Average Error: 54.2 → 9.1
Time: 5.2s
Precision: binary64
\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]
\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.7850136495907486 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]
(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
 (if (<= beta 2.7850136495907486e+158)
   0.0625
   (/ (/ i beta) (/ beta (+ i alpha)))))
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 tmp;
	if (beta <= 2.7850136495907486e+158) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) / (beta / (i + alpha));
	}
	return tmp;
}

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) :: tmp
    if (beta <= 2.7850136495907486d+158) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) / (beta / (i + alpha))
    end if
    code = tmp
end function

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 tmp;
	if (beta <= 2.7850136495907486e+158) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) / (beta / (i + alpha));
	}
	return tmp;
}

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):
	tmp = 0
	if beta <= 2.7850136495907486e+158:
		tmp = 0.0625
	else:
		tmp = (i / beta) / (beta / (i + alpha))
	return tmp

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)
	tmp = 0.0
	if (beta <= 2.7850136495907486e+158)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) / Float64(beta / Float64(i + alpha)));
	end
	return tmp
end

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)
	tmp = 0.0;
	if (beta <= 2.7850136495907486e+158)
		tmp = 0.0625;
	else
		tmp = (i / beta) / (beta / (i + alpha));
	end
	tmp_2 = tmp;
end

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_] := If[LessEqual[beta, 2.7850136495907486e+158], 0.0625, N[(N[(i / beta), $MachinePrecision] / N[(beta / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\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}
\mathbf{if}\;\beta \leq 2.7850136495907486 \cdot 10^{+158}:\\
\;\;\;\;0.0625\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\


\end{array}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if beta < 2.7850136495907486e158

    1. Initial program 50.0

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

    2. Simplified45.3

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

    3. Taylor expanded in i around inf 6.4

      \[\leadsto \color{blue}{0.0625} \]

    if 2.7850136495907486e158 < beta

    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. Simplified56.9

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

    3. Applied egg-rr56.9

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

    4. Taylor expanded in beta around inf 47.3

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

    5. Simplified15.6

      \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}} \]

    6. Applied egg-rr15.6

      \[\leadsto \color{blue}{\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7850136495907486 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022178 
(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)))