Average Error: 3.9 → 1.8
Time: 20.4s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \sqrt{t + a} \cdot \frac{z}{t}\right)}, x\right)}\]
\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}{\mathsf{fma}\left(y, e^{2 \cdot \mathsf{fma}\left(c - b, \frac{5}{6} + \left(a - \frac{\frac{2}{t}}{3}\right), \sqrt{t + a} \cdot \frac{z}{t}\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r3137459 = x;
        double r3137460 = y;
        double r3137461 = 2.0;
        double r3137462 = z;
        double r3137463 = t;
        double r3137464 = a;
        double r3137465 = r3137463 + r3137464;
        double r3137466 = sqrt(r3137465);
        double r3137467 = r3137462 * r3137466;
        double r3137468 = r3137467 / r3137463;
        double r3137469 = b;
        double r3137470 = c;
        double r3137471 = r3137469 - r3137470;
        double r3137472 = 5.0;
        double r3137473 = 6.0;
        double r3137474 = r3137472 / r3137473;
        double r3137475 = r3137464 + r3137474;
        double r3137476 = 3.0;
        double r3137477 = r3137463 * r3137476;
        double r3137478 = r3137461 / r3137477;
        double r3137479 = r3137475 - r3137478;
        double r3137480 = r3137471 * r3137479;
        double r3137481 = r3137468 - r3137480;
        double r3137482 = r3137461 * r3137481;
        double r3137483 = exp(r3137482);
        double r3137484 = r3137460 * r3137483;
        double r3137485 = r3137459 + r3137484;
        double r3137486 = r3137459 / r3137485;
        return r3137486;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r3137487 = x;
        double r3137488 = y;
        double r3137489 = 2.0;
        double r3137490 = c;
        double r3137491 = b;
        double r3137492 = r3137490 - r3137491;
        double r3137493 = 5.0;
        double r3137494 = 6.0;
        double r3137495 = r3137493 / r3137494;
        double r3137496 = a;
        double r3137497 = t;
        double r3137498 = r3137489 / r3137497;
        double r3137499 = 3.0;
        double r3137500 = r3137498 / r3137499;
        double r3137501 = r3137496 - r3137500;
        double r3137502 = r3137495 + r3137501;
        double r3137503 = r3137497 + r3137496;
        double r3137504 = sqrt(r3137503);
        double r3137505 = z;
        double r3137506 = r3137505 / r3137497;
        double r3137507 = r3137504 * r3137506;
        double r3137508 = fma(r3137492, r3137502, r3137507);
        double r3137509 = r3137489 * r3137508;
        double r3137510 = exp(r3137509);
        double r3137511 = fma(r3137488, r3137510, r3137487);
        double r3137512 = r3137487 / r3137511;
        return r3137512;
}

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. Simplified1.8

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

  3. Final simplification1.8

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

Reproduce

herbie shell --seed 2019192 +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)))))))))))