Average Error: 4.0 → 1.6
Time: 26.0s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt{t + a}}{\sqrt[3]{t}}\right)\right)}, x\right)}\]
\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}{t}}{3} - \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 r52456 = x;
        double r52457 = y;
        double r52458 = 2.0;
        double r52459 = z;
        double r52460 = t;
        double r52461 = a;
        double r52462 = r52460 + r52461;
        double r52463 = sqrt(r52462);
        double r52464 = r52459 * r52463;
        double r52465 = r52464 / r52460;
        double r52466 = b;
        double r52467 = c;
        double r52468 = r52466 - r52467;
        double r52469 = 5.0;
        double r52470 = 6.0;
        double r52471 = r52469 / r52470;
        double r52472 = r52461 + r52471;
        double r52473 = 3.0;
        double r52474 = r52460 * r52473;
        double r52475 = r52458 / r52474;
        double r52476 = r52472 - r52475;
        double r52477 = r52468 * r52476;
        double r52478 = r52465 - r52477;
        double r52479 = r52458 * r52478;
        double r52480 = exp(r52479);
        double r52481 = r52457 * r52480;
        double r52482 = r52456 + r52481;
        double r52483 = r52456 / r52482;
        return r52483;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r52484 = x;
        double r52485 = y;
        double r52486 = 2.0;
        double r52487 = exp(r52486);
        double r52488 = t;
        double r52489 = r52486 / r52488;
        double r52490 = 3.0;
        double r52491 = r52489 / r52490;
        double r52492 = a;
        double r52493 = 5.0;
        double r52494 = 6.0;
        double r52495 = r52493 / r52494;
        double r52496 = r52492 + r52495;
        double r52497 = r52491 - r52496;
        double r52498 = b;
        double r52499 = c;
        double r52500 = r52498 - r52499;
        double r52501 = z;
        double r52502 = cbrt(r52488);
        double r52503 = r52502 * r52502;
        double r52504 = r52501 / r52503;
        double r52505 = r52488 + r52492;
        double r52506 = sqrt(r52505);
        double r52507 = r52506 / r52502;
        double r52508 = r52504 * r52507;
        double r52509 = fma(r52497, r52500, r52508);
        double r52510 = pow(r52487, r52509);
        double r52511 = fma(r52485, r52510, r52484);
        double r52512 = r52484 / r52511;
        return r52512;
}

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

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

  4. Applied add-cube-cbrt2.6

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

  5. Applied times-frac1.6

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

  6. Final simplification1.6

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

Reproduce

herbie shell --seed 2019198 +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)))))))))))