Octave 3.8, jcobi/4

Average Error: 54.1 → 10.5

Time: 25.3s

Precision: binary64

Cost: 24196

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

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

Math

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

↓

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

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

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) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double t_3 = pow((beta + (i * 2.0)), 2.0);
	double tmp;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= ((double) INFINITY)) {
		tmp = ((i * i) / t_3) * (pow((i + beta), 2.0) / (t_3 + -1.0));
	} else {
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	}
	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) + (i * 2.0);
	double t_1 = t_0 * t_0;
	double t_2 = i * (i + (alpha + beta));
	double t_3 = Math.pow((beta + (i * 2.0)), 2.0);
	double tmp;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= Double.POSITIVE_INFINITY) {
		tmp = ((i * i) / t_3) * (Math.pow((i + beta), 2.0) / (t_3 + -1.0));
	} else {
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	}
	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) + (i * 2.0)
	t_1 = t_0 * t_0
	t_2 = i * (i + (alpha + beta))
	t_3 = math.pow((beta + (i * 2.0)), 2.0)
	tmp = 0
	if (((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= math.inf:
		tmp = ((i * i) / t_3) * (math.pow((i + beta), 2.0) / (t_3 + -1.0))
	else:
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125)
	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(i * 2.0))
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(i * Float64(i + Float64(alpha + beta)))
	t_3 = Float64(beta + Float64(i * 2.0)) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(Float64(t_2 * Float64(t_2 + Float64(alpha * beta))) / t_1) / Float64(t_1 + -1.0)) <= Inf)
		tmp = Float64(Float64(Float64(i * i) / t_3) * Float64((Float64(i + beta) ^ 2.0) / Float64(t_3 + -1.0)));
	else
		tmp = Float64(Float64(0.0625 + Float64(0.125 * Float64(beta / i))) + Float64(Float64(beta / i) * -0.125));
	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) + (i * 2.0);
	t_1 = t_0 * t_0;
	t_2 = i * (i + (alpha + beta));
	t_3 = (beta + (i * 2.0)) ^ 2.0;
	tmp = 0.0;
	if ((((t_2 * (t_2 + (alpha * beta))) / t_1) / (t_1 + -1.0)) <= Inf)
		tmp = ((i * i) / t_3) * (((i + beta) ^ 2.0) / (t_3 + -1.0));
	else
		tmp = (0.0625 + (0.125 * (beta / i))) + ((beta / i) * -0.125);
	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[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(i * N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$2 * N[(t$95$2 + N[(alpha * beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(i * i), $MachinePrecision] / t$95$3), $MachinePrecision] * N[(N[Power[N[(i + beta), $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$3 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 + N[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(beta / i), $MachinePrecision] * -0.125), $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 34.9

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

   2. Simplified 0.2

      \[\leadsto \color{blue}{\left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}} \]
      Proof
      (*.f64 (*.f64 (/.f64 i (fma.f64 i 2 (+.f64 alpha beta))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta)))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 21 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 21 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 21 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 23 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (fma.f64 i (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (fma.f64 i (+.f64 (+.f64 alpha beta) i) (Rewrite<= *-commutative_binary64 (*.f64 beta alpha))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 2 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 beta alpha))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 13 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 23 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 21 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) -1))): 21 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)))): 0 points increase in error, 21 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r*_binary64 (/.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))): 21 points increase in error, 0 points decrease in error

   3. Taylor expanded in alpha around 0 37.7

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

   4. Simplified 1.1

      \[\leadsto \color{blue}{\frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} + -1}} \]
      Proof
      (*.f64 (/.f64 (*.f64 i i) (pow.f64 (+.f64 beta (*.f64 2 i)) 2)) (/.f64 (pow.f64 (+.f64 beta i) 2) (+.f64 (pow.f64 (+.f64 beta (*.f64 2 i)) 2) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 i 2)) (pow.f64 (+.f64 beta (*.f64 2 i)) 2)) (/.f64 (pow.f64 (+.f64 beta i) 2) (+.f64 (pow.f64 (+.f64 beta (*.f64 2 i)) 2) -1))): 0 points increase in error, 4 points decrease in error
      (*.f64 (/.f64 (pow.f64 i 2) (pow.f64 (+.f64 beta (*.f64 2 i)) 2)) (/.f64 (pow.f64 (+.f64 beta i) 2) (+.f64 (pow.f64 (+.f64 beta (*.f64 2 i)) 2) (Rewrite<= metadata-eval (neg.f64 1))))): 4 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (pow.f64 i 2) (pow.f64 (+.f64 beta (*.f64 2 i)) 2)) (/.f64 (pow.f64 (+.f64 beta i) 2) (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 (+.f64 beta (*.f64 2 i)) 2) 1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (pow.f64 i 2) (pow.f64 (+.f64 beta i) 2)) (*.f64 (pow.f64 (+.f64 beta (*.f64 2 i)) 2) (-.f64 (pow.f64 (+.f64 beta (*.f64 2 i)) 2) 1)))): 0 points increase in error, 0 points decrease in error

   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 61.3

      \[\leadsto \color{blue}{\left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}} \]
      Proof
      (*.f64 (*.f64 (/.f64 i (fma.f64 i 2 (+.f64 alpha beta))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta)))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 21 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 21 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (+.f64 i (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 21 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i)) (fma.f64 i 2 (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 23 points decrease in error
      (*.f64 (*.f64 (/.f64 i (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (/.f64 (+.f64 (+.f64 alpha beta) i) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (/.f64 (fma.f64 i (+.f64 i (+.f64 alpha beta)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (fma.f64 i (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) i)) (*.f64 alpha beta)) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (fma.f64 i (+.f64 (+.f64 alpha beta) i) (Rewrite<= *-commutative_binary64 (*.f64 beta alpha))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 2 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 beta alpha))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 13 points increase in error, 2 points decrease in error
      (*.f64 (/.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (fma.f64 (fma.f64 i 2 (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 23 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (fma.f64 i 2 (+.f64 alpha beta)) -1))): 0 points increase in error, 21 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 i 2) (+.f64 alpha beta))) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 i)) (+.f64 alpha beta)) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) -1))): 21 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (fma.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.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 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)))): 0 points increase in error, 21 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r*_binary64 (/.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))): 21 points increase in error, 0 points decrease in error

   3. Taylor expanded in i around inf 15.4

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

   4. Taylor expanded in alpha around 0 15.3

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

   5. Taylor expanded in beta around inf 15.3

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

4. Final simplification 10.5

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

Alternatives

Alternative 1
Error	10.5
Cost	18372

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

Alternative 2
Error	10.0
Cost	14532

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 8.4 \cdot 10^{+210}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \left(\alpha + \beta\right)}{t_0}}{\frac{t_0}{i}} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 3
Error	10.0
Cost	7364

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.85 \cdot 10^{+211}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 4
Error	10.0
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.4 \cdot 10^{+210}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 5
Error	10.0
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.4 \cdot 10^{+210}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta + i \cdot 2}\\ \end{array} \]

Alternative 6
Error	10.5
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.36 \cdot 10^{+212}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 7
Error	10.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{+210}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 8
Error	15.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+229}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 9
Error	11.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{+210}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 10
Error	16.5
Cost	324

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.7 \cdot 10^{+229}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]

Alternative 11
Error	18.4
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022340 
(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)))