Octave 3.8, jcobi/4

Average Error: 53.9 → 12.8

Time: 22.0s

Precision: binary64

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := \mathsf{fma}\left(t_0, t_0, -1\right)\\ \mathbf{if}\;i \leq 2.660517665891581 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{i \cdot i + i \cdot \beta}}}{t_1}\\ \mathbf{elif}\;i \leq 8.934596183311938 \cdot 10^{+125}:\\ \;\;\;\;e^{\log \left(\frac{\left(i \cdot i\right) \cdot 0.25}{t_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

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
 (let* ((t_0 (fma i 2.0 (+ alpha beta))) (t_1 (fma t_0 t_0 -1.0)))
   (if (<= i 2.660517665891581e+36)
     (/
      (/
       (* i (+ i (+ alpha beta)))
       (/ (pow (+ beta (* i 2.0)) 2.0) (+ (* i i) (* i beta))))
      t_1)
     (if (<= i 8.934596183311938e+125)
       (exp (log (/ (* (* i i) 0.25) t_1)))
       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 t_0 = fma(i, 2.0, (alpha + beta));
	double t_1 = fma(t_0, t_0, -1.0);
	double tmp;
	if (i <= 2.660517665891581e+36) {
		tmp = ((i * (i + (alpha + beta))) / (pow((beta + (i * 2.0)), 2.0) / ((i * i) + (i * beta)))) / t_1;
	} else if (i <= 8.934596183311938e+125) {
		tmp = exp(log(((i * i) * 0.25) / t_1));
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if i < 2.6605176658915813e36

   1. Initial program 22.6

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

   2. Simplified 22.6

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

   3. Applied associate-/l*_binary64 8.4

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

   4. Simplified 8.4

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

   5. Taylor expanded in alpha around 0 8.2

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

   6. Simplified 8.2

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

   if 2.6605176658915813e36 < i < 8.9345961833119378e125

   1. Initial program 46.2

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

   2. Simplified 46.2

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

   3. Applied associate-/l*_binary64 17.2

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

   4. Simplified 17.2

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

   5. Taylor expanded in i around inf 18.0

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

   6. Simplified 18.0

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

   7. Applied add-exp-log_binary64 21.3

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

   8. Applied add-exp-log_binary64 20.9

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

   9. Applied add-exp-log_binary64 20.9

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

   10. Applied prod-exp_binary64 20.9

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

   11. Applied add-exp-log_binary64 20.9

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

   12. Applied prod-exp_binary64 21.4

       \[\leadsto \frac{\color{blue}{e^{\log 0.25 + \left(\log i + \log i\right)}}}{e^{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)\right)}} \]

   13. Applied div-exp_binary64 21.4

       \[\leadsto \color{blue}{e^{\left(\log 0.25 + \left(\log i + \log i\right)\right) - \log \left(\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)\right)}} \]

   14. Simplified 18.0

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

   if 8.9345961833119378e125 < 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. Simplified 64.0

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

   3. Taylor expanded in i around inf 11.0

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 2.660517665891581 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\beta + i \cdot 2\right)}^{2}}{i \cdot i + i \cdot \beta}}}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}\\ \mathbf{elif}\;i \leq 8.934596183311938 \cdot 10^{+125}:\\ \;\;\;\;e^{\log \left(\frac{\left(i \cdot i\right) \cdot 0.25}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (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)))