Octave 3.8, jcobi/4

Average Error: 54.2 → 11.2

Time: 14.4s

Precision: binary64

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

\[[alpha, beta]=\mathsf{sort}([alpha, beta])\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \leq 1.825204277267042 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{i}{\frac{\left(\beta + i \cdot 2\right) + \alpha}{\alpha + \left(i + \beta\right)}}}{\left(\left(\beta + i \cdot 2\right) + \alpha\right) + 1} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\alpha + \left(i + \beta\right)\right)}{\left(\beta + i \cdot 2\right) + \alpha}}{\left(\left(\beta + i \cdot 2\right) + \alpha\right) - 1}\\ \mathbf{elif}\;i \leq 4.2936690958191245 \cdot 10^{+91}:\\ \;\;\;\;\frac{0.25 \cdot {i}^{2}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) - 1}\\ \mathbf{elif}\;i \leq 3.244261990187702 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\alpha + \left(i + \beta\right)\right)}{\left(\beta + i \cdot 2\right) + \alpha}}{\left(\left(\beta + i \cdot 2\right) + \alpha\right) + 1} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{i + \beta}}}{\left(\left(\beta + i \cdot 2\right) + \alpha\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (/
   (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
   (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))))
  (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))

↓

(FPCore (alpha beta i)
 :precision binary64
 (if (<= i 1.825204277267042e+82)
   (*
    (/
     (/ i (/ (+ (+ beta (* i 2.0)) alpha) (+ alpha (+ i beta))))
     (+ (+ (+ beta (* i 2.0)) alpha) 1.0))
    (/
     (/
      (+ (* beta alpha) (* i (+ alpha (+ i beta))))
      (+ (+ beta (* i 2.0)) alpha))
     (- (+ (+ beta (* i 2.0)) alpha) 1.0)))
   (if (<= i 4.2936690958191245e+91)
     (/
      (* 0.25 (pow i 2.0))
      (- (* (+ (* i 2.0) (+ beta alpha)) (+ (* i 2.0) (+ beta alpha))) 1.0))
     (if (<= i 3.244261990187702e+138)
       (*
        (/
         (/ (* i (+ alpha (+ i beta))) (+ (+ beta (* i 2.0)) alpha))
         (+ (+ (+ beta (* i 2.0)) alpha) 1.0))
        (/
         (/ i (/ (+ beta (* i 2.0)) (+ i beta)))
         (- (+ (+ beta (* i 2.0)) alpha) 1.0)))
       0.0625))))

C

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

↓

double code(double alpha, double beta, double i) {
	double tmp;
	if (i <= 1.825204277267042e+82) {
		tmp = ((i / (((beta + (i * 2.0)) + alpha) / (alpha + (i + beta)))) / (((beta + (i * 2.0)) + alpha) + 1.0)) * ((((beta * alpha) + (i * (alpha + (i + beta)))) / ((beta + (i * 2.0)) + alpha)) / (((beta + (i * 2.0)) + alpha) - 1.0));
	} else if (i <= 4.2936690958191245e+91) {
		tmp = (0.25 * pow(i, 2.0)) / ((((i * 2.0) + (beta + alpha)) * ((i * 2.0) + (beta + alpha))) - 1.0);
	} else if (i <= 3.244261990187702e+138) {
		tmp = (((i * (alpha + (i + beta))) / ((beta + (i * 2.0)) + alpha)) / (((beta + (i * 2.0)) + alpha) + 1.0)) * ((i / ((beta + (i * 2.0)) / (i + beta))) / (((beta + (i * 2.0)) + alpha) - 1.0));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 4 regimes

2. if i < 1.82520427726704184e82

   1. Initial program 27.0

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

   3. Applied difference-of-sqr-1_binary64 27.0

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

   4. Applied times-frac_binary64 12.7

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

   5. Applied times-frac_binary64 7.9

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

   6. Simplified 7.9

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

   7. Simplified 7.9

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

   9. Applied associate-/l*_binary64 7.9

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

   10. Simplified 7.9

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

   if 1.82520427726704184e82 < i < 4.29366909581912448e91

   1. Initial program 64.0

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

   2. Taylor expanded around inf 19.6

      \[\leadsto \frac{\color{blue}{0.25 \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]

   if 4.29366909581912448e91 < i < 3.244261990187702e138

   1. Initial program 64.0

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

   3. Applied difference-of-sqr-1_binary64 64.0

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

   4. Applied times-frac_binary64 19.1

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

   5. Applied times-frac_binary64 15.3

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

   6. Simplified 15.3

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

   7. Simplified 15.3

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

   8. Taylor expanded around 0 15.5

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

   9. Simplified 15.4

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

   if 3.244261990187702e138 < i

   1. Initial program 64.0

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

   2. Taylor expanded around inf 11.2

      \[\leadsto \color{blue}{0.0625}\]
3. Recombined 4 regimes into one program.

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 1.825204277267042 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{i}{\frac{\left(\beta + i \cdot 2\right) + \alpha}{\alpha + \left(i + \beta\right)}}}{\left(\left(\beta + i \cdot 2\right) + \alpha\right) + 1} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\alpha + \left(i + \beta\right)\right)}{\left(\beta + i \cdot 2\right) + \alpha}}{\left(\left(\beta + i \cdot 2\right) + \alpha\right) - 1}\\ \mathbf{elif}\;i \leq 4.2936690958191245 \cdot 10^{+91}:\\ \;\;\;\;\frac{0.25 \cdot {i}^{2}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) - 1}\\ \mathbf{elif}\;i \leq 3.244261990187702 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{i \cdot \left(\alpha + \left(i + \beta\right)\right)}{\left(\beta + i \cdot 2\right) + \alpha}}{\left(\left(\beta + i \cdot 2\right) + \alpha\right) + 1} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{i + \beta}}}{\left(\left(\beta + i \cdot 2\right) + \alpha\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array}\]

Reproduce

herbie shell --seed 2021174 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))