Octave 3.8, jcobi/2

Average Error: 23.6 → 7.3

Time: 18.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r125398 = alpha;
        double r125399 = beta;
        double r125400 = r125398 + r125399;
        double r125401 = r125399 - r125398;
        double r125402 = r125400 * r125401;
        double r125403 = 2.0;
        double r125404 = i;
        double r125405 = r125403 * r125404;
        double r125406 = r125400 + r125405;
        double r125407 = r125402 / r125406;
        double r125408 = r125406 + r125403;
        double r125409 = r125407 / r125408;
        double r125410 = 1.0;
        double r125411 = r125409 + r125410;
        double r125412 = r125411 / r125403;
        return r125412;
}

↓

double f(double alpha, double beta, double i) {
        double r125413 = alpha;
        double r125414 = beta;
        double r125415 = r125413 + r125414;
        double r125416 = r125414 - r125413;
        double r125417 = r125415 * r125416;
        double r125418 = 2.0;
        double r125419 = i;
        double r125420 = r125418 * r125419;
        double r125421 = r125415 + r125420;
        double r125422 = r125417 / r125421;
        double r125423 = r125421 + r125418;
        double r125424 = r125422 / r125423;
        double r125425 = -0.9999999999999999;
        bool r125426 = r125424 <= r125425;
        double r125427 = r125418 / r125413;
        double r125428 = 8.0;
        double r125429 = 3.0;
        double r125430 = pow(r125413, r125429);
        double r125431 = r125428 / r125430;
        double r125432 = 4.0;
        double r125433 = 2.0;
        double r125434 = pow(r125413, r125433);
        double r125435 = r125432 / r125434;
        double r125436 = r125431 - r125435;
        double r125437 = r125427 + r125436;
        double r125438 = r125437 / r125418;
        double r125439 = sqrt(r125423);
        double r125440 = r125415 / r125439;
        double r125441 = r125416 / r125421;
        double r125442 = r125441 / r125439;
        double r125443 = r125440 * r125442;
        double r125444 = 1.0;
        double r125445 = r125443 + r125444;
        double r125446 = pow(r125445, r125429);
        double r125447 = cbrt(r125446);
        double r125448 = r125447 / r125418;
        double r125449 = r125426 ? r125438 : r125448;
        return r125449;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

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 add-cube-cbrt 63.0

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

   4. Applied *-un-lft-identity 63.0

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

   5. Applied times-frac 54.6

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

   6. Applied times-frac 54.6

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

   7. Simplified 54.6

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

   8. Taylor expanded around inf 32.0

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

   9. Simplified 32.0

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

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

   1. Initial program 12.6

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

   3. Applied add-cube-cbrt 12.7

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

   4. Applied *-un-lft-identity 12.7

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

   5. Applied times-frac 0.6

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

   6. Applied times-frac 0.6

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

   7. Simplified 0.6

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

   9. Applied add-cbrt-cube 0.5

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

   10. Simplified 0.4

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

   12. Applied add-sqr-sqrt 0.4

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

   13. Applied times-frac 0.4

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

4. Final simplification 7.3

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

Reproduce

herbie shell --seed 2020042 
(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))