Octave 3.8, jcobi/2

Average Error: 23.8 → 1.3

Time: 12.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999999753411:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + \frac{2}{\alpha}\right)\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(12 \cdot \frac{i}{\alpha \cdot \alpha} + \left(12 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{i}{\alpha} + \frac{i}{\alpha} \cdot \frac{i}{\alpha}\right) + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \end{array}\]

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))))
      -0.9999999999753411)
   (/
    (-
     (+ (* 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)
   (/
    (+
     (*
      (+ alpha beta)
      (/
       (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i)))
       (+ 2.0 (+ (+ alpha beta) (* 2.0 i)))))
     1.0)
    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 + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (2.0 + ((alpha + beta) + (2.0 * i)))) <= -0.9999999999753411) {
		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 = (((alpha + beta) * (((beta - alpha) / ((alpha + beta) + (2.0 * i))) / (2.0 + ((alpha + beta) + (2.0 * i))))) + 1.0) / 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 (/.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)) < -0.99999999997534106

   1. Initial program 62.8

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

   2. Taylor expanded around inf 12.6

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

   3. Simplified 5.4

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

   if -0.99999999997534106 < (/.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 12.9

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

   3. Applied *-un-lft-identity_binary64 12.9

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

   4. Applied *-un-lft-identity_binary64 12.9

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

   5. Applied times-frac_binary64 0.1

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

   6. Applied times-frac_binary64 0.1

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

   7. Simplified 0.1

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

   8. Simplified 0.1

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999999753411:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + \frac{2}{\alpha}\right)\right) - \left(6 \cdot \frac{\beta}{\alpha \cdot \alpha} + \left(12 \cdot \frac{i}{\alpha \cdot \alpha} + \left(12 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{i}{\alpha} + \frac{i}{\alpha} \cdot \frac{i}{\alpha}\right) + \left(\frac{4}{\alpha \cdot \alpha} + 2 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021173 
(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))