Average Error: 4.2 → 2.3
Time: 7.8s
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 r83453 = x;
        double r83454 = y;
        double r83455 = 2.0;
        double r83456 = z;
        double r83457 = t;
        double r83458 = a;
        double r83459 = r83457 + r83458;
        double r83460 = sqrt(r83459);
        double r83461 = r83456 * r83460;
        double r83462 = r83461 / r83457;
        double r83463 = b;
        double r83464 = c;
        double r83465 = r83463 - r83464;
        double r83466 = 5.0;
        double r83467 = 6.0;
        double r83468 = r83466 / r83467;
        double r83469 = r83458 + r83468;
        double r83470 = 3.0;
        double r83471 = r83457 * r83470;
        double r83472 = r83455 / r83471;
        double r83473 = r83469 - r83472;
        double r83474 = r83465 * r83473;
        double r83475 = r83462 - r83474;
        double r83476 = r83455 * r83475;
        double r83477 = exp(r83476);
        double r83478 = r83454 * r83477;
        double r83479 = r83453 + r83478;
        double r83480 = r83453 / r83479;
        return r83480;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r83481 = x;
        double r83482 = y;
        double r83483 = 2.0;
        double r83484 = z;
        double r83485 = t;
        double r83486 = a;
        double r83487 = r83485 + r83486;
        double r83488 = sqrt(r83487);
        double r83489 = r83488 / r83485;
        double r83490 = b;
        double r83491 = c;
        double r83492 = r83490 - r83491;
        double r83493 = 5.0;
        double r83494 = 6.0;
        double r83495 = r83493 / r83494;
        double r83496 = r83486 + r83495;
        double r83497 = 3.0;
        double r83498 = r83485 * r83497;
        double r83499 = r83483 / r83498;
        double r83500 = r83496 - r83499;
        double r83501 = r83492 * r83500;
        double r83502 = -r83501;
        double r83503 = fma(r83484, r83489, r83502);
        double r83504 = -r83492;
        double r83505 = r83504 + r83492;
        double r83506 = r83500 * r83505;
        double r83507 = r83503 + r83506;
        double r83508 = r83483 * r83507;
        double r83509 = exp(r83508);
        double r83510 = r83482 * r83509;
        double r83511 = r83481 + r83510;
        double r83512 = r83481 / r83511;
        return r83512;
}

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 4.2

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

  3. Applied *-un-lft-identity4.2

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

  4. Applied times-frac3.5

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

  5. Applied prod-diff22.5

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

  6. Simplified22.5

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

  7. Simplified2.3

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

  8. Final simplification2.3

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

Reproduce

herbie shell --seed 2020083 +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)))))))))))