Octave 3.8, jcobi/2

Average Error: 24.1 → 1.3

Time: 18.5min

Precision: binary64

Cost: 6145

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999999997727:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + \frac{2}{\alpha}\right)\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(12 \cdot \frac{i}{\alpha \cdot \alpha} + \left(12 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{i}{\alpha} + \frac{i}{\alpha} \cdot \frac{i}{\alpha}\right) + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\beta - \alpha}} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot i}}{2}\\ \end{array}\]

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 beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
       (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))
      -0.9999999999997727)
   (/
    (-
     (+ (* 2.0 (/ beta alpha)) (+ (* 4.0 (/ i alpha)) (/ 2.0 alpha)))
     (+
      (* 6.0 (/ beta (* alpha alpha)))
      (+
       (* 12.0 (/ i (* alpha alpha)))
       (+
        (* 12.0 (+ (* (/ beta alpha) (/ i alpha)) (* (/ i alpha) (/ i alpha))))
        (+
         (/ 4.0 (* alpha alpha))
         (* 2.0 (* (/ beta alpha) (/ beta alpha))))))))
    2.0)
   (/
    (+
     1.0
     (*
      (/
       (+ alpha beta)
       (/ (+ 2.0 (+ (+ alpha beta) (* 2.0 i))) (- beta alpha)))
      (/ 1.0 (+ (+ alpha beta) (* 2.0 i)))))
    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 + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (2.0 + ((alpha + beta) + (2.0 * i)))) <= -0.9999999999997727) {
		tmp = (((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + (2.0 / alpha))) - ((6.0 * (beta / (alpha * alpha))) + ((12.0 * (i / (alpha * alpha))) + ((12.0 * (((beta / alpha) * (i / alpha)) + ((i / alpha) * (i / alpha)))) + ((4.0 / (alpha * alpha)) + (2.0 * ((beta / alpha) * (beta / alpha)))))))) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) / ((2.0 + ((alpha + beta) + (2.0 * i))) / (beta - alpha))) * (1.0 / ((alpha + beta) + (2.0 * i))))) / 2.0;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Alternatives

Alternative 1
Error	1.4
Cost	3841

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999999997727:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\beta - \alpha}} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot i}}{2}\\ \end{array}\]

Alternative 2
Error	1.4
Cost	3713

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999999997727:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array}\]

Alternative 3
Error	6.9
Cost	1921

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.2424984140172976 \cdot 10^{+45}:\\ \;\;\;\;\frac{1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta}{\beta + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array}\]

Alternative 4
Error	6.9
Cost	1409

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.6015243294856198 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array}\]

Alternative 5
Error	7.2
Cost	1281

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.949902580252124 \cdot 10^{+47}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array}\]

Alternative 6
Error	10.9
Cost	1153

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 22846195777.341858:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array}\]

Alternative 7
Error	16.3
Cost	1153

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7.295694539430045 \cdot 10^{+19}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array}\]

Alternative 8
Error	17.8
Cost	385

\[\begin{array}{l} \mathbf{if}\;\beta \leq 214997879345.9953:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Alternative 9
Error	43.5
Cost	64

\[1\]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.999999999999772737

   1. Initial program 63.1

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

   2. Taylor expanded around inf 14.3

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)\right) - \left(6 \cdot \frac{\beta}{{\alpha}^{2}} + \left(12 \cdot \frac{i}{{\alpha}^{2}} + \left(12 \cdot \frac{i \cdot \beta}{{\alpha}^{2}} + \left(12 \cdot \frac{{i}^{2}}{{\alpha}^{2}} + \left(4 \cdot \frac{1}{{\alpha}^{2}} + 2 \cdot \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right)\right)\right)\right)}}{2}\]

   3. Simplified 5.2

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + \frac{2}{\alpha}\right)\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(12 \cdot \frac{i}{\alpha \cdot \alpha} + \left(12 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{i}{\alpha} + \frac{i}{\alpha} \cdot \frac{i}{\alpha}\right) + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right)\right)}}{2}\]

   4. Simplified 5.2

      \[\leadsto \color{blue}{\frac{\left(2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + \frac{2}{\alpha}\right)\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(12 \cdot \frac{i}{\alpha \cdot \alpha} + \left(12 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{i}{\alpha} + \frac{i}{\alpha} \cdot \frac{i}{\alpha}\right) + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right)\right)}{2}}\]

   if -0.999999999999772737 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 12.7

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

   3. Applied *-un-lft-identity_binary64_1783 12.7

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

   4. Applied *-un-lft-identity_binary64_1783 12.7

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

   5. Applied times-frac_binary64_1789 0.2

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

   6. Applied times-frac_binary64_1789 0.2

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

   7. Simplified 0.2

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

   8. Simplified 0.2

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

   10. Applied *-un-lft-identity_binary64_1783 0.2

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

   11. Applied *-un-lft-identity_binary64_1783 0.2

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

   12. Applied times-frac_binary64_1789 0.2

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

   13. Applied associate-/l*_binary64_1728 0.2

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

   15. Applied associate-/r/_binary64_1729 0.2

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

   16. Applied add-sqr-sqrt_binary64_1805 0.2

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

   17. Applied times-frac_binary64_1789 0.2

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

   18. Applied associate-*r*_binary64_1723 0.2

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

   19. Simplified 0.2

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

   20. Simplified 0.2

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999999997727:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + \frac{2}{\alpha}\right)\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(12 \cdot \frac{i}{\alpha \cdot \alpha} + \left(12 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{i}{\alpha} + \frac{i}{\alpha} \cdot \frac{i}{\alpha}\right) + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\frac{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}{\beta - \alpha}} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot i}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021065 
(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))