Average Error: 53.9 → 11.7
Time: 7.8s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
↓
\[\begin{array}{l} t_0 := i \cdot \left(i + \left(\beta + \alpha\right)\right)\\ t_1 := \alpha + \left(\beta + i \cdot 2\right)\\ t_2 := \frac{\frac{t_0}{t_1}}{t_1 + 1} \cdot \frac{\frac{t_0 + \beta \cdot \alpha}{t_1}}{t_1 - 1}\\ \mathbf{if}\;i \leq 2.4897970452779856 \cdot 10^{+51}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;i \leq 6.665594334552595 \cdot 10^{+87}:\\ \;\;\;\;\begin{array}{l} t_3 := \left(\beta + \alpha\right) + i \cdot 2\\ \frac{\left(i \cdot i\right) \cdot 0.25}{t_3 \cdot t_3 - 1} \end{array}\\ \mathbf{elif}\;i \leq 1.724874057153644 \cdot 10^{+133}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
\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 := i \cdot \left(i + \left(\beta + \alpha\right)\right)\\
t_1 := \alpha + \left(\beta + i \cdot 2\right)\\
t_2 := \frac{\frac{t_0}{t_1}}{t_1 + 1} \cdot \frac{\frac{t_0 + \beta \cdot \alpha}{t_1}}{t_1 - 1}\\
\mathbf{if}\;i \leq 2.4897970452779856 \cdot 10^{+51}:\\
\;\;\;\;t_2\\

\mathbf{elif}\;i \leq 6.665594334552595 \cdot 10^{+87}:\\
\;\;\;\;\begin{array}{l}
t_3 := \left(\beta + \alpha\right) + i \cdot 2\\
\frac{\left(i \cdot i\right) \cdot 0.25}{t_3 \cdot t_3 - 1}
\end{array}\\

\mathbf{elif}\;i \leq 1.724874057153644 \cdot 10^{+133}:\\
\;\;\;\;t_2\\

\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 (* i (+ i (+ beta alpha))))
        (t_1 (+ alpha (+ beta (* i 2.0))))
        (t_2
         (*
          (/ (/ t_0 t_1) (+ t_1 1.0))
          (/ (/ (+ t_0 (* beta alpha)) t_1) (- t_1 1.0)))))
   (if (<= i 2.4897970452779856e+51)
     t_2
     (if (<= i 6.665594334552595e+87)
       (let* ((t_3 (+ (+ beta alpha) (* i 2.0))))
         (/ (* (* i i) 0.25) (- (* t_3 t_3) 1.0)))
       (if (<= i 1.724874057153644e+133) t_2 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 = i * (i + (beta + alpha));
	double t_1 = alpha + (beta + (i * 2.0));
	double t_2 = ((t_0 / t_1) / (t_1 + 1.0)) * (((t_0 + (beta * alpha)) / t_1) / (t_1 - 1.0));
	double tmp;
	if (i <= 2.4897970452779856e+51) {
		tmp = t_2;
	} else if (i <= 6.665594334552595e+87) {
		double t_3 = (beta + alpha) + (i * 2.0);
		tmp = ((i * i) * 0.25) / ((t_3 * t_3) - 1.0);
	} else if (i <= 1.724874057153644e+133) {
		tmp = t_2;
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

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 3 regimes

  2. if i < 2.4897970452779856e51 or 6.66559433455259541e87 < i < 1.7248740571536441e133

    1. Initial program 42.9

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

    3. Applied difference-of-sqr-1_binary6442.9

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

    4. Applied times-frac_binary6413.8

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

    5. Applied times-frac_binary6410.0

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

    6. Simplified10.0

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

    7. Simplified10.0

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

    if 2.4897970452779856e51 < i < 6.66559433455259541e87

    1. Initial program 37.1

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

    2. Taylor expanded around inf 17.7

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

    3. Simplified17.7

      \[\leadsto \frac{\color{blue}{\left(i \cdot i\right) \cdot 0.25}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

    if 1.7248740571536441e133 < 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. Taylor expanded around inf 11.3

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

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.4897970452779856 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \beta \cdot \alpha}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}\\ \mathbf{elif}\;i \leq 6.665594334552595 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) \cdot \left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;i \leq 1.724874057153644 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right) + \beta \cdot \alpha}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (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)))