Octave 3.8, jcobi/4

Average Error: 53.6 → 10.2

Time: 38.9s

Precision: binary64

Cost: 6468

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

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

Math

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

↓

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

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 (* i (+ (+ alpha beta) i)))
        (t_3 (* 0.125 (/ beta i)))
        (t_4 (+ alpha (+ (+ beta i) i))))
   (if (<= (/ (/ (* t_2 (+ (* beta alpha) t_2)) t_1) (- t_1 1.0)) INFINITY)
     (/
      (/ (+ (* beta alpha) (* i (+ beta (+ alpha i)))) (- 1.0 (* t_4 t_4)))
      (* -1.0 (* (+ (* i 2.0) beta) (+ (/ 1.0 (+ i beta)) (/ 1.0 i)))))
     (- (+ 0.0625 t_3) t_3))))

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 = i * ((alpha + beta) + i);
	double t_3 = 0.125 * (beta / i);
	double t_4 = alpha + ((beta + i) + i);
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= ((double) INFINITY)) {
		tmp = (((beta * alpha) + (i * (beta + (alpha + i)))) / (1.0 - (t_4 * t_4))) / (-1.0 * (((i * 2.0) + beta) * ((1.0 / (i + beta)) + (1.0 / i))));
	} else {
		tmp = (0.0625 + t_3) - t_3;
	}
	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 = i * ((alpha + beta) + i);
	double t_3 = 0.125 * (beta / i);
	double t_4 = alpha + ((beta + i) + i);
	double tmp;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= Double.POSITIVE_INFINITY) {
		tmp = (((beta * alpha) + (i * (beta + (alpha + i)))) / (1.0 - (t_4 * t_4))) / (-1.0 * (((i * 2.0) + beta) * ((1.0 / (i + beta)) + (1.0 / i))));
	} else {
		tmp = (0.0625 + t_3) - t_3;
	}
	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 = i * ((alpha + beta) + i)
	t_3 = 0.125 * (beta / i)
	t_4 = alpha + ((beta + i) + i)
	tmp = 0
	if (((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= math.inf:
		tmp = (((beta * alpha) + (i * (beta + (alpha + i)))) / (1.0 - (t_4 * t_4))) / (-1.0 * (((i * 2.0) + beta) * ((1.0 / (i + beta)) + (1.0 / i))))
	else:
		tmp = (0.0625 + t_3) - t_3
	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(i * Float64(Float64(alpha + beta) + i))
	t_3 = Float64(0.125 * Float64(beta / i))
	t_4 = Float64(alpha + Float64(Float64(beta + i) + i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(Float64(beta * alpha) + t_2)) / t_1) / Float64(t_1 - 1.0)) <= Inf)
		tmp = Float64(Float64(Float64(Float64(beta * alpha) + Float64(i * Float64(beta + Float64(alpha + i)))) / Float64(1.0 - Float64(t_4 * t_4))) / Float64(-1.0 * Float64(Float64(Float64(i * 2.0) + beta) * Float64(Float64(1.0 / Float64(i + beta)) + Float64(1.0 / i)))));
	else
		tmp = Float64(Float64(0.0625 + t_3) - t_3);
	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 = i * ((alpha + beta) + i);
	t_3 = 0.125 * (beta / i);
	t_4 = alpha + ((beta + i) + i);
	tmp = 0.0;
	if ((((t_2 * ((beta * alpha) + t_2)) / t_1) / (t_1 - 1.0)) <= Inf)
		tmp = (((beta * alpha) + (i * (beta + (alpha + i)))) / (1.0 - (t_4 * t_4))) / (-1.0 * (((i * 2.0) + beta) * ((1.0 / (i + beta)) + (1.0 / i))));
	else
		tmp = (0.0625 + t_3) - t_3;
	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[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(alpha + N[(N[(beta + i), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(N[(beta * alpha), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(beta * alpha), $MachinePrecision] + N[(i * N[(beta + N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(N[(N[(i * 2.0), $MachinePrecision] + beta), $MachinePrecision] * N[(N[(1.0 / N[(i + beta), $MachinePrecision]), $MachinePrecision] + N[(1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + t$95$3), $MachinePrecision] - t$95$3), $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 34.9

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

   2. Simplified 36.6

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

      [Start] 34.9	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational.json-simplify-47 [=>] 37.7	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      rational.json-simplify-2 [=>] 37.7	\[ \frac{\color{blue}{\left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)} \]
      rational.json-simplify-49 [=>] 36.6	\[ \color{blue}{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      rational.json-simplify-1 [=>] 36.6	\[ \left(i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}\right) \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)} \]

