Average Error: 54.1 → 37.4
Time: 7.2s
Precision: binary64
\[\alpha > -1 \land \beta > -1 \land i > 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \leq 6.11547452069048 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;i \leq 2.2832363808567277 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;i \leq 1.3577553450061928 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 6.11547452069048 \cdot 10^{+66}:\\
\;\;\;\;\frac{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\

\mathbf{elif}\;i \leq 2.2832363808567277 \cdot 10^{+94}:\\
\;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\

\mathbf{elif}\;i \leq 1.3577553450061928 \cdot 10^{+154}:\\
\;\;\;\;\frac{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
(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 6.11547452069048e+66)
   (/
    (*
     (/
      (+ (* alpha beta) (* i (+ i (+ alpha beta))))
      (+ (+ alpha beta) (* i 2.0)))
     (/
      (/ (* i (+ i (+ alpha beta))) (+ (+ alpha beta) (* i 2.0)))
      (+ (+ (+ alpha beta) (* i 2.0)) 1.0)))
    (- (+ (+ alpha beta) (* i 2.0)) 1.0))
   (if (<= i 2.2832363808567277e+94)
     (/
      (* (* i i) 0.25)
      (- (* (+ (+ alpha beta) (* i 2.0)) (+ (+ alpha beta) (* i 2.0))) 1.0))
     (if (<= i 1.3577553450061928e+154)
       (/
        (*
         (/
          (+ (* alpha beta) (* i (+ i (+ alpha beta))))
          (+ (+ alpha beta) (* i 2.0)))
         (/
          (/ (* i (+ i (+ alpha beta))) (+ (+ alpha beta) (* i 2.0)))
          (+ (+ (+ alpha beta) (* i 2.0)) 1.0)))
        (- (+ (+ alpha beta) (* i 2.0)) 1.0))
       0.0))))
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 <= 6.11547452069048e+66) {
		tmp = ((((alpha * beta) + (i * (i + (alpha + beta)))) / ((alpha + beta) + (i * 2.0))) * (((i * (i + (alpha + beta))) / ((alpha + beta) + (i * 2.0))) / (((alpha + beta) + (i * 2.0)) + 1.0))) / (((alpha + beta) + (i * 2.0)) - 1.0);
	} else if (i <= 2.2832363808567277e+94) {
		tmp = ((i * i) * 0.25) / ((((alpha + beta) + (i * 2.0)) * ((alpha + beta) + (i * 2.0))) - 1.0);
	} else if (i <= 1.3577553450061928e+154) {
		tmp = ((((alpha * beta) + (i * (i + (alpha + beta)))) / ((alpha + beta) + (i * 2.0))) * (((i * (i + (alpha + beta))) / ((alpha + beta) + (i * 2.0))) / (((alpha + beta) + (i * 2.0)) + 1.0))) / (((alpha + beta) + (i * 2.0)) - 1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if i < 6.11547452069048e66 or 2.2832363808567277e94 < i < 1.3577553450061928e154

    1. Initial program 43.1

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

    3. Applied add-sqr-sqrt_binary6443.1

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

    4. Applied difference-of-squares_binary6443.1

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

    5. Applied times-frac_binary6416.2

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

    6. Applied times-frac_binary6412.3

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

    7. Simplified12.3

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

    8. Simplified12.3

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

    10. Applied associate-*r/_binary6412.3

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

    11. Simplified12.3

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

    if 6.11547452069048e66 < i < 2.2832363808567277e94

    1. Initial program 51.2

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

    2. Taylor expanded around inf 21.2

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

    3. Simplified21.2

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

    if 1.3577553450061928e154 < 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 61.9

      \[\leadsto \color{blue}{0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification37.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq 6.11547452069048 \cdot 10^{+66}:\\ \;\;\;\;\frac{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;i \leq 2.2832363808567277 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(i \cdot i\right) \cdot 0.25}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{elif}\;i \leq 1.3577553450061928 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\alpha \cdot \beta + i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2} \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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)))