Octave 3.8, jcobi/4

Average Error: 54.0 → 12.8

Time: 15.6s

Precision: binary64

\[\alpha > -1 \land \beta > -1 \land i > 1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6436317829303763 \cdot 10^{+83} \lor \neg \left(\beta \leq 9.537261936518884 \cdot 10^{+128}\right) \land \beta \leq 5.193139048227797 \cdot 10^{+150}:\\ \;\;\;\;\sqrt[3]{{\left(i \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\right)}^{3}} \cdot \frac{\alpha \cdot 0.25 + \left(i \cdot 0.5 + \beta \cdot 0.25\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\right) \cdot \frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \end{array}\]

FPCore

(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 (or (<= beta 2.6436317829303763e+83)
         (and (not (<= beta 9.537261936518884e+128))
              (<= beta 5.193139048227797e+150)))
   (*
    (cbrt
     (pow
      (*
       i
       (/
        (/ (+ i (+ beta alpha)) (+ (+ beta alpha) (* i 2.0)))
        (+ (+ (+ beta alpha) (* i 2.0)) 1.0)))
      3.0))
    (/
     (+ (* alpha 0.25) (+ (* i 0.5) (* beta 0.25)))
     (- (+ (+ beta alpha) (* i 2.0)) 1.0)))
   (*
    (*
     i
     (/
      (/ (+ i (+ beta alpha)) (+ (+ beta alpha) (* i 2.0)))
      (+ (+ (+ beta alpha) (* i 2.0)) 1.0)))
    (/ i (- (+ (+ beta alpha) (* i 2.0)) 1.0)))))

C

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 <= 2.6436317829303763e+83) || (!(beta <= 9.537261936518884e+128) && (beta <= 5.193139048227797e+150))) {
		tmp = cbrt(pow((i * (((i + (beta + alpha)) / ((beta + alpha) + (i * 2.0))) / (((beta + alpha) + (i * 2.0)) + 1.0))), 3.0)) * (((alpha * 0.25) + ((i * 0.5) + (beta * 0.25))) / (((beta + alpha) + (i * 2.0)) - 1.0));
	} else {
		tmp = (i * (((i + (beta + alpha)) / ((beta + alpha) + (i * 2.0))) / (((beta + alpha) + (i * 2.0)) + 1.0))) * (i / (((beta + alpha) + (i * 2.0)) - 1.0));
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if beta < 2.64363178293037635e83 or 9.5372619365188842e128 < beta < 5.19313904822779687e150

   1. Initial program 51.5

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

   3. Applied difference-of-sqr-1_binary64_2457 51.5

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

   4. Applied times-frac_binary64_2493 35.8

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

   5. Applied times-frac_binary64_2493 34.6

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

   6. Simplified 34.6

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

   7. Simplified 34.6

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

   9. Applied *-un-lft-identity_binary64_2487 34.6

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

   10. Applied *-un-lft-identity_binary64_2487 34.6

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

   11. Applied times-frac_binary64_2493 34.5

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

   12. Applied times-frac_binary64_2493 34.6

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

   13. Simplified 34.6

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

   14. Taylor expanded around 0 11.3

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

   15. Simplified 11.3

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

   17. Applied add-cbrt-cube_binary64_2523 9.6

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

   18. Simplified 9.6

       \[\leadsto \sqrt[3]{\color{blue}{{\left(i \cdot \frac{\frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}\right)}^{3}}} \cdot \frac{\alpha \cdot 0.25 + \left(i \cdot 0.5 + \beta \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\]

   if 2.64363178293037635e83 < beta < 9.5372619365188842e128 or 5.19313904822779687e150 < beta

   1. Initial program 62.5

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

   3. Applied difference-of-sqr-1_binary64_2457 62.5

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

   4. Applied times-frac_binary64_2493 52.9

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

   5. Applied times-frac_binary64_2493 47.9

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

   6. Simplified 47.9

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

   7. Simplified 47.9

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

   9. Applied *-un-lft-identity_binary64_2487 47.9

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

   10. Applied *-un-lft-identity_binary64_2487 47.9

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

   11. Applied times-frac_binary64_2493 47.8

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

   12. Applied times-frac_binary64_2493 47.8

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

   13. Simplified 47.8

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

   14. Taylor expanded around inf 24.0

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6436317829303763 \cdot 10^{+83} \lor \neg \left(\beta \leq 9.537261936518884 \cdot 10^{+128}\right) \land \beta \leq 5.193139048227797 \cdot 10^{+150}:\\ \;\;\;\;\sqrt[3]{{\left(i \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\right)}^{3}} \cdot \frac{\alpha \cdot 0.25 + \left(i \cdot 0.5 + \beta \cdot 0.25\right)}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot \frac{\frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 1}\right) \cdot \frac{i}{\left(\left(\beta + \alpha\right) + i \cdot 2\right) - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020277 
(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)))