Average Error: 53.8 → 12.4
Time: 33.9s
Precision: binary64
\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq 4.539218113043745 \cdot 10^{+135}:\\ \;\;\;\;\begin{array}{l} t_0 := i + \left(\alpha + \beta\right)\\ t_1 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \frac{\frac{i \cdot t_0}{\frac{\mathsf{fma}\left(4, i \cdot \alpha, \alpha \cdot \alpha + \left(\beta \cdot \beta + \mathsf{fma}\left(4, i \cdot i, \mathsf{fma}\left(2, \alpha \cdot \beta, 4 \cdot \left(i \cdot \beta\right)\right)\right)\right)\right)}{\mathsf{fma}\left(i, t_0, \alpha \cdot \beta\right)}}}{\mathsf{fma}\left(t_1, t_1, -1\right)} \end{array}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
\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}\;i \leq 4.539218113043745 \cdot 10^{+135}:\\
\;\;\;\;\begin{array}{l}
t_0 := i + \left(\alpha + \beta\right)\\
t_1 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\
\frac{\frac{i \cdot t_0}{\frac{\mathsf{fma}\left(4, i \cdot \alpha, \alpha \cdot \alpha + \left(\beta \cdot \beta + \mathsf{fma}\left(4, i \cdot i, \mathsf{fma}\left(2, \alpha \cdot \beta, 4 \cdot \left(i \cdot \beta\right)\right)\right)\right)\right)}{\mathsf{fma}\left(i, t_0, \alpha \cdot \beta\right)}}}{\mathsf{fma}\left(t_1, t_1, -1\right)}
\end{array}\\

\mathbf{else}:\\
\;\;\;\;0.0625\\


\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 (<= i 4.539218113043745e+135)
   (let* ((t_0 (+ i (+ alpha beta))) (t_1 (fma i 2.0 (+ alpha beta))))
     (/
      (/
       (* i t_0)
       (/
        (fma
         4.0
         (* i alpha)
         (+
          (* alpha alpha)
          (+
           (* beta beta)
           (fma 4.0 (* i i) (fma 2.0 (* alpha beta) (* 4.0 (* i beta)))))))
        (fma i t_0 (* alpha beta))))
      (fma t_1 t_1 -1.0)))
   0.0625))
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 (i <= 4.539218113043745e+135) {
		double t_0_1 = i + (alpha + beta);
		double t_1_2 = fma(i, 2.0, (alpha + beta));
		tmp = ((i * t_0_1) / (fma(4.0, (i * alpha), ((alpha * alpha) + ((beta * beta) + fma(4.0, (i * i), fma(2.0, (alpha * beta), (4.0 * (i * beta))))))) / fma(i, t_0_1, (alpha * beta)))) / fma(t_1_2, t_1_2, -1.0);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 4.5392181130437448e135

    1. Initial program 41.1

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

    2. Simplified41.1

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

    3. Applied associate-/l*_binary6414.7

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

    4. Taylor expanded in i around 0 14.7

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

    5. Simplified14.7

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

    if 4.5392181130437448e135 < i

    1. Initial program 63.9

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

    2. Simplified63.9

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

    3. Taylor expanded in i around inf 10.5

      \[\leadsto \color{blue}{0.0625} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 4.539218113043745 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{\mathsf{fma}\left(4, i \cdot \alpha, \alpha \cdot \alpha + \left(\beta \cdot \beta + \mathsf{fma}\left(4, i \cdot i, \mathsf{fma}\left(2, \alpha \cdot \beta, 4 \cdot \left(i \cdot \beta\right)\right)\right)\right)\right)}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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)))