Average Error: 54.0 → 10.7
Time: 16.7s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq 3.2748075795360256 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{\sqrt{\left(\alpha + \beta\right) + i \cdot 2}}{i}} \cdot \frac{\frac{i + \left(\alpha + \beta\right)}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}\right) \cdot \frac{i \cdot 0.5 + \left(\alpha + \beta\right) \cdot 0.25}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \end{array}\]
\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 3.2748075795360256 \cdot 10^{+133}:\\
\;\;\;\;\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{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{\frac{\sqrt{\left(\alpha + \beta\right) + i \cdot 2}}{i}} \cdot \frac{\frac{i + \left(\alpha + \beta\right)}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}\right) \cdot \frac{i \cdot 0.5 + \left(\alpha + \beta\right) \cdot 0.25}{\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 3.2748075795360256e+133)
   (*
    (/
     (/ (* i (+ i (+ alpha beta))) (+ (+ alpha beta) (* i 2.0)))
     (+ (+ (+ alpha beta) (* i 2.0)) 1.0))
    (/
     (/
      (+ (* i (+ i (+ alpha beta))) (* alpha beta))
      (+ (+ alpha beta) (* i 2.0)))
     (- (+ (+ alpha beta) (* i 2.0)) 1.0)))
   (*
    (*
     (/ 1.0 (/ (sqrt (+ (+ alpha beta) (* i 2.0))) i))
     (/
      (/ (+ i (+ alpha beta)) (sqrt (+ (+ alpha beta) (* i 2.0))))
      (+ (+ (+ alpha beta) (* i 2.0)) 1.0)))
    (/
     (+ (* i 0.5) (* (+ alpha 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 <= 3.2748075795360256e+133) {
		tmp = (((i * (i + (alpha + beta))) / ((alpha + beta) + (i * 2.0))) / (((alpha + beta) + (i * 2.0)) + 1.0)) * ((((i * (i + (alpha + beta))) + (alpha * beta)) / ((alpha + beta) + (i * 2.0))) / (((alpha + beta) + (i * 2.0)) - 1.0));
	} else {
		tmp = ((1.0 / (sqrt((alpha + beta) + (i * 2.0)) / i)) * (((i + (alpha + beta)) / sqrt((alpha + beta) + (i * 2.0))) / (((alpha + beta) + (i * 2.0)) + 1.0))) * (((i * 0.5) + ((alpha + 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 < 3.2748075795360256e133

    1. Initial program 40.7

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

    3. Applied difference-of-sqr-1_binary64_173240.7

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

    4. Applied times-frac_binary64_176814.6

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

    5. Applied times-frac_binary64_176810.2

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

    6. Simplified10.2

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

    7. Simplified10.2

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

    if 3.2748075795360256e133 < 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_173264.0

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

    4. Applied times-frac_binary64_176859.0

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

    5. Applied times-frac_binary64_176858.8

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

    6. Simplified58.8

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

    7. Simplified58.8

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

    9. Applied *-un-lft-identity_binary64_176258.8

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

    10. Applied add-sqr-sqrt_binary64_178358.8

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

    11. Applied times-frac_binary64_176858.8

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

    12. Applied times-frac_binary64_176858.8

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

    13. Simplified58.8

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

    14. Taylor expanded around 0 11.2

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

    15. Simplified11.2

      \[\leadsto \left(\frac{i}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2}} \cdot \frac{\frac{i + \left(\alpha + \beta\right)}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}\right) \cdot \frac{\color{blue}{i \cdot 0.5 + 0.25 \cdot \left(\alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]
    16. Using strategy rm

    17. Applied clear-num_binary64_176111.1

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 3.2748075795360256 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{\sqrt{\left(\alpha + \beta\right) + i \cdot 2}}{i}} \cdot \frac{\frac{i + \left(\alpha + \beta\right)}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}\right) \cdot \frac{i \cdot 0.5 + \left(\alpha + \beta\right) \cdot 0.25}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020292 
(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)))