Octave 3.8, jcobi/2

Average Error: 24.1 → 12.1

Time: 5.7s

Precision: binary64

\[\alpha > -1 \land \beta > -1 \land i > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.0611776262030834 \cdot 10^{+92}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\left(\alpha + \beta\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \leq 1.6459537248978528 \cdot 10^{+211} \lor \neg \left(\alpha \leq 1.6308915972802799 \cdot 10^{+255}\right):\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \sqrt{\alpha + \beta} \cdot \left(\sqrt{\alpha + \beta} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}^{3}}}{2}\\ \end{array}\]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (+
   (/
    (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
    (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))
   1.0)
  2.0))

↓

(FPCore (alpha beta i)
 :precision binary64
 (if (<= alpha 3.0611776262030834e+92)
   (/
    (cbrt
     (pow
      (+
       (*
        (*
         (+ alpha beta)
         (/
          (*
           (cbrt (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i))))
           (cbrt (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i)))))
          (sqrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))
        (/
         (cbrt (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i))))
         (sqrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))
       1.0)
      3.0))
    2.0)
   (if (or (<= alpha 1.6459537248978528e+211)
           (not (<= alpha 1.6308915972802799e+255)))
     (/
      (- (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0))) (/ 4.0 (* alpha alpha)))
      2.0)
     (/
      (cbrt
       (pow
        (+
         1.0
         (*
          (sqrt (+ alpha beta))
          (*
           (sqrt (+ alpha beta))
           (/
            (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i)))
            (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))))))
        3.0))
      2.0))))

C

double code(double alpha, double beta, double i) {
	return (((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (((alpha + beta) + (2.0 * i)) + 2.0)) + 1.0) / 2.0;
}

↓

double code(double alpha, double beta, double i) {
	double tmp;
	if (alpha <= 3.0611776262030834e+92) {
		tmp = cbrt(pow(((((alpha + beta) * ((cbrt((beta - alpha) / ((alpha + beta) + (2.0 * i))) * cbrt((beta - alpha) / ((alpha + beta) + (2.0 * i)))) / sqrt(2.0 + ((alpha + beta) + (2.0 * i))))) * (cbrt((beta - alpha) / ((alpha + beta) + (2.0 * i))) / sqrt(2.0 + ((alpha + beta) + (2.0 * i))))) + 1.0), 3.0)) / 2.0;
	} else if ((alpha <= 1.6459537248978528e+211) || !(alpha <= 1.6308915972802799e+255)) {
		tmp = (((2.0 / alpha) + (8.0 / pow(alpha, 3.0))) - (4.0 / (alpha * alpha))) / 2.0;
	} else {
		tmp = cbrt(pow((1.0 + (sqrt(alpha + beta) * (sqrt(alpha + beta) * (((beta - alpha) / ((alpha + beta) + (2.0 * i))) / (2.0 + ((alpha + beta) + (2.0 * i))))))), 3.0)) / 2.0;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 3.06117762620308344e92

   1. Initial program 13.8

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

   3. Applied *-un-lft-identity_binary64 13.8

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

   4. Applied *-un-lft-identity_binary64 13.8

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

   5. Applied times-frac_binary64 2.9

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

   6. Applied times-frac_binary64 2.9

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

   7. Simplified 2.9

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

   8. Simplified 2.9

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

   10. Applied add-cbrt-cube_binary64 2.9

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

   11. Simplified 2.9

       \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1\right)}^{3}}}}{2}\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt_binary64 2.9

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

   14. Applied add-cube-cbrt_binary64 2.9

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

   15. Applied times-frac_binary64 2.9

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

   16. Applied associate-*r*_binary64 2.9

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

   if 3.06117762620308344e92 < alpha < 1.6459537248978528e211 or 1.63089159728027987e255 < alpha

   1. Initial program 56.4

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

   2. Taylor expanded around inf 41.3

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

   3. Simplified 41.3

      \[\leadsto \frac{\color{blue}{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}}{2}\]

   if 1.6459537248978528e211 < alpha < 1.63089159728027987e255

   1. Initial program 64.0

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

   3. Applied *-un-lft-identity_binary64 64.0

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

   4. Applied *-un-lft-identity_binary64 64.0

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

   5. Applied times-frac_binary64 46.9

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

   6. Applied times-frac_binary64 46.8

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

   7. Simplified 46.8

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

   8. Simplified 46.8

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

   10. Applied add-cbrt-cube_binary64 46.8

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

   11. Simplified 46.8

       \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1\right)}^{3}}}}{2}\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt_binary64 47.1

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

   14. Applied associate-*l*_binary64 47.1

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

   15. Simplified 47.1

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

4. Final simplification 12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.0611776262030834 \cdot 10^{+92}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\left(\alpha + \beta\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \leq 1.6459537248978528 \cdot 10^{+211} \lor \neg \left(\alpha \leq 1.6308915972802799 \cdot 10^{+255}\right):\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \sqrt{\alpha + \beta} \cdot \left(\sqrt{\alpha + \beta} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}^{3}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))