Average Error: 54.0 → 12.7
Time: 22.7s
Precision: binary64
\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := i + \left(\alpha + \beta\right)\\ \mathbf{if}\;i \leq 1.4297873900624532 \cdot 10^{+134}:\\ \;\;\;\;\frac{i \cdot \left(-\frac{\mathsf{fma}\left(i, t_1, \alpha \cdot \beta\right)}{t_0}\right)}{\left(1 - {t_0}^{2}\right) \cdot \frac{t_0}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
\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}
t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\
t_1 := i + \left(\alpha + \beta\right)\\
\mathbf{if}\;i \leq 1.4297873900624532 \cdot 10^{+134}:\\
\;\;\;\;\frac{i \cdot \left(-\frac{\mathsf{fma}\left(i, t_1, \alpha \cdot \beta\right)}{t_0}\right)}{\left(1 - {t_0}^{2}\right) \cdot \frac{t_0}{t_1}}\\

\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
 (let* ((t_0 (fma i 2.0 (+ alpha beta))) (t_1 (+ i (+ alpha beta))))
   (if (<= i 1.4297873900624532e+134)
     (/
      (* i (- (/ (fma i t_1 (* alpha beta)) t_0)))
      (* (- 1.0 (pow t_0 2.0)) (/ t_0 t_1)))
     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 t_0 = fma(i, 2.0, (alpha + beta));
	double t_1 = i + (alpha + beta);
	double tmp;
	if (i <= 1.4297873900624532e+134) {
		tmp = (i * -(fma(i, t_1, (alpha * beta)) / t_0)) / ((1.0 - pow(t_0, 2.0)) * (t_0 / t_1));
	} 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 < 1.42978739006245322e134

    1. Initial program 41.2

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

    2. Simplified41.2

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

    3. Applied egg-rr15.1

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

    4. Applied egg-rr15.0

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

    if 1.42978739006245322e134 < i

    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. Simplified64.0

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

    3. Taylor expanded in i around inf 11.0

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

  4. Final simplification12.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.4297873900624532 \cdot 10^{+134}:\\ \;\;\;\;\frac{i \cdot \left(-\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}{\left(1 - {\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}\right) \cdot \frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(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)))