Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2

Average Error: 3.9 → 2.6

Time: 9.0s

Precision: 64

Math

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

↓

\[\frac{x}{x + y \cdot e^{2 \cdot \sqrt[3]{{\left(\mathsf{fma}\left(-\left(b - c\right), \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}^{3}}}}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r107280 = x;
        double r107281 = y;
        double r107282 = 2.0;
        double r107283 = z;
        double r107284 = t;
        double r107285 = a;
        double r107286 = r107284 + r107285;
        double r107287 = sqrt(r107286);
        double r107288 = r107283 * r107287;
        double r107289 = r107288 / r107284;
        double r107290 = b;
        double r107291 = c;
        double r107292 = r107290 - r107291;
        double r107293 = 5.0;
        double r107294 = 6.0;
        double r107295 = r107293 / r107294;
        double r107296 = r107285 + r107295;
        double r107297 = 3.0;
        double r107298 = r107284 * r107297;
        double r107299 = r107282 / r107298;
        double r107300 = r107296 - r107299;
        double r107301 = r107292 * r107300;
        double r107302 = r107289 - r107301;
        double r107303 = r107282 * r107302;
        double r107304 = exp(r107303);
        double r107305 = r107281 * r107304;
        double r107306 = r107280 + r107305;
        double r107307 = r107280 / r107306;
        return r107307;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r107308 = x;
        double r107309 = y;
        double r107310 = 2.0;
        double r107311 = b;
        double r107312 = c;
        double r107313 = r107311 - r107312;
        double r107314 = -r107313;
        double r107315 = a;
        double r107316 = 5.0;
        double r107317 = 6.0;
        double r107318 = r107316 / r107317;
        double r107319 = r107315 + r107318;
        double r107320 = t;
        double r107321 = 3.0;
        double r107322 = r107320 * r107321;
        double r107323 = r107310 / r107322;
        double r107324 = r107319 - r107323;
        double r107325 = z;
        double r107326 = r107320 + r107315;
        double r107327 = sqrt(r107326);
        double r107328 = r107325 * r107327;
        double r107329 = r107328 / r107320;
        double r107330 = fma(r107314, r107324, r107329);
        double r107331 = 3.0;
        double r107332 = pow(r107330, r107331);
        double r107333 = cbrt(r107332);
        double r107334 = r107310 * r107333;
        double r107335 = exp(r107334);
        double r107336 = r107309 * r107335;
        double r107337 = r107308 + r107336;
        double r107338 = r107308 / r107337;
        return r107338;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 3.9

   \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
2. Using strategy rm

3. Applied add-cbrt-cube 3.9

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot \color{blue}{\sqrt[3]{\left(3 \cdot 3\right) \cdot 3}}}\right)\right)}}\]

4. Applied add-cbrt-cube 7.0

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{\color{blue}{\sqrt[3]{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\left(3 \cdot 3\right) \cdot 3}}\right)\right)}}\]

5. Applied cbrt-unprod 7.0

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{\color{blue}{\sqrt[3]{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\left(3 \cdot 3\right) \cdot 3\right)}}}\right)\right)}}\]

6. Applied add-cbrt-cube 7.0

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{\color{blue}{\sqrt[3]{\left(2 \cdot 2\right) \cdot 2}}}{\sqrt[3]{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\left(3 \cdot 3\right) \cdot 3\right)}}\right)\right)}}\]

7. Applied cbrt-undiv 7.1

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \color{blue}{\sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(t \cdot t\right) \cdot t\right) \cdot \left(\left(3 \cdot 3\right) \cdot 3\right)}}}\right)\right)}}\]

8. Simplified 7.1

   \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \sqrt[3]{\color{blue}{{\left(\frac{2}{t \cdot 3}\right)}^{3}}}\right)\right)}}\]
9. Using strategy rm

10. Applied add-cube-cbrt 7.1

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \sqrt[3]{{\left(\frac{2}{t \cdot 3}\right)}^{3}}\right)\right)}}\]

11. Applied times-frac 5.9

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\color{blue}{\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \sqrt[3]{{\left(\frac{2}{t \cdot 3}\right)}^{3}}\right)\right)}}\]

12. Applied fma-neg 5.2

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \sqrt[3]{{\left(\frac{2}{t \cdot 3}\right)}^{3}}\right)\right)}}}\]

13. Simplified 1.9

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, \color{blue}{\left(-\mathsf{fma}\left(1, a + \frac{5}{6}, -\frac{2}{t \cdot 3}\right)\right) \cdot \left(b - c\right)}\right)}}\]
14. Using strategy rm

15. Applied add-cbrt-cube 1.9

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, \left(-\mathsf{fma}\left(1, a + \frac{5}{6}, -\frac{2}{t \cdot 3}\right)\right) \cdot \left(b - c\right)\right) \cdot \mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, \left(-\mathsf{fma}\left(1, a + \frac{5}{6}, -\frac{2}{t \cdot 3}\right)\right) \cdot \left(b - c\right)\right)\right) \cdot \mathsf{fma}\left(\frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}}, \frac{\sqrt{t + a}}{\sqrt[3]{t}}, \left(-\mathsf{fma}\left(1, a + \frac{5}{6}, -\frac{2}{t \cdot 3}\right)\right) \cdot \left(b - c\right)\right)}}}}\]

16. Simplified 2.6

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(-\left(b - c\right), \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}^{3}}}}}\]

17. Final simplification 2.6

    \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \sqrt[3]{{\left(\mathsf{fma}\left(-\left(b - c\right), \left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}, \frac{z \cdot \sqrt{t + a}}{t}\right)\right)}^{3}}}}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ x (+ x (* y (exp (* 2 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5 6)) (/ 2 (* t 3)))))))))))