Average Error: 24.2 → 8.0
Time: 8.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \le -0.999999999999999889:\\ \;\;\;\;\frac{\left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1\right)}^{3}}}{2}\\ \end{array}\]
\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}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \le -0.999999999999999889:\\
\;\;\;\;\frac{\left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}{2}\\

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

\end{array}
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 VAR;
	if ((((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (((alpha + beta) + (2.0 * i)) + 2.0)) <= -0.9999999999999999)) {
		VAR = ((((8.0 * (1.0 / pow(alpha, 3.0))) - (4.0 * (1.0 / pow(alpha, 2.0)))) + (2.0 / alpha)) / 2.0);
	} else {
		VAR = (cbrt(pow(((((alpha + beta) / sqrt((((alpha + beta) + (2.0 * i)) + 2.0))) * (((beta - alpha) / ((alpha + beta) + (2.0 * i))) / sqrt((((alpha + beta) + (2.0 * i)) + 2.0)))) + 1.0), 3.0)) / 2.0);
	}
	return VAR;
}

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 2 regimes

  2. if (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) < -0.9999999999999999

    1. Initial program 63.3

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

    3. Applied *-un-lft-identity63.3

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

    4. Applied times-frac54.3

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

    5. Simplified54.3

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

    6. Taylor expanded around inf 34.1

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

    7. Simplified34.1

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

    if -0.9999999999999999 < (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0))

    1. Initial program 12.8

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

    3. Applied *-un-lft-identity12.8

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

    4. Applied *-un-lft-identity12.8

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

    5. Applied times-frac0.3

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

    6. Applied times-frac0.3

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

    7. Simplified0.3

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

    9. Applied add-cbrt-cube0.3

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

    10. Simplified0.3

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1\right)}^{3}}}}{2}\]
    11. Using strategy rm

    12. Applied add-sqr-sqrt0.4

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

    13. Applied *-un-lft-identity0.4

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

    14. Applied *-un-lft-identity0.4

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

    15. Applied times-frac0.4

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

    16. Applied times-frac0.4

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

    17. Applied associate-*r*0.4

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

    18. Simplified0.4

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

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} \le -0.999999999999999889:\\ \;\;\;\;\frac{\left(8 \cdot \frac{1}{{\alpha}^{3}} - 4 \cdot \frac{1}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1\right)}^{3}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))