Octave 3.8, jcobi/2

Average Error: 23.3 → 7.4

Time: 5.2min

Precision: binary64

Cost: 3713

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999911246198:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\frac{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}{\frac{\alpha + \beta}{-2 - \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}}{2}\\ \end{array}\]

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))))
      -0.9999999911246198)
   (/
    (- (+ (/ 2.0 alpha) (/ 8.0 (pow alpha 3.0))) (/ 4.0 (* alpha alpha)))
    2.0)
   (/
    (+
     1.0
     (/
      -1.0
      (/
       (/ (+ (+ alpha beta) (* 2.0 i)) (- beta alpha))
       (/ (+ alpha beta) (- -2.0 (+ (+ 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)))) <= -0.9999999911246198) {
		tmp = (((2.0 / alpha) + (8.0 / pow(alpha, 3.0))) - (4.0 / (alpha * alpha))) / 2.0;
	} else {
		tmp = (1.0 + (-1.0 / ((((alpha + beta) + (2.0 * i)) / (beta - alpha)) / ((alpha + beta) / (-2.0 - ((alpha + beta) + (2.0 * i))))))) / 2.0;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Alternative 1
Accuracy	7.4
Cost	3585

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999911246198:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array}\]

Alternative 2
Accuracy	12.3
Cost	3392

\[\frac{\frac{0}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha} \cdot \left(-2 - \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} - \left(-1 + \frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha} \cdot \left(-2 - \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}\right)}{2}\]

Alternative 3
Accuracy	12.3
Cost	3776

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

Alternative 4
Accuracy	12.3
Cost	3904

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

Alternative 5
Accuracy	13.1
Cost	3585

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.72612383697892 \cdot 10^{+210}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt[3]{\beta + \alpha} \cdot \sqrt[3]{\beta + \alpha}}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\sqrt[3]{\beta + \alpha} \cdot \left(\beta - \alpha\right)\right)}{2 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta + \alpha}{\sqrt{\left(2 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)\right) \cdot \frac{\left(\beta + \alpha\right) + 2 \cdot i}{\beta - \alpha}} \cdot \sqrt{\left(2 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)\right) \cdot \frac{\left(\beta + \alpha\right) + 2 \cdot i}{\beta - \alpha}}}}{2}\\ \end{array}\]

Alternative 6
Accuracy	17.9
Cost	2560

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

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)) < -0.99999999112461979

   1. Initial program 62.6

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

   2. Taylor expanded around inf 32.6

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

   3. Simplified 32.6

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

   if -0.99999999112461979 < (/.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 11.9

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

   3. Applied associate-/l*_binary64_3433 0.1

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

   5. Applied associate-/l/_binary64_3435 0.1

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

   6. Simplified 0.1

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

   8. Applied frac-2neg_binary64_3499 0.1

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

   9. Simplified 0.1

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

   11. Applied *-un-lft-identity_binary64_3488 0.1

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

   12. Applied distribute-lft-neg-in_binary64_3445 0.1

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

   13. Applied associate-/l*_binary64_3433 0.1

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

   14. Simplified 0.1

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

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999911246198:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-1}{\frac{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}{\frac{\alpha + \beta}{-2 - \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020322 
(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))