Average Error: 54.1 → 35.9
Time: 9.9s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.151273590251243 \cdot 10^{+206}:\\ \;\;\;\;\left(\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\right) \cdot \frac{1}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 4.151273590251243 \cdot 10^{+206}:\\
\;\;\;\;\left(\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\right) \cdot \frac{1}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\

\mathbf{else}:\\
\;\;\;\;0\\

\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 4.151273590251243e+206)
   (*
    (*
     (/
      (+ (* beta alpha) (* i (+ i (+ beta alpha))))
      (+ (+ beta alpha) (* i 2.0)))
     (/
      (/ (* i (+ i (+ beta alpha))) (+ (+ beta alpha) (* i 2.0)))
      (+ (+ (+ beta alpha) (* i 2.0)) 1.0)))
    (/ 1.0 (- (+ (+ beta alpha) (* i 2.0)) 1.0)))
   0.0))
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 <= 4.151273590251243e+206) {
		tmp = ((((beta * alpha) + (i * (i + (beta + alpha)))) / ((beta + alpha) + (i * 2.0))) * (((i * (i + (beta + alpha))) / ((beta + alpha) + (i * 2.0))) / (((beta + alpha) + (i * 2.0)) + 1.0))) * (1.0 / (((beta + alpha) + (i * 2.0)) - 1.0));
	} else {
		tmp = 0.0;
	}
	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 2 regimes

  2. if beta < 4.1512735902512431e206

    1. Initial program 52.9

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

    3. Applied difference-of-sqr-1_binary6452.9

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

    4. Applied times-frac_binary6437.3

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

    5. Applied times-frac_binary6434.9

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

    6. Simplified34.9

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

    7. Simplified34.9

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

    9. Applied div-inv_binary6434.9

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

    10. Applied associate-*r*_binary6434.9

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

    11. Simplified34.9

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

    if 4.1512735902512431e206 < 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. Taylor expanded around inf 43.7

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification35.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.151273590251243 \cdot 10^{+206}:\\ \;\;\;\;\left(\frac{\beta \cdot \alpha + i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2} \cdot \frac{\frac{i \cdot \left(i + \left(\beta + \alpha\right)\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\right) \cdot \frac{1}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(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)))