Average Error: 3.8 → 1.5
Time: 17.3s
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 r53269 = x;
        double r53270 = y;
        double r53271 = 2.0;
        double r53272 = z;
        double r53273 = t;
        double r53274 = a;
        double r53275 = r53273 + r53274;
        double r53276 = sqrt(r53275);
        double r53277 = r53272 * r53276;
        double r53278 = r53277 / r53273;
        double r53279 = b;
        double r53280 = c;
        double r53281 = r53279 - r53280;
        double r53282 = 5.0;
        double r53283 = 6.0;
        double r53284 = r53282 / r53283;
        double r53285 = r53274 + r53284;
        double r53286 = 3.0;
        double r53287 = r53273 * r53286;
        double r53288 = r53271 / r53287;
        double r53289 = r53285 - r53288;
        double r53290 = r53281 * r53289;
        double r53291 = r53278 - r53290;
        double r53292 = r53271 * r53291;
        double r53293 = exp(r53292);
        double r53294 = r53270 * r53293;
        double r53295 = r53269 + r53294;
        double r53296 = r53269 / r53295;
        return r53296;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r53297 = x;
        double r53298 = y;
        double r53299 = 2.0;
        double r53300 = exp(r53299);
        double r53301 = t;
        double r53302 = r53299 / r53301;
        double r53303 = 3.0;
        double r53304 = r53302 / r53303;
        double r53305 = a;
        double r53306 = 5.0;
        double r53307 = 6.0;
        double r53308 = r53306 / r53307;
        double r53309 = r53305 + r53308;
        double r53310 = r53304 - r53309;
        double r53311 = b;
        double r53312 = c;
        double r53313 = r53311 - r53312;
        double r53314 = z;
        double r53315 = cbrt(r53301);
        double r53316 = r53315 * r53315;
        double r53317 = r53314 / r53316;
        double r53318 = r53301 + r53305;
        double r53319 = sqrt(r53318);
        double r53320 = r53319 / r53315;
        double r53321 = r53317 * r53320;
        double r53322 = fma(r53310, r53313, r53321);
        double r53323 = pow(r53300, r53322);
        double r53324 = fma(r53298, r53323, r53297);
        double r53325 = r53297 / r53324;
        return r53325;
}

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 3.8

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

  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-frac1.5

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

  6. Final simplification1.5

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

Reproduce

herbie shell --seed 2019347 +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)))))))))))