Octave 3.8, jcobi/4

Average Error: 53.9 → 10.0

Time: 30.8s

Precision: binary64

Cost: 6852

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

Math

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

↓

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

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 (* i (+ (+ alpha beta) i)))
        (t_2 (+ i (+ alpha beta)))
        (t_3 (* t_0 t_0))
        (t_4 (+ (+ alpha beta) (+ i i))))
   (if (<= (/ (/ (* t_1 (+ (* beta alpha) t_1)) t_3) (- t_3 1.0)) INFINITY)
     (/
      (/
       (+ (* i t_2) (* alpha beta))
       (* (+ (* t_4 t_4) -1.0) (/ (/ t_4 i) t_2)))
      t_4)
     (-
      (+ 0.0625 (* 0.0625 (/ (+ (* 2.0 beta) (* 2.0 alpha)) i)))
      (* 0.125 (/ (+ beta alpha) i))))))

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 = i * ((alpha + beta) + i);
	double t_2 = i + (alpha + beta);
	double t_3 = t_0 * t_0;
	double t_4 = (alpha + beta) + (i + i);
	double tmp;
	if ((((t_1 * ((beta * alpha) + t_1)) / t_3) / (t_3 - 1.0)) <= ((double) INFINITY)) {
		tmp = (((i * t_2) + (alpha * beta)) / (((t_4 * t_4) + -1.0) * ((t_4 / i) / t_2))) / t_4;
	} else {
		tmp = (0.0625 + (0.0625 * (((2.0 * beta) + (2.0 * alpha)) / i))) - (0.125 * ((beta + alpha) / i));
	}
	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 = i * ((alpha + beta) + i);
	double t_2 = i + (alpha + beta);
	double t_3 = t_0 * t_0;
	double t_4 = (alpha + beta) + (i + i);
	double tmp;
	if ((((t_1 * ((beta * alpha) + t_1)) / t_3) / (t_3 - 1.0)) <= Double.POSITIVE_INFINITY) {
		tmp = (((i * t_2) + (alpha * beta)) / (((t_4 * t_4) + -1.0) * ((t_4 / i) / t_2))) / t_4;
	} else {
		tmp = (0.0625 + (0.0625 * (((2.0 * beta) + (2.0 * alpha)) / i))) - (0.125 * ((beta + alpha) / i));
	}
	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 = i * ((alpha + beta) + i)
	t_2 = i + (alpha + beta)
	t_3 = t_0 * t_0
	t_4 = (alpha + beta) + (i + i)
	tmp = 0
	if (((t_1 * ((beta * alpha) + t_1)) / t_3) / (t_3 - 1.0)) <= math.inf:
		tmp = (((i * t_2) + (alpha * beta)) / (((t_4 * t_4) + -1.0) * ((t_4 / i) / t_2))) / t_4
	else:
		tmp = (0.0625 + (0.0625 * (((2.0 * beta) + (2.0 * alpha)) / i))) - (0.125 * ((beta + alpha) / i))
	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(i * Float64(Float64(alpha + beta) + i))
	t_2 = Float64(i + Float64(alpha + beta))
	t_3 = Float64(t_0 * t_0)
	t_4 = Float64(Float64(alpha + beta) + Float64(i + i))
	tmp = 0.0
	if (Float64(Float64(Float64(t_1 * Float64(Float64(beta * alpha) + t_1)) / t_3) / Float64(t_3 - 1.0)) <= Inf)
		tmp = Float64(Float64(Float64(Float64(i * t_2) + Float64(alpha * beta)) / Float64(Float64(Float64(t_4 * t_4) + -1.0) * Float64(Float64(t_4 / i) / t_2))) / t_4);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(Float64(Float64(2.0 * beta) + Float64(2.0 * alpha)) / i))) - Float64(0.125 * Float64(Float64(beta + alpha) / i)));
	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 = i * ((alpha + beta) + i);
	t_2 = i + (alpha + beta);
	t_3 = t_0 * t_0;
	t_4 = (alpha + beta) + (i + i);
	tmp = 0.0;
	if ((((t_1 * ((beta * alpha) + t_1)) / t_3) / (t_3 - 1.0)) <= Inf)
		tmp = (((i * t_2) + (alpha * beta)) / (((t_4 * t_4) + -1.0) * ((t_4 / i) / t_2))) / t_4;
	else
		tmp = (0.0625 + (0.0625 * (((2.0 * beta) + (2.0 * alpha)) / i))) - (0.125 * ((beta + alpha) / i));
	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[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(alpha + beta), $MachinePrecision] + N[(i + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$1 * N[(N[(beta * alpha), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / N[(t$95$3 - 1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(N[(i * t$95$2), $MachinePrecision] + N[(alpha * beta), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$4 * t$95$4), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(t$95$4 / i), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[(0.0625 + N[(0.0625 * N[(N[(N[(2.0 * beta), $MachinePrecision] + N[(2.0 * alpha), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 35.2

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

   2. Simplified 25.6

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

      [Start] 35.2	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational.json-simplify-15 [=>] 35.3	\[ \frac{\color{blue}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational.json-simplify-16 [=>] 36.5	\[ \color{blue}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      rational.json-simplify-39 [=>] 36.5	\[ \frac{\frac{\color{blue}{\left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)} \]
      rational.json-simplify-19 [=>] 26.1	\[ \frac{\color{blue}{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)} \]

   3. Applied egg-rr 0.4

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

   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 61.0

      \[\leadsto \color{blue}{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1\right)}} \]
      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.json-simplify-15 [=>] 64.0	\[ \frac{\color{blue}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational.json-simplify-16 [=>] 64.0	\[ \color{blue}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      rational.json-simplify-39 [=>] 64.0	\[ \frac{\frac{\color{blue}{\left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)} \]
      rational.json-simplify-19 [=>] 62.2	\[ \frac{\color{blue}{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)} \]

   3. Taylor expanded in i around inf 15.2

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

4. Final simplification 10.0

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

Alternatives

Alternative 1
Error	10.0
Cost	6852

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

Alternative 2
Error	13.4
Cost	3012

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + \left(i + i\right)\\ \mathbf{if}\;i \leq 6.4 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \frac{\frac{i + \beta}{t_0}}{t_0 \cdot t_0 + -1}\right)}{\beta + \left(i + i\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}\\ \end{array} \]

Alternative 3
Error	14.1
Cost	1344

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

Alternative 4
Error	16.7
Cost	1088

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

Alternative 5
Error	17.4
Cost	960

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

Alternative 6
Error	15.9
Cost	960

\[\left(0.125 + \frac{\beta}{\frac{i}{0.25}}\right) - \left(0.0625 + \frac{\beta}{i} \cdot 0.25\right) \]

Alternative 7
Error	17.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.6 \cdot 10^{+241}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) - \left(\alpha + \beta\right)}{i \cdot 8}\\ \end{array} \]

Alternative 8
Error	17.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+241}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\left(\beta - \left(\alpha + \beta\right)\right) \cdot \frac{0.125}{i}\\ \end{array} \]

Alternative 9
Error	18.3
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023066 
(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)))