Octave 3.8, jcobi/4

Average Error: 53.8 → 37.4

Time: 18.0s

Precision: 64

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

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

Math

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

↓

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

C

double f(double alpha, double beta, double i) {
        double r163267 = i;
        double r163268 = alpha;
        double r163269 = beta;
        double r163270 = r163268 + r163269;
        double r163271 = r163270 + r163267;
        double r163272 = r163267 * r163271;
        double r163273 = r163269 * r163268;
        double r163274 = r163273 + r163272;
        double r163275 = r163272 * r163274;
        double r163276 = 2.0;
        double r163277 = r163276 * r163267;
        double r163278 = r163270 + r163277;
        double r163279 = r163278 * r163278;
        double r163280 = r163275 / r163279;
        double r163281 = 1.0;
        double r163282 = r163279 - r163281;
        double r163283 = r163280 / r163282;
        return r163283;
}

↓

double f(double alpha, double beta, double i) {
        double r163284 = i;
        double r163285 = alpha;
        double r163286 = beta;
        double r163287 = r163285 + r163286;
        double r163288 = r163287 + r163284;
        double r163289 = r163284 * r163288;
        double r163290 = 2.0;
        double r163291 = fma(r163290, r163284, r163287);
        double r163292 = r163289 / r163291;
        double r163293 = fma(r163286, r163285, r163289);
        double r163294 = r163293 / r163291;
        double r163295 = 1.0;
        double r163296 = sqrt(r163295);
        double r163297 = r163296 + r163291;
        double r163298 = r163294 / r163297;
        double r163299 = r163292 * r163298;
        double r163300 = r163290 * r163284;
        double r163301 = r163287 + r163300;
        double r163302 = r163301 - r163296;
        double r163303 = r163299 / r163302;
        return r163303;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Initial program 53.8

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

3. Applied add-sqr-sqrt 53.8

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

4. Applied difference-of-squares 53.8

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

5. Applied associate-/r* 53.9

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

6. Simplified 37.4

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

8. Applied *-un-lft-identity 37.4

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

9. Applied add-sqr-sqrt 37.4

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

10. Applied times-frac 37.4

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

11. Simplified 37.4

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

12. Simplified 37.4

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

14. Applied *-un-lft-identity 37.4

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

15. Applied times-frac 37.4

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

16. Simplified 37.4

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

17. Simplified 37.4

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

18. Final simplification 37.4

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1)))