Average Error: 4.0 → 1.4
Time: 12.9s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{3}}{t} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]
\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, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{3}}{t} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r89448 = x;
        double r89449 = y;
        double r89450 = 2.0;
        double r89451 = z;
        double r89452 = t;
        double r89453 = a;
        double r89454 = r89452 + r89453;
        double r89455 = sqrt(r89454);
        double r89456 = r89451 * r89455;
        double r89457 = r89456 / r89452;
        double r89458 = b;
        double r89459 = c;
        double r89460 = r89458 - r89459;
        double r89461 = 5.0;
        double r89462 = 6.0;
        double r89463 = r89461 / r89462;
        double r89464 = r89453 + r89463;
        double r89465 = 3.0;
        double r89466 = r89452 * r89465;
        double r89467 = r89450 / r89466;
        double r89468 = r89464 - r89467;
        double r89469 = r89460 * r89468;
        double r89470 = r89457 - r89469;
        double r89471 = r89450 * r89470;
        double r89472 = exp(r89471);
        double r89473 = r89449 * r89472;
        double r89474 = r89448 + r89473;
        double r89475 = r89448 / r89474;
        return r89475;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r89476 = x;
        double r89477 = y;
        double r89478 = 2.0;
        double r89479 = exp(r89478);
        double r89480 = 3.0;
        double r89481 = r89478 / r89480;
        double r89482 = t;
        double r89483 = r89481 / r89482;
        double r89484 = a;
        double r89485 = 5.0;
        double r89486 = 6.0;
        double r89487 = r89485 / r89486;
        double r89488 = r89484 + r89487;
        double r89489 = r89483 - r89488;
        double r89490 = b;
        double r89491 = c;
        double r89492 = r89490 - r89491;
        double r89493 = z;
        double r89494 = cbrt(r89482);
        double r89495 = r89494 * r89494;
        double r89496 = r89493 / r89495;
        double r89497 = r89482 + r89484;
        double r89498 = sqrt(r89497);
        double r89499 = r89498 / r89494;
        double r89500 = r89496 * r89499;
        double r89501 = fma(r89489, r89492, r89500);
        double r89502 = pow(r89479, r89501);
        double r89503 = fma(r89477, r89502, r89476);
        double r89504 = r89476 / r89503;
        return r89504;
}

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 4.0

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

  2. Simplified2.6

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

  4. Applied add-cube-cbrt2.6

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

  5. Applied times-frac1.4

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

  6. Final simplification1.4

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

Reproduce

herbie shell --seed 2020047 +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)))))))))))