Average Error: 53.9 → 13.7
Time: 7.3s
Precision: binary64
\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]
\[ \begin{array}{c}[alpha, beta] = \mathsf{sort}([alpha, beta])\\ \end{array} \]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
↓
\[\begin{array}{l} t_0 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ t_1 := \frac{i}{t_0} \cdot \left(i + \left(\beta + \alpha\right)\right)\\ \mathbf{if}\;\beta \leq 10^{+160}:\\ \;\;\;\;t_1 \cdot \left(\mathsf{fma}\left(0.125, \frac{\beta \cdot \alpha}{{i}^{3}}, \mathsf{fma}\left(0.125, \frac{1}{i}, 0.03125 \cdot \frac{1}{{i}^{3}}\right)\right) + \left(\frac{\alpha}{i \cdot i} + \frac{\beta}{i \cdot i}\right) \cdot -0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\frac{1}{t_0} \cdot \frac{i + \alpha}{\beta}\right)\\ \end{array} \]
(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
 (let* ((t_0 (+ alpha (fma i 2.0 beta)))
        (t_1 (* (/ i t_0) (+ i (+ beta alpha)))))
   (if (<= beta 1e+160)
     (*
      t_1
      (+
       (fma
        0.125
        (/ (* beta alpha) (pow i 3.0))
        (fma 0.125 (/ 1.0 i) (* 0.03125 (/ 1.0 (pow i 3.0)))))
       (* (+ (/ alpha (* i i)) (/ beta (* i i))) -0.0625)))
     (* t_1 (* (/ 1.0 t_0) (/ (+ i alpha) beta))))))
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 + fma(i, 2.0, beta);
	double t_1 = (i / t_0) * (i + (beta + alpha));
	double tmp;
	if (beta <= 1e+160) {
		tmp = t_1 * (fma(0.125, ((beta * alpha) / pow(i, 3.0)), fma(0.125, (1.0 / i), (0.03125 * (1.0 / pow(i, 3.0))))) + (((alpha / (i * i)) + (beta / (i * i))) * -0.0625));
	} else {
		tmp = t_1 * ((1.0 / t_0) * ((i + alpha) / beta));
	}
	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)
	t_0 = Float64(alpha + fma(i, 2.0, beta))
	t_1 = Float64(Float64(i / t_0) * Float64(i + Float64(beta + alpha)))
	tmp = 0.0
	if (beta <= 1e+160)
		tmp = Float64(t_1 * Float64(fma(0.125, Float64(Float64(beta * alpha) / (i ^ 3.0)), fma(0.125, Float64(1.0 / i), Float64(0.03125 * Float64(1.0 / (i ^ 3.0))))) + Float64(Float64(Float64(alpha / Float64(i * i)) + Float64(beta / Float64(i * i))) * -0.0625)));
	else
		tmp = Float64(t_1 * Float64(Float64(1.0 / t_0) * Float64(Float64(i + alpha) / beta)));
	end
	return 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_] := Block[{t$95$0 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(i / t$95$0), $MachinePrecision] * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1e+160], N[(t$95$1 * N[(N[(0.125 * N[(N[(beta * alpha), $MachinePrecision] / N[Power[i, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[(1.0 / i), $MachinePrecision] + N[(0.03125 * N[(1.0 / N[Power[i, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(alpha / N[(i * i), $MachinePrecision]), $MachinePrecision] + N[(beta / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $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}
t_0 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\
t_1 := \frac{i}{t_0} \cdot \left(i + \left(\beta + \alpha\right)\right)\\
\mathbf{if}\;\beta \leq 10^{+160}:\\
\;\;\;\;t_1 \cdot \left(\mathsf{fma}\left(0.125, \frac{\beta \cdot \alpha}{{i}^{3}}, \mathsf{fma}\left(0.125, \frac{1}{i}, 0.03125 \cdot \frac{1}{{i}^{3}}\right)\right) + \left(\frac{\alpha}{i \cdot i} + \frac{\beta}{i \cdot i}\right) \cdot -0.0625\right)\\

\mathbf{else}:\\
\;\;\;\;t_1 \cdot \left(\frac{1}{t_0} \cdot \frac{i + \alpha}{\beta}\right)\\


\end{array}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if beta < 1.00000000000000001e160

    1. Initial program 49.7

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

    2. Simplified45.1

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

    3. Taylor expanded in i around inf 6.8

      \[\leadsto \left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \color{blue}{\left(\left(0.125 \cdot \frac{\beta \cdot \alpha}{{i}^{3}} + \left(0.125 \cdot \frac{1}{i} + 0.03125 \cdot \frac{1}{{i}^{3}}\right)\right) - \left(0.0625 \cdot \frac{\alpha}{{i}^{2}} + 0.0625 \cdot \frac{\beta}{{i}^{2}}\right)\right)} \]

    4. Simplified6.8

      \[\leadsto \left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(0.125, \frac{\beta \cdot \alpha}{{i}^{3}}, \mathsf{fma}\left(0.125, \frac{1}{i}, 0.03125 \cdot \frac{1}{{i}^{3}}\right)\right) - 0.0625 \cdot \left(\frac{\alpha}{i \cdot i} + \frac{\beta}{i \cdot i}\right)\right)} \]

    if 1.00000000000000001e160 < 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. Simplified57.1

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

    3. Applied egg-rr57.1

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

    4. Taylor expanded in beta around inf 30.2

      \[\leadsto \left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(\frac{1}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \color{blue}{\frac{i + \alpha}{\beta}}\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 10^{+160}:\\ \;\;\;\;\left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(\mathsf{fma}\left(0.125, \frac{\beta \cdot \alpha}{{i}^{3}}, \mathsf{fma}\left(0.125, \frac{1}{i}, 0.03125 \cdot \frac{1}{{i}^{3}}\right)\right) + \left(\frac{\alpha}{i \cdot i} + \frac{\beta}{i \cdot i}\right) \cdot -0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(\frac{1}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \alpha}{\beta}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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)))