Octave 3.8, jcobi/4

Average Error: 53.8 → 12.7

Time: 32.8s

Precision: binary64

Cost: 19908

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t_0 \cdot t_0\\ t_2 := t_1 - 1\\ t_3 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_4 := t_3 \cdot \left(\beta \cdot \alpha + t_3\right)\\ \mathbf{if}\;\frac{\frac{t_4}{t_1}}{t_2} \leq 0.06249999999999998:\\ \;\;\;\;\frac{\frac{t_4}{{\left(\alpha + i \cdot 2\right)}^{2} + \left({\beta}^{2} + \beta \cdot \left(\alpha \cdot 2 + i \cdot 4\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta + \alpha}{i}\right) - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \]

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 (* t_0 t_0))
        (t_2 (- t_1 1.0))
        (t_3 (* i (+ (+ alpha beta) i)))
        (t_4 (* t_3 (+ (* beta alpha) t_3))))
   (if (<= (/ (/ t_4 t_1) t_2) 0.06249999999999998)
     (/
      (/
       t_4
       (+
        (pow (+ alpha (* i 2.0)) 2.0)
        (+ (pow beta 2.0) (* beta (+ (* alpha 2.0) (* i 4.0))))))
      t_2)
     (- (+ 0.0625 (* 0.125 (/ (+ beta alpha) i))) (* 0.125 (/ beta 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 = t_0 * t_0;
	double t_2 = t_1 - 1.0;
	double t_3 = i * ((alpha + beta) + i);
	double t_4 = t_3 * ((beta * alpha) + t_3);
	double tmp;
	if (((t_4 / t_1) / t_2) <= 0.06249999999999998) {
		tmp = (t_4 / (pow((alpha + (i * 2.0)), 2.0) + (pow(beta, 2.0) + (beta * ((alpha * 2.0) + (i * 4.0)))))) / t_2;
	} else {
		tmp = (0.0625 + (0.125 * ((beta + alpha) / i))) - (0.125 * (beta / i));
	}
	return tmp;
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = (((i * ((alpha + beta) + i)) * ((beta * alpha) + (i * ((alpha + beta) + i)))) / (((alpha + beta) + (2.0d0 * i)) * ((alpha + beta) + (2.0d0 * i)))) / ((((alpha + beta) + (2.0d0 * i)) * ((alpha + beta) + (2.0d0 * i))) - 1.0d0)
end function

↓

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (alpha + beta) + (2.0d0 * i)
    t_1 = t_0 * t_0
    t_2 = t_1 - 1.0d0
    t_3 = i * ((alpha + beta) + i)
    t_4 = t_3 * ((beta * alpha) + t_3)
    if (((t_4 / t_1) / t_2) <= 0.06249999999999998d0) then
        tmp = (t_4 / (((alpha + (i * 2.0d0)) ** 2.0d0) + ((beta ** 2.0d0) + (beta * ((alpha * 2.0d0) + (i * 4.0d0)))))) / t_2
    else
        tmp = (0.0625d0 + (0.125d0 * ((beta + alpha) / i))) - (0.125d0 * (beta / i))
    end if
    code = tmp
end function

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 = t_3 * ((beta * alpha) + t_3);
	double tmp;
	if (((t_4 / t_1) / t_2) <= 0.06249999999999998) {
		tmp = (t_4 / (Math.pow((alpha + (i * 2.0)), 2.0) + (Math.pow(beta, 2.0) + (beta * ((alpha * 2.0) + (i * 4.0)))))) / t_2;
	} else {
		tmp = (0.0625 + (0.125 * ((beta + alpha) / i))) - (0.125 * (beta / 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 = t_0 * t_0
	t_2 = t_1 - 1.0
	t_3 = i * ((alpha + beta) + i)
	t_4 = t_3 * ((beta * alpha) + t_3)
	tmp = 0
	if ((t_4 / t_1) / t_2) <= 0.06249999999999998:
		tmp = (t_4 / (math.pow((alpha + (i * 2.0)), 2.0) + (math.pow(beta, 2.0) + (beta * ((alpha * 2.0) + (i * 4.0)))))) / t_2
	else:
		tmp = (0.0625 + (0.125 * ((beta + alpha) / i))) - (0.125 * (beta / 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(t_0 * t_0)
	t_2 = Float64(t_1 - 1.0)
	t_3 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_4 = Float64(t_3 * Float64(Float64(beta * alpha) + t_3))
	tmp = 0.0
	if (Float64(Float64(t_4 / t_1) / t_2) <= 0.06249999999999998)
		tmp = Float64(Float64(t_4 / Float64((Float64(alpha + Float64(i * 2.0)) ^ 2.0) + Float64((beta ^ 2.0) + Float64(beta * Float64(Float64(alpha * 2.0) + Float64(i * 4.0)))))) / t_2);
	else
		tmp = Float64(Float64(0.0625 + Float64(0.125 * Float64(Float64(beta + alpha) / i))) - Float64(0.125 * Float64(beta / 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 = t_0 * t_0;
	t_2 = t_1 - 1.0;
	t_3 = i * ((alpha + beta) + i);
	t_4 = t_3 * ((beta * alpha) + t_3);
	tmp = 0.0;
	if (((t_4 / t_1) / t_2) <= 0.06249999999999998)
		tmp = (t_4 / (((alpha + (i * 2.0)) ^ 2.0) + ((beta ^ 2.0) + (beta * ((alpha * 2.0) + (i * 4.0)))))) / t_2;
	else
		tmp = (0.0625 + (0.125 * ((beta + alpha) / i))) - (0.125 * (beta / 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[(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[(t$95$3 * N[(N[(beta * alpha), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$4 / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], 0.06249999999999998], N[(N[(t$95$4 / N[(N[Power[N[(alpha + N[(i * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[beta, 2.0], $MachinePrecision] + N[(beta * N[(N[(alpha * 2.0), $MachinePrecision] + N[(i * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(0.0625 + N[(0.125 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.125 * N[(beta / 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)) < 0.0624999999999999792

