Octave 3.8, jcobi/2

Average Error: 23.7 → 12.6

Time: 34.6s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]

Math

\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

↓

\[\frac{\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta}, \left(\frac{\alpha + \beta}{\left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta}\right) \cdot \left(\frac{\alpha + \beta}{\left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta}\right), \left(1.0 \cdot 1.0\right) \cdot 1.0\right)}{\mathsf{fma}\left(1.0, 1.0, \left(\frac{\alpha + \beta}{\left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta}\right) \cdot \left(\sqrt[3]{\frac{\alpha + \beta}{\left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} - 1.0} \cdot \left(\sqrt[3]{\frac{\alpha + \beta}{\left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} - 1.0} \cdot \sqrt[3]{\frac{\alpha + \beta}{\left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} - 1.0}\right)\right)\right)}}{2.0}\]

C

double f(double alpha, double beta, double i) {
        double r4216289 = alpha;
        double r4216290 = beta;
        double r4216291 = r4216289 + r4216290;
        double r4216292 = r4216290 - r4216289;
        double r4216293 = r4216291 * r4216292;
        double r4216294 = 2.0;
        double r4216295 = i;
        double r4216296 = r4216294 * r4216295;
        double r4216297 = r4216291 + r4216296;
        double r4216298 = r4216293 / r4216297;
        double r4216299 = 2.0;
        double r4216300 = r4216297 + r4216299;
        double r4216301 = r4216298 / r4216300;
        double r4216302 = 1.0;
        double r4216303 = r4216301 + r4216302;
        double r4216304 = r4216303 / r4216299;
        return r4216304;
}

↓

double f(double alpha, double beta, double i) {
        double r4216305 = alpha;
        double r4216306 = beta;
        double r4216307 = r4216305 + r4216306;
        double r4216308 = i;
        double r4216309 = 2.0;
        double r4216310 = fma(r4216308, r4216309, r4216305);
        double r4216311 = r4216310 + r4216306;
        double r4216312 = 2.0;
        double r4216313 = r4216311 + r4216312;
        double r4216314 = r4216307 / r4216313;
        double r4216315 = r4216306 - r4216305;
        double r4216316 = r4216315 / r4216311;
        double r4216317 = r4216314 * r4216316;
        double r4216318 = r4216317 * r4216317;
        double r4216319 = 1.0;
        double r4216320 = r4216319 * r4216319;
        double r4216321 = r4216320 * r4216319;
        double r4216322 = fma(r4216317, r4216318, r4216321);
        double r4216323 = r4216317 - r4216319;
        double r4216324 = cbrt(r4216323);
        double r4216325 = r4216324 * r4216324;
        double r4216326 = r4216324 * r4216325;
        double r4216327 = r4216317 * r4216326;
        double r4216328 = fma(r4216319, r4216319, r4216327);
        double r4216329 = r4216322 / r4216328;
        double r4216330 = r4216329 / r4216312;
        return r4216330;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Initial program 23.7

   \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

2. Simplified 23.5

   \[\leadsto \color{blue}{\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)} + 1.0}{2.0}}\]
3. Using strategy rm

4. Applied flip3-+ 23.5

   \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}\right)}^{3} + {1.0}^{3}}{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)} \cdot \frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)} + \left(1.0 \cdot 1.0 - \frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)} \cdot 1.0\right)}}}{2.0}\]

5. Simplified 23.5

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)}, \left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)}\right) \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)}\right), 1.0 \cdot \left(1.0 \cdot 1.0\right)\right)}}{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)} \cdot \frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)} + \left(1.0 \cdot 1.0 - \frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)} \cdot 1.0\right)}}{2.0}\]

6. Simplified 12.6

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)}, \left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)}\right) \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)}\right), 1.0 \cdot \left(1.0 \cdot 1.0\right)\right)}{\color{blue}{\mathsf{fma}\left(1.0, 1.0, \left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)}\right) \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)} - 1.0\right)\right)}}}{2.0}\]
7. Using strategy rm

8. Applied add-cube-cbrt 12.6

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)}, \left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)}\right) \cdot \left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)}\right), 1.0 \cdot \left(1.0 \cdot 1.0\right)\right)}{\mathsf{fma}\left(1.0, 1.0, \left(\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)} - 1.0} \cdot \sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)} - 1.0}\right) \cdot \sqrt[3]{\frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} \cdot \frac{\alpha + \beta}{2.0 + \left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right)} - 1.0}\right)}\right)}}{2.0}\]

9. Final simplification 12.6

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta}, \left(\frac{\alpha + \beta}{\left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta}\right) \cdot \left(\frac{\alpha + \beta}{\left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta}\right), \left(1.0 \cdot 1.0\right) \cdot 1.0\right)}{\mathsf{fma}\left(1.0, 1.0, \left(\frac{\alpha + \beta}{\left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta}\right) \cdot \left(\sqrt[3]{\frac{\alpha + \beta}{\left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} - 1.0} \cdot \left(\sqrt[3]{\frac{\alpha + \beta}{\left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} - 1.0} \cdot \sqrt[3]{\frac{\alpha + \beta}{\left(\mathsf{fma}\left(i, 2, \alpha\right) + \beta\right) + 2.0} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(i, 2, \alpha\right) + \beta} - 1.0}\right)\right)\right)}}{2.0}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))