Average Error: 4.0 → 1.4
Time: 3.5m
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \mathsf{fma}\left(\left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right), \left(\frac{\sqrt{t + a}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(\left(-\left(b - c\right)\right) \cdot \left(\left(a - \frac{2.0}{3.0 \cdot t}\right) + \frac{5.0}{6.0}\right)\right)\right)}\right), x\right)}\]
\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}
↓
\frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \mathsf{fma}\left(\left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right), \left(\frac{\sqrt{t + a}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(\left(-\left(b - c\right)\right) \cdot \left(\left(a - \frac{2.0}{3.0 \cdot t}\right) + \frac{5.0}{6.0}\right)\right)\right)}\right), x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r19859376 = x;
        double r19859377 = y;
        double r19859378 = 2.0;
        double r19859379 = z;
        double r19859380 = t;
        double r19859381 = a;
        double r19859382 = r19859380 + r19859381;
        double r19859383 = sqrt(r19859382);
        double r19859384 = r19859379 * r19859383;
        double r19859385 = r19859384 / r19859380;
        double r19859386 = b;
        double r19859387 = c;
        double r19859388 = r19859386 - r19859387;
        double r19859389 = 5.0;
        double r19859390 = 6.0;
        double r19859391 = r19859389 / r19859390;
        double r19859392 = r19859381 + r19859391;
        double r19859393 = 3.0;
        double r19859394 = r19859380 * r19859393;
        double r19859395 = r19859378 / r19859394;
        double r19859396 = r19859392 - r19859395;
        double r19859397 = r19859388 * r19859396;
        double r19859398 = r19859385 - r19859397;
        double r19859399 = r19859378 * r19859398;
        double r19859400 = exp(r19859399);
        double r19859401 = r19859377 * r19859400;
        double r19859402 = r19859376 + r19859401;
        double r19859403 = r19859376 / r19859402;
        return r19859403;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r19859404 = x;
        double r19859405 = y;
        double r19859406 = 2.0;
        double r19859407 = z;
        double r19859408 = cbrt(r19859407);
        double r19859409 = t;
        double r19859410 = cbrt(r19859409);
        double r19859411 = r19859408 / r19859410;
        double r19859412 = r19859411 * r19859411;
        double r19859413 = a;
        double r19859414 = r19859409 + r19859413;
        double r19859415 = sqrt(r19859414);
        double r19859416 = r19859410 / r19859408;
        double r19859417 = r19859415 / r19859416;
        double r19859418 = b;
        double r19859419 = c;
        double r19859420 = r19859418 - r19859419;
        double r19859421 = -r19859420;
        double r19859422 = 3.0;
        double r19859423 = r19859422 * r19859409;
        double r19859424 = r19859406 / r19859423;
        double r19859425 = r19859413 - r19859424;
        double r19859426 = 5.0;
        double r19859427 = 6.0;
        double r19859428 = r19859426 / r19859427;
        double r19859429 = r19859425 + r19859428;
        double r19859430 = r19859421 * r19859429;
        double r19859431 = fma(r19859412, r19859417, r19859430);
        double r19859432 = r19859406 * r19859431;
        double r19859433 = exp(r19859432);
        double r19859434 = fma(r19859405, r19859433, r19859404);
        double r19859435 = r19859404 / r19859434;
        return r19859435;
}

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

    \[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]

  2. Simplified3.2

    \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\sqrt{a + t}}{\frac{t}{z}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt3.2

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\sqrt{a + t}}{\frac{t}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

  5. Applied add-cube-cbrt3.2

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\sqrt{a + t}}{\frac{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

  6. Applied times-frac3.2

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\sqrt{a + t}}{\color{blue}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{z}}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

  7. Applied *-un-lft-identity3.2

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\frac{\color{blue}{1 \cdot \sqrt{a + t}}}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{z}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

  8. Applied times-frac2.8

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \left(\color{blue}{\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}} - \left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)}\right), x\right)}\]

  9. Applied fma-neg1.4

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}\right), \left(\frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(-\left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)\right)}}\right), x\right)}\]

  10. Simplified1.4

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right)}, \left(\frac{\sqrt{a + t}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(-\left(\left(a - \frac{2.0}{t \cdot 3.0}\right) + \frac{5.0}{6.0}\right) \cdot \left(b - c\right)\right)\right)}\right), x\right)}\]

  11. Final simplification1.4

    \[\leadsto \frac{x}{\mathsf{fma}\left(y, \left(e^{2.0 \cdot \mathsf{fma}\left(\left(\frac{\sqrt[3]{z}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right), \left(\frac{\sqrt{t + a}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\right), \left(\left(-\left(b - c\right)\right) \cdot \left(\left(a - \frac{2.0}{3.0 \cdot t}\right) + \frac{5.0}{6.0}\right)\right)\right)}\right), x\right)}\]

Reproduce

herbie shell --seed 2019124 +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)))))))))))