right)\right)\right) \cdot \left(\alpha + \left(\beta + \left(i + i\right)\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-47 [=>] 0.2	\[ \frac{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\left(\alpha + \color{blue}{\left(i + \left(i + \beta\right)\right)}\right) \cdot \left(\alpha + \left(\beta + \left(i + i\right)\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-3 [<=] 0.2	\[ \frac{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\left(\alpha + \left(i + \color{blue}{\left(\beta + i\right)}\right)\right) \cdot \left(\alpha + \left(\beta + \left(i + i\right)\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-47 [=>] 0.2	\[ \frac{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\left(\alpha + \left(i + \left(\beta + i\right)\right)\right) \cdot \left(\alpha + \color{blue}{\left(i + \left(i + \beta\right)\right)}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-3 [<=] 0.2	\[ \frac{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\left(\alpha + \left(i + \left(\beta + i\right)\right)\right) \cdot \left(\alpha + \left(i + \color{blue}{\left(\beta + i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Applied egg-rr 0.2

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

   5. Simplified 0.2

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

      [Start] 0.2	\[ \frac{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\left(\alpha + \frac{\left(\beta + \beta\right) + i \cdot 4}{2}\right) \cdot \left(\alpha + \left(i + \left(\beta + i\right)\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-3 [=>] 0.2	\[ \frac{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\left(\alpha + \frac{\color{blue}{i \cdot 4 + \left(\beta + \beta\right)}}{2}\right) \cdot \left(\alpha + \left(i + \left(\beta + i\right)\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-47 [=>] 0.2	\[ \frac{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\left(\alpha + \frac{\color{blue}{\beta + \left(\beta + i \cdot 4\right)}}{2}\right) \cdot \left(\alpha + \left(i + \left(\beta + i\right)\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational_best-simplify-53 [=>] 64.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      rational_best-simplify-1 [=>] 64.0	\[ \frac{\color{blue}{\left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)} \]
      rational_best-simplify-1 [=>] 64.0	\[ \frac{\left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)} \]
      rational_best-simplify-50 [=>] 64.0	\[ \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)} \]
      rational_best-simplify-3 [=>] 64.0	\[ \frac{i \cdot \left(\color{blue}{\left(i + \left(\alpha + \beta\right)\right)} \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)} \]
      rational_best-simplify-3 [=>] 64.0	\[ \frac{i \cdot \left(\left(i + \left(\alpha + \beta\right)\right) \cdot \color{blue}{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right) + \beta \cdot \alpha\right)}\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)} \]
      rational_best-simplify-3 [=>] 64.0	\[ \frac{i \cdot \left(\left(i + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)} + \beta \cdot \alpha\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)} \]
      rational_best-simplify-1 [=>] 64.0	\[ \frac{i \cdot \left(\left(i + \left(\alpha + \beta\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \color{blue}{\alpha \cdot \beta}\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)} \]

   3. Taylor expanded in i around inf 16.1

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

   4. Simplified 16.1

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

      [Start] 16.1	\[ \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      rational_best-simplify-3 [<=] 16.1	\[ \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{2 \cdot \alpha + 2 \cdot \beta}}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      rational_best-simplify-55 [=>] 17.9	\[ \left(0.0625 + \color{blue}{\left(2 \cdot \alpha + 2 \cdot \beta\right) \cdot \frac{0.0625}{i}}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      rational_best-simplify-3 [=>] 17.9	\[ \left(0.0625 + \color{blue}{\left(2 \cdot \beta + 2 \cdot \alpha\right)} \cdot \frac{0.0625}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      rational_best-simplify-1 [=>] 17.9	\[ \left(0.0625 + \left(\color{blue}{\beta \cdot 2} + 2 \cdot \alpha\right) \cdot \frac{0.0625}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      rational_best-simplify-1 [=>] 17.9	\[ \left(0.0625 + \left(\beta \cdot 2 + \color{blue}{\alpha \cdot 2}\right) \cdot \frac{0.0625}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      rational_best-simplify-55 [=>] 16.1	\[ \left(0.0625 + \left(\beta \cdot 2 + \alpha \cdot 2\right) \cdot \frac{0.0625}{i}\right) - \color{blue}{\left(\beta + \alpha\right) \cdot \frac{0.125}{i}} \]

