Average Error: 4.0 → 2.4
Time: 27.4s
Precision: 64
\[\frac{x}{x + y \cdot e^{2.0 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5.0}{6.0}\right) - \frac{2.0}{t \cdot 3.0}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{t} \cdot z\right)}, x\right)}\]
\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, e^{2.0 \cdot \mathsf{fma}\left(c - b, \frac{5.0}{6.0} - \left(\frac{\frac{2.0}{t}}{3.0} - a\right), \frac{\sqrt{a + t}}{t} \cdot z\right)}, x\right)}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2024459 = x;
        double r2024460 = y;
        double r2024461 = 2.0;
        double r2024462 = z;
        double r2024463 = t;
        double r2024464 = a;
        double r2024465 = r2024463 + r2024464;
        double r2024466 = sqrt(r2024465);
        double r2024467 = r2024462 * r2024466;
        double r2024468 = r2024467 / r2024463;
        double r2024469 = b;
        double r2024470 = c;
        double r2024471 = r2024469 - r2024470;
        double r2024472 = 5.0;
        double r2024473 = 6.0;
        double r2024474 = r2024472 / r2024473;
        double r2024475 = r2024464 + r2024474;
        double r2024476 = 3.0;
        double r2024477 = r2024463 * r2024476;
        double r2024478 = r2024461 / r2024477;
        double r2024479 = r2024475 - r2024478;
        double r2024480 = r2024471 * r2024479;
        double r2024481 = r2024468 - r2024480;
        double r2024482 = r2024461 * r2024481;
        double r2024483 = exp(r2024482);
        double r2024484 = r2024460 * r2024483;
        double r2024485 = r2024459 + r2024484;
        double r2024486 = r2024459 / r2024485;
        return r2024486;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r2024487 = x;
        double r2024488 = y;
        double r2024489 = 2.0;
        double r2024490 = c;
        double r2024491 = b;
        double r2024492 = r2024490 - r2024491;
        double r2024493 = 5.0;
        double r2024494 = 6.0;
        double r2024495 = r2024493 / r2024494;
        double r2024496 = t;
        double r2024497 = r2024489 / r2024496;
        double r2024498 = 3.0;
        double r2024499 = r2024497 / r2024498;
        double r2024500 = a;
        double r2024501 = r2024499 - r2024500;
        double r2024502 = r2024495 - r2024501;
        double r2024503 = r2024500 + r2024496;
        double r2024504 = sqrt(r2024503);
        double r2024505 = r2024504 / r2024496;
        double r2024506 = z;
        double r2024507 = r2024505 * r2024506;
        double r2024508 = fma(r2024492, r2024502, r2024507);
        double r2024509 = r2024489 * r2024508;
        double r2024510 = exp(r2024509);
        double r2024511 = fma(r2024488, r2024510, r2024487);
        double r2024512 = r2024487 / r2024511;
        return r2024512;
}

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. Simplified2.0

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

  4. Applied associate-/r/2.4

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

  5. Final simplification2.4

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

Reproduce

herbie shell --seed 2019164 +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)))))))))))