Average Error: 3.9 → 2.1
Time: 47.0s
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \mathsf{fma}\left(\left(\frac{\sqrt{a + t}}{t}\right), z, \left(\left(c - b\right) \cdot \left(\frac{5.0}{6.0} - \left(\frac{0.6666666666666666}{t} - a\right)\right)\right)\right)}\right), x\right)}\]
\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}
↓
\frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \mathsf{fma}\left(\left(\frac{\sqrt{a + t}}{t}\right), z, \left(\left(c - b\right) \cdot \left(\frac{5.0}{6.0} - \left(\frac{0.6666666666666666}{t} - a\right)\right)\right)\right)}\right), x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2534316 = x;
        double r2534317 = y;
        double r2534318 = 2.0;
        double r2534319 = z;
        double r2534320 = t;
        double r2534321 = a;
        double r2534322 = r2534320 + r2534321;
        double r2534323 = sqrt(r2534322);
        double r2534324 = r2534319 * r2534323;
        double r2534325 = r2534324 / r2534320;
        double r2534326 = b;
        double r2534327 = c;
        double r2534328 = r2534326 - r2534327;
        double r2534329 = 5.0;
        double r2534330 = 6.0;
        double r2534331 = r2534329 / r2534330;
        double r2534332 = r2534321 + r2534331;
        double r2534333 = 3.0;
        double r2534334 = r2534320 * r2534333;
        double r2534335 = r2534318 / r2534334;
        double r2534336 = r2534332 - r2534335;
        double r2534337 = r2534328 * r2534336;
        double r2534338 = r2534325 - r2534337;
        double r2534339 = r2534318 * r2534338;
        double r2534340 = exp(r2534339);
        double r2534341 = r2534317 * r2534340;
        double r2534342 = r2534316 + r2534341;
        double r2534343 = r2534316 / r2534342;
        return r2534343;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2534344 = x;
        double r2534345 = y;
        double r2534346 = 2.0;
        double r2534347 = a;
        double r2534348 = t;
        double r2534349 = r2534347 + r2534348;
        double r2534350 = sqrt(r2534349);
        double r2534351 = r2534350 / r2534348;
        double r2534352 = z;
        double r2534353 = c;
        double r2534354 = b;
        double r2534355 = r2534353 - r2534354;
        double r2534356 = 5.0;
        double r2534357 = 6.0;
        double r2534358 = r2534356 / r2534357;
        double r2534359 = 0.6666666666666666;
        double r2534360 = r2534359 / r2534348;
        double r2534361 = r2534360 - r2534347;
        double r2534362 = r2534358 - r2534361;
        double r2534363 = r2534355 * r2534362;
        double r2534364 = fma(r2534351, r2534352, r2534363);
        double r2534365 = r2534346 * r2534364;
        double r2534366 = exp(r2534365);
        double r2534367 = fma(r2534345, r2534366, r2534344);
        double r2534368 = r2534344 / r2534367;
        return r2534368;
}

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

    \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

  2. Simplified2.1

    \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \mathsf{fma}\left(\left(\frac{\sqrt{a + t}}{t}\right), z, \left(\left(c - b\right) \cdot \left(\frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right)\right)\right)\right)}\right), x\right)}}\]

  3. Taylor expanded around -inf 2.1

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \mathsf{fma}\left(\left(\frac{\sqrt{a + t}}{t}\right), z, \left(\left(c - b\right) \cdot \left(\frac{5.0}{6.0} - \left(\color{blue}{\frac{0.6666666666666666}{t}} - a\right)\right)\right)\right)}\right), x\right)}\]

  4. Final simplification2.1

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \mathsf{fma}\left(\left(\frac{\sqrt{a + t}}{t}\right), z, \left(\left(c - b\right) \cdot \left(\frac{5.0}{6.0} - \left(\frac{0.6666666666666666}{t} - a\right)\right)\right)\right)}\right), x\right)}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2"
  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))