   5. Taylor expanded in beta around inf 17.9

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

   6. Applied egg-rr 17.9

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

   7. Simplified 16.1

      \[\leadsto \color{blue}{\left(0.0625 + \frac{\frac{\beta}{i}}{16}\right) + \frac{-0.125 \cdot \frac{\beta + \left(\alpha + \alpha\right)}{i}}{2}} \]
      Proof

      [Start] 17.9	\[ \frac{-\left(\beta + \alpha\right) \cdot \frac{0.125}{i}}{2} + \frac{0.0625 + \left(\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + 0.125 \cdot \left(-\frac{\alpha}{i}\right)\right)}{2} \]
      rational_best-simplify-64 [=>] 17.9	\[ \color{blue}{\frac{\left(-\left(\beta + \alpha\right) \cdot \frac{0.125}{i}\right) + \left(0.0625 + \left(\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + 0.125 \cdot \left(-\frac{\alpha}{i}\right)\right)\right)}{2}} \]
      rational_best-simplify-47 [=>] 17.9	\[ \frac{\left(-\left(\beta + \alpha\right) \cdot \frac{0.125}{i}\right) + \color{blue}{\left(0.125 \cdot \left(-\frac{\alpha}{i}\right) + \left(\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + 0.0625\right)\right)}}{2} \]
      rational_best-simplify-47 [=>] 17.9	\[ \frac{\color{blue}{\left(\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + 0.0625\right) + \left(0.125 \cdot \left(-\frac{\alpha}{i}\right) + \left(-\left(\beta + \alpha\right) \cdot \frac{0.125}{i}\right)\right)}}{2} \]
      rational_best-simplify-65 [=>] 17.9	\[ \color{blue}{\frac{\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + 0.0625}{2} + \frac{0.125 \cdot \left(-\frac{\alpha}{i}\right) + \left(-\left(\beta + \alpha\right) \cdot \frac{0.125}{i}\right)}{2}} \]

   8. Taylor expanded in beta around inf 16.1

      \[\leadsto \left(0.0625 + \frac{\frac{\beta}{i}}{16}\right) + \frac{-0.125 \cdot \color{blue}{\frac{\beta}{i}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \leq \infty:\\ \;\;\;\;\frac{\left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right) \cdot \frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\left(\alpha + \frac{\beta + \left(\beta + i \cdot 4\right)}{2}\right) \cdot \left(\alpha + \left(i + \left(\beta + i\right)\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + \frac{\frac{\beta}{i}}{16}\right) + \frac{-0.125 \cdot \frac{\beta}{i}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.5
Cost	6852

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t_0 \cdot t_0\\ t_2 := t_1 - 1\\ t_3 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_4 := i \cdot \left(\beta + \left(i + \alpha\right)\right)\\ t_5 := \alpha + \left(i + \left(\beta + i\right)\right)\\ \mathbf{if}\;\frac{\frac{t_3 \cdot \left(\beta \cdot \alpha + t_3\right)}{t_1}}{t_2} \leq \infty:\\ \;\;\;\;\frac{t_4 \cdot \frac{\beta \cdot \alpha + t_4}{t_5 \cdot t_5}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + \frac{\frac{\beta}{i}}{16}\right) + \frac{-0.125 \cdot \frac{\beta}{i}}{2}\\ \end{array} \]

Alternative 2
Error	13.1
Cost	3076

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;i \leq 9 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \beta\right)}{i \cdot 2 + \beta} \cdot \frac{-i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{i}{-0.5} - \left(\beta + \alpha\right)}}{t_0 \cdot t_0 - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + \frac{\frac{\beta}{i}}{16}\right) + \frac{-0.125 \cdot \frac{\beta}{i}}{2}\\ \end{array} \]

Alternative 3
Error	14.8
Cost	960

\[\left(0.0625 + \frac{\frac{\beta}{i}}{16}\right) + \frac{-0.125 \cdot \frac{\beta}{i}}{2} \]

Alternative 4
Error	16.8
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.4 \cdot 10^{+212}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	57.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023099 
(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)))