Average Error: 4.2 → 2.1
Time: 28.8s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, z \cdot \frac{\sqrt{t + a}}{t}\right)\right)}, x\right)}\]
\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, z \cdot \frac{\sqrt{t + a}}{t}\right)\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r67371 = x;
        double r67372 = y;
        double r67373 = 2.0;
        double r67374 = z;
        double r67375 = t;
        double r67376 = a;
        double r67377 = r67375 + r67376;
        double r67378 = sqrt(r67377);
        double r67379 = r67374 * r67378;
        double r67380 = r67379 / r67375;
        double r67381 = b;
        double r67382 = c;
        double r67383 = r67381 - r67382;
        double r67384 = 5.0;
        double r67385 = 6.0;
        double r67386 = r67384 / r67385;
        double r67387 = r67376 + r67386;
        double r67388 = 3.0;
        double r67389 = r67375 * r67388;
        double r67390 = r67373 / r67389;
        double r67391 = r67387 - r67390;
        double r67392 = r67383 * r67391;
        double r67393 = r67380 - r67392;
        double r67394 = r67373 * r67393;
        double r67395 = exp(r67394);
        double r67396 = r67372 * r67395;
        double r67397 = r67371 + r67396;
        double r67398 = r67371 / r67397;
        return r67398;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r67399 = x;
        double r67400 = y;
        double r67401 = 2.0;
        double r67402 = exp(r67401);
        double r67403 = t;
        double r67404 = r67401 / r67403;
        double r67405 = 3.0;
        double r67406 = r67404 / r67405;
        double r67407 = a;
        double r67408 = 5.0;
        double r67409 = 6.0;
        double r67410 = r67408 / r67409;
        double r67411 = r67407 + r67410;
        double r67412 = r67406 - r67411;
        double r67413 = b;
        double r67414 = c;
        double r67415 = r67413 - r67414;
        double r67416 = z;
        double r67417 = r67403 + r67407;
        double r67418 = sqrt(r67417);
        double r67419 = r67418 / r67403;
        double r67420 = r67416 * r67419;
        double r67421 = fma(r67412, r67415, r67420);
        double r67422 = pow(r67402, r67421);
        double r67423 = fma(r67400, r67422, r67399);
        double r67424 = r67399 / r67423;
        return r67424;
}

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 4.2

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

  2. Simplified2.7

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

  4. Applied *-un-lft-identity2.7

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \frac{z \cdot \sqrt{t + a}}{\color{blue}{1 \cdot t}}\right)\right)}, x\right)}\]

  5. Applied times-frac2.1

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, \color{blue}{\frac{z}{1} \cdot \frac{\sqrt{t + a}}{t}}\right)\right)}, x\right)}\]

  6. Simplified2.1

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

  7. Final simplification2.1

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

Reproduce

herbie shell --seed 2019326 +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)))))))))))