Octave 3.8, jcobi/3

Average Error: 3.8 → 2.8

Time: 9.9s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 7.9290885316467188 \cdot 10^{91}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 \cdot \frac{1}{{\alpha}^{2}} + 1\right) - 1 \cdot \frac{1}{\alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

C

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

↓

double code(double alpha, double beta) {
	double temp;
	if ((alpha <= 7.929088531646719e+91)) {
		temp = ((sqrt(sqrt((((alpha + beta) + (beta * alpha)) + 1.0))) / sqrt((((alpha + beta) + (2.0 * 1.0)) * (((alpha + beta) + (2.0 * 1.0)) / sqrt((((alpha + beta) + (beta * alpha)) + 1.0)))))) / ((((alpha + beta) + (2.0 * 1.0)) + 1.0) / sqrt((((((alpha + beta) + (beta * alpha)) + 1.0) / ((alpha + beta) + (2.0 * 1.0))) / ((alpha + beta) + (2.0 * 1.0))))));
	} else {
		temp = (((((2.0 * (1.0 / pow(alpha, 2.0))) + 1.0) - (1.0 * (1.0 / alpha))) / ((alpha + beta) + (2.0 * 1.0))) / (((alpha + beta) + (2.0 * 1.0)) + 1.0));
	}
	return temp;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 7.929088531646719e+91

   1. Initial program 0.6

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

   3. Applied add-sqr-sqrt 0.7

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

   4. Applied associate-/l* 0.7

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

   6. Applied *-un-lft-identity 0.7

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

   7. Applied add-sqr-sqrt 0.7

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

   8. Applied times-frac 0.6

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

   9. Applied associate-/l* 0.6

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

   10. Applied sqrt-div 0.7

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

   11. Simplified 0.7

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

   12. Simplified 0.7

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

   if 7.929088531646719e+91 < alpha

   1. Initial program 14.2

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

   2. Taylor expanded around inf 9.7

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

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 7.9290885316467188 \cdot 10^{91}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) \cdot \frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}}}}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}{\sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(2 \cdot \frac{1}{{\alpha}^{2}} + 1\right) - 1 \cdot \frac{1}{\alpha}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1)))