Average Error: 3.7 → 1.3
Time: 30.5s
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 r87325 = x;
        double r87326 = y;
        double r87327 = 2.0;
        double r87328 = z;
        double r87329 = t;
        double r87330 = a;
        double r87331 = r87329 + r87330;
        double r87332 = sqrt(r87331);
        double r87333 = r87328 * r87332;
        double r87334 = r87333 / r87329;
        double r87335 = b;
        double r87336 = c;
        double r87337 = r87335 - r87336;
        double r87338 = 5.0;
        double r87339 = 6.0;
        double r87340 = r87338 / r87339;
        double r87341 = r87330 + r87340;
        double r87342 = 3.0;
        double r87343 = r87329 * r87342;
        double r87344 = r87327 / r87343;
        double r87345 = r87341 - r87344;
        double r87346 = r87337 * r87345;
        double r87347 = r87334 - r87346;
        double r87348 = r87327 * r87347;
        double r87349 = exp(r87348);
        double r87350 = r87326 * r87349;
        double r87351 = r87325 + r87350;
        double r87352 = r87325 / r87351;
        return r87352;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r87353 = x;
        double r87354 = y;
        double r87355 = 2.0;
        double r87356 = exp(r87355);
        double r87357 = t;
        double r87358 = r87355 / r87357;
        double r87359 = 3.0;
        double r87360 = r87358 / r87359;
        double r87361 = a;
        double r87362 = 5.0;
        double r87363 = 6.0;
        double r87364 = r87362 / r87363;
        double r87365 = r87361 + r87364;
        double r87366 = r87360 - r87365;
        double r87367 = b;
        double r87368 = c;
        double r87369 = r87367 - r87368;
        double r87370 = z;
        double r87371 = cbrt(r87357);
        double r87372 = r87371 * r87371;
        double r87373 = r87370 / r87372;
        double r87374 = r87357 + r87361;
        double r87375 = sqrt(r87374);
        double r87376 = r87375 / r87371;
        double r87377 = r87373 * r87376;
        double r87378 = fma(r87366, r87369, r87377);
        double r87379 = pow(r87356, r87378);
        double r87380 = fma(r87354, r87379, r87353);
        double r87381 = r87353 / r87380;
        return r87381;
}

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 3.7

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

  2. Simplified2.4

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

  4. Applied add-cube-cbrt2.4

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

  5. Applied times-frac1.3

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

  6. Final simplification1.3

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

Reproduce

herbie shell --seed 2019209 +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)))))))))))