Average Error: 3.9 → 2.1
Time: 14.7s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}\]
\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 \left(\mathsf{fma}\left(z, \frac{\sqrt{t + a}}{t}, -\left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right) + \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right) \cdot \left(\left(-\left(b - c\right)\right) + \left(b - c\right)\right)\right)}}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r140315 = x;
        double r140316 = y;
        double r140317 = 2.0;
        double r140318 = z;
        double r140319 = t;
        double r140320 = a;
        double r140321 = r140319 + r140320;
        double r140322 = sqrt(r140321);
        double r140323 = r140318 * r140322;
        double r140324 = r140323 / r140319;
        double r140325 = b;
        double r140326 = c;
        double r140327 = r140325 - r140326;
        double r140328 = 5.0;
        double r140329 = 6.0;
        double r140330 = r140328 / r140329;
        double r140331 = r140320 + r140330;
        double r140332 = 3.0;
        double r140333 = r140319 * r140332;
        double r140334 = r140317 / r140333;
        double r140335 = r140331 - r140334;
        double r140336 = r140327 * r140335;
        double r140337 = r140324 - r140336;
        double r140338 = r140317 * r140337;
        double r140339 = exp(r140338);
        double r140340 = r140316 * r140339;
        double r140341 = r140315 + r140340;
        double r140342 = r140315 / r140341;
        return r140342;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r140343 = x;
        double r140344 = y;
        double r140345 = 2.0;
        double r140346 = z;
        double r140347 = t;
        double r140348 = a;
        double r140349 = r140347 + r140348;
        double r140350 = sqrt(r140349);
        double r140351 = r140350 / r140347;
        double r140352 = b;
        double r140353 = c;
        double r140354 = r140352 - r140353;
        double r140355 = 5.0;
        double r140356 = 6.0;
        double r140357 = r140355 / r140356;
        double r140358 = r140348 + r140357;
        double r140359 = 3.0;
        double r140360 = r140347 * r140359;
        double r140361 = r140345 / r140360;
        double r140362 = r140358 - r140361;
        double r140363 = r140354 * r140362;
        double r140364 = -r140363;
        double r140365 = fma(r140346, r140351, r140364);
        double r140366 = -r140354;
        double r140367 = r140366 + r140354;
        double r140368 = r140362 * r140367;
        double r140369 = r140365 + r140368;
        double r140370 = r140345 * r140369;
        double r140371 = exp(r140370);
        double r140372 = r140344 * r140371;
        double r140373 = r140343 + r140372;
        double r140374 = r140343 / r140373;
        return r140374;
}

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 *-un-lft-identity3.9

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

  4. Applied times-frac3.4

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

  5. Applied prod-diff22.0

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

  6. Simplified22.0

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

  7. Simplified2.1

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

  8. Final simplification2.1

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

Reproduce

herbie shell --seed 2020002 +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)))))))))))