   3. Applied egg-rr 0.2

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

   4. Simplified 0.3

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

      [Start] 0.2

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

   5. Taylor expanded in alpha around 0 0.4

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

   6. Simplified 0.4

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

      [Start] 0.4	\[ \frac{\frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(\alpha + i\right)\right)}{1 - \left(\alpha + \left(\left(\beta + i\right) + i\right)\right) \cdot \left(\alpha + \left(\left(\beta + i\right) + i\right)\right)}}{-1 \cdot \left(\left(\beta + 2 \cdot i\right) \cdot \left(\frac{1}{\beta + i} + \frac{1}{i}\right)\right)} \]
      rational.json-simplify-1 [=>] 0.4	\[ \frac{\frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(\alpha + i\right)\right)}{1 - \left(\alpha + \left(\left(\beta + i\right) + i\right)\right) \cdot \left(\alpha + \left(\left(\beta + i\right) + i\right)\right)}}{-1 \cdot \left(\color{blue}{\left(2 \cdot i + \beta\right)} \cdot \left(\frac{1}{\beta + i} + \frac{1}{i}\right)\right)} \]
      rational.json-simplify-2 [=>] 0.4	\[ \frac{\frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(\alpha + i\right)\right)}{1 - \left(\alpha + \left(\left(\beta + i\right) + i\right)\right) \cdot \left(\alpha + \left(\left(\beta + i\right) + i\right)\right)}}{-1 \cdot \left(\left(\color{blue}{i \cdot 2} + \beta\right) \cdot \left(\frac{1}{\beta + i} + \frac{1}{i}\right)\right)} \]
      rational.json-simplify-1 [=>] 0.4	\[ \frac{\frac{\beta \cdot \alpha + i \cdot \left(\beta + \left(\alpha + i\right)\right)}{1 - \left(\alpha + \left(\left(\beta + i\right) + i\right)\right) \cdot \left(\alpha + \left(\left(\beta + i\right) + i\right)\right)}}{-1 \cdot \left(\left(i \cdot 2 + \beta\right) \cdot \left(\frac{1}{\color{blue}{i + \beta}} + \frac{1}{i}\right)\right)} \]

   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 60.8

      \[\leadsto \color{blue}{\left(i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}\right) \cdot \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}} \]
      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.json-simplify-49 [=>] 60.8	\[ \frac{\color{blue}{\left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational.json-simplify-49 [=>] 60.8	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 15.7

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

   4. Simplified 15.7

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

      [Start] 15.7	\[ \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      rational.json-simplify-2 [=>] 15.7	\[ \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{\beta \cdot 2} + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      rational.json-simplify-2 [=>] 15.7	\[ \left(0.0625 + 0.0625 \cdot \frac{\beta \cdot 2 + \color{blue}{\alpha \cdot 2}}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]

   5. Taylor expanded in beta around inf 15.7

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

   6. Taylor expanded in i around 0 15.7

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

   7. Taylor expanded in beta around inf 15.7

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

4. Final simplification 10.2

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

Alternatives

Alternative 1
Error	10.2
Cost	2252

\[\begin{array}{l} t_0 := \beta + \left(i + \left(i + \alpha\right)\right)\\ t_1 := 0.125 \cdot \frac{\beta}{i}\\ t_2 := \alpha + \left(i + \beta\right)\\ t_3 := i + t_2\\ \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+130}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.55 \cdot 10^{+162}:\\ \;\;\;\;i \cdot \frac{\frac{t_2}{t_3}}{t_3 \cdot \frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 6.6 \cdot 10^{+212}:\\ \;\;\;\;\left(0.0625 + t_1\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{t_0 \cdot \frac{t_0}{i + \left(\alpha + \beta\right)}} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 2
Error	10.3
Cost	2188

\[\begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \mathbf{if}\;\beta \leq 2 \cdot 10^{+132}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.95 \cdot 10^{+162}:\\ \;\;\;\;\frac{i + \alpha}{\left(\beta + \alpha\right) \cdot \frac{\beta}{i} + \left(\beta \cdot 4 - \beta\right)}\\ \mathbf{elif}\;\beta \leq 6.3 \cdot 10^{+212}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(i + \alpha\right)}{\beta}}{-1 \cdot \left(\left(\alpha + \left(\left(\beta + i\right) + i\right)\right) \cdot \left(\frac{1}{i} + \frac{1}{\beta + \left(\alpha + i\right)}\right)\right)}\\ \end{array} \]

Alternative 3
Error	10.2
Cost	2188

\[\begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ t_1 := \alpha + \left(i + \beta\right)\\ t_2 := i + t_1\\ \mathbf{if}\;\beta \leq 2.2 \cdot 10^{+130}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 5.6 \cdot 10^{+162}:\\ \;\;\;\;i \cdot \frac{\frac{t_1}{t_2}}{t_2 \cdot \frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 7 \cdot 10^{+212}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(i + \alpha\right)}{\beta}}{-1 \cdot \left(\left(\alpha + \left(\left(\beta + i\right) + i\right)\right) \cdot \left(\frac{1}{i} + \frac{1}{\beta + \left(\alpha + i\right)}\right)\right)}\\ \end{array} \]

Alternative 4
Error	10.4
Cost	1352

\[\begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \mathbf{if}\;\beta \leq 1.7 \cdot 10^{+132}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.7 \cdot 10^{+162}:\\ \;\;\;\;\frac{i + \alpha}{\left(\beta + \alpha\right) \cdot \frac{\beta}{i} + \left(\beta \cdot 4 - \beta\right)}\\ \mathbf{elif}\;\beta \leq 7 \cdot 10^{+212}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 5
Error	10.4
Cost	1228

\[\begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \mathbf{if}\;\beta \leq 1.7 \cdot 10^{+132}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.25 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{\frac{i}{\beta}}{\beta}}{\frac{1}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{+212}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 6
Error	16.1
Cost	844

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.9 \cdot 10^{+132}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+148}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 2.5 \cdot 10^{+239}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 7
Error	9.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.05 \cdot 10^{+132}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot \frac{1}{\beta}\right) \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 8
Error	9.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.05 \cdot 10^{+132}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 9
Error	16.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+239}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 10
Error	16.7
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.45 \cdot 10^{+239}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]

Alternative 11
Error	19.0
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023073 
(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)))