Octave 3.8, jcobi/2

Average Error: 23.9 → 11.2

Time: 7.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.962350371619421 \cdot 10^{+114}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\alpha + \beta}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]

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 3.962350371619421e+114)
   (/
    (cbrt
     (pow
      (+
       (*
        (/ (+ alpha beta) (sqrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))))
        (/
         (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i)))
         (sqrt (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))
       1.0)
      3.0))
    2.0)
   (/ (- (/ 2.0 alpha) (/ 4.0 (* alpha alpha))) 2.0)))

C

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

↓

double code(double alpha, double beta, double i) {
	double tmp;
	if ((alpha <= 3.962350371619421e+114)) {
		tmp = (((double) cbrt(((double) pow(((double) (((double) ((((double) (alpha + beta)) / ((double) sqrt(((double) (2.0 + ((double) (((double) (alpha + beta)) + ((double) (2.0 * i))))))))) * ((((double) (beta - alpha)) / ((double) (((double) (alpha + beta)) + ((double) (2.0 * i))))) / ((double) sqrt(((double) (2.0 + ((double) (((double) (alpha + beta)) + ((double) (2.0 * i))))))))))) + 1.0)), 3.0)))) / 2.0);
	} else {
		tmp = (((double) ((2.0 / alpha) - (4.0 / ((double) (alpha * alpha))))) / 2.0);
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 3.9623503716194211e114

   1. Initial program 14.7

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

   3. Applied *-un-lft-identity_binary64 14.7

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

   4. Applied *-un-lft-identity_binary64 14.7

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

   5. Applied times-frac_binary64 3.7

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

   6. Applied times-frac_binary64 3.7

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

   7. Simplified 3.7

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

   8. Simplified 3.7

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

   10. Applied add-cbrt-cube_binary64 3.7

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

   11. Simplified 3.7

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

   13. Applied add-sqr-sqrt_binary64 3.7

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

   14. Applied *-un-lft-identity_binary64 3.7

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

   15. Applied *-un-lft-identity_binary64 3.7

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

   16. Applied times-frac_binary64 3.7

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

   17. Applied times-frac_binary64 3.7

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

   18. Applied associate-*r*_binary64 3.7

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

   19. Simplified 3.7

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

   if 3.9623503716194211e114 < alpha

   1. Initial program 60.6

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

   3. Applied *-un-lft-identity_binary64 60.6

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

   4. Applied *-un-lft-identity_binary64 60.6

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

   5. Applied times-frac_binary64 45.3

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

   6. Applied times-frac_binary64 45.4

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

   7. Simplified 45.4

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

   8. Simplified 45.4

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

   10. Applied add-log-exp_binary64 45.4

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

   11. Applied add-log-exp_binary64 45.4

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

   12. Applied sum-log_binary64 45.4

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

   13. Simplified 45.4

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

   14. Taylor expanded around inf 40.7

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

   15. Simplified 40.7

       \[\leadsto \frac{\color{blue}{\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.962350371619421 \cdot 10^{+114}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\alpha + \beta}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} + 1\right)}^{3}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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))