   1. Initial program 0.9

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

   2. Taylor expanded in beta around 0 0.8

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

   3. Simplified 0.8

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

      [Start] 0.8	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\beta \cdot \left(4 \cdot i + 2 \cdot \alpha\right) + \left({\beta}^{2} + {\left(\alpha + 2 \cdot i\right)}^{2}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational.json-simplify-1 [=>] 0.8	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\beta \cdot \left(4 \cdot i + 2 \cdot \alpha\right) + \color{blue}{\left({\left(\alpha + 2 \cdot i\right)}^{2} + {\beta}^{2}\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational.json-simplify-41 [=>] 0.8	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\color{blue}{{\left(\alpha + 2 \cdot i\right)}^{2} + \left({\beta}^{2} + \beta \cdot \left(4 \cdot i + 2 \cdot \alpha\right)\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational.json-simplify-2 [=>] 0.8	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{{\left(\alpha + \color{blue}{i \cdot 2}\right)}^{2} + \left({\beta}^{2} + \beta \cdot \left(4 \cdot i + 2 \cdot \alpha\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational.json-simplify-1 [=>] 0.8	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{{\left(\alpha + i \cdot 2\right)}^{2} + \left({\beta}^{2} + \beta \cdot \color{blue}{\left(2 \cdot \alpha + 4 \cdot i\right)}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational.json-simplify-2 [=>] 0.8	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{{\left(\alpha + i \cdot 2\right)}^{2} + \left({\beta}^{2} + \beta \cdot \left(\color{blue}{\alpha \cdot 2} + 4 \cdot i\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      rational.json-simplify-2 [=>] 0.8	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{{\left(\alpha + i \cdot 2\right)}^{2} + \left({\beta}^{2} + \beta \cdot \left(\alpha \cdot 2 + \color{blue}{i \cdot 4}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   if 0.0624999999999999792 < (/.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 55.6

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

   2. Taylor expanded in i around inf 13.1

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

   3. Simplified 13.1

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

      [Start] 13.1	\[ \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      rational.json-simplify-1 [=>] 13.1	\[ \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{2 \cdot \alpha + 2 \cdot \beta}}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      rational.json-simplify-2 [=>] 13.1	\[ \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \alpha + \color{blue}{\beta \cdot 2}}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      rational.json-simplify-47 [=>] 13.1	\[ \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{2 \cdot \left(\beta + \alpha\right)}}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      rational.json-simplify-2 [<=] 13.1	\[ \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{\left(\beta + \alpha\right) \cdot 2}}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]

   4. Taylor expanded in beta around inf 13.1

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

   5. Simplified 13.1

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

      [Start] 13.1	\[ \left(0.0625 + 0.0625 \cdot \frac{\left(\beta + \alpha\right) \cdot 2}{i}\right) - 0.125 \cdot \frac{\beta}{i} \]
      rational.json-simplify-2 [=>] 13.1	\[ \left(0.0625 + 0.0625 \cdot \frac{\left(\beta + \alpha\right) \cdot 2}{i}\right) - \color{blue}{\frac{\beta}{i} \cdot 0.125} \]

   6. Taylor expanded in i around 0 13.1

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

4. Final simplification 12.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \leq 0.06249999999999998:\\ \;\;\;\;\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{{\left(\alpha + i \cdot 2\right)}^{2} + \left({\beta}^{2} + \beta \cdot \left(\alpha \cdot 2 + i \cdot 4\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta + \alpha}{i}\right) - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.7
Cost	13316

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t_0 \cdot t_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \frac{t_2 \cdot \left(\beta \cdot \alpha + t_2\right)}{t_1}\\ \mathbf{if}\;\frac{t_3}{t_1 - 1} \leq 0.06249999999999998:\\ \;\;\;\;\frac{t_3}{\left|\left(\left(\left(-i\right) - i\right) - \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \left(i + \left(i + \beta\right)\right)\right)\right| - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta + \alpha}{i}\right) - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \]

Alternative 2
Error	12.7
Cost	6852

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := t_0 \cdot t_0\\ t_2 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_3 := \frac{\frac{t_2 \cdot \left(\beta \cdot \alpha + t_2\right)}{t_1}}{t_1 - 1}\\ \mathbf{if}\;t_3 \leq 0.06249999999999998:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta + \alpha}{i}\right) - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \]

Alternative 3
Error	14.6
Cost	1736

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\beta \leq 8.6 \cdot 10^{+120}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{i \cdot \left(i + \alpha\right)}{t_0 \cdot t_0 - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta + \alpha}{i}\right) - 0.125 \cdot \frac{\beta}{i}\\ \end{array} \]

Alternative 4
Error	14.5
Cost	960

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

Alternative 5
Error	14.5
Cost	832

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

Alternative 6
Error	14.5
Cost	832

\[\begin{array}{l} t_0 := 0.0625 \cdot \frac{\beta}{i}\\ \left(t_0 + 0.0625\right) - t_0 \end{array} \]

Alternative 7
Error	16.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.6 \cdot 10^{+202}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125 \cdot \left(\left(\alpha + \beta\right) - \beta\right)}{i}\\ \end{array} \]

Alternative 8
Error	18.5
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023077 
(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)))