Average Error: 53.6 → 10.3
Time: 20.8s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq 4.235675601254117 \cdot 10^{+130}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\alpha \cdot 0.25 + \left(i \cdot 0.5 + \beta \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \end{array}\]
\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 4.235675601254117 \cdot 10^{+130}:\\
\;\;\;\;\frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \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}\\

\mathbf{else}:\\
\;\;\;\;\frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\alpha \cdot 0.25 + \left(i \cdot 0.5 + \beta \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\

\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 4.235675601254117e+130)
   (*
    (/
     (* i (/ (+ i (+ alpha beta)) (+ (+ alpha beta) (* i 2.0))))
     (+ (+ (+ alpha beta) (* i 2.0)) 1.0))
    (/
     (/
      (+ (* alpha beta) (* i (+ i (+ alpha beta))))
      (+ (+ alpha beta) (* i 2.0)))
     (- (+ (+ alpha beta) (* i 2.0)) 1.0)))
   (*
    (/
     (* i (/ (+ i (+ alpha beta)) (+ (+ alpha beta) (* i 2.0))))
     (+ (+ (+ alpha beta) (* i 2.0)) 1.0))
    (/
     (+ (* alpha 0.25) (+ (* i 0.5) (* beta 0.25)))
     (- (+ (+ alpha beta) (* i 2.0)) 1.0)))))
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 (i <= 4.235675601254117e+130) {
		tmp = ((i * ((i + (alpha + beta)) / ((alpha + beta) + (i * 2.0)))) / (((alpha + beta) + (i * 2.0)) + 1.0)) * ((((alpha * beta) + (i * (i + (alpha + beta)))) / ((alpha + beta) + (i * 2.0))) / (((alpha + beta) + (i * 2.0)) - 1.0));
	} else {
		tmp = ((i * ((i + (alpha + beta)) / ((alpha + beta) + (i * 2.0)))) / (((alpha + beta) + (i * 2.0)) + 1.0)) * (((alpha * 0.25) + ((i * 0.5) + (beta * 0.25))) / (((alpha + beta) + (i * 2.0)) - 1.0));
	}
	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 i < 4.23567560125411713e130

    1. Initial program 40.0

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

    3. Applied difference-of-sqr-1_binary64_240640.0

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

    4. Applied times-frac_binary64_244214.5

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

    5. Applied times-frac_binary64_24429.9

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

    6. Simplified9.9

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

    7. Simplified9.9

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

    9. Applied *-un-lft-identity_binary64_24369.9

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

    10. Applied times-frac_binary64_24429.8

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

    11. Simplified9.8

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

    if 4.23567560125411713e130 < 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. Using strategy rm

    3. Applied difference-of-sqr-1_binary64_240664.0

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

    4. Applied times-frac_binary64_244258.2

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

    5. Applied times-frac_binary64_244258.0

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

    6. Simplified58.0

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

    7. Simplified58.0

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

    9. Applied *-un-lft-identity_binary64_243658.0

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

    10. Applied times-frac_binary64_244258.0

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

    11. Simplified58.0

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

    12. Taylor expanded around 0 10.6

      \[\leadsto \frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\color{blue}{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]

    13. Simplified10.6

      \[\leadsto \frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\color{blue}{\alpha \cdot 0.25 + \left(i \cdot 0.5 + \beta \cdot 0.25\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 4.235675601254117 \cdot 10^{+130}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\alpha \cdot 0.25 + \left(i \cdot 0.5 + \beta \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020288 
(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)))