Average Error: 4.2 → 2.1
Time: 29.3s
Precision: 64
\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]
↓
\[\frac{x}{\mathsf{fma}\left(y, {\left(e^{2}\right)}^{\left(\mathsf{fma}\left(\frac{\frac{2}{t}}{3} - \left(a + \frac{5}{6}\right), b - c, z \cdot \frac{\sqrt{t + a}}{t}\right)\right)}, x\right)}\]
\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 r67496 = x;
        double r67497 = y;
        double r67498 = 2.0;
        double r67499 = z;
        double r67500 = t;
        double r67501 = a;
        double r67502 = r67500 + r67501;
        double r67503 = sqrt(r67502);
        double r67504 = r67499 * r67503;
        double r67505 = r67504 / r67500;
        double r67506 = b;
        double r67507 = c;
        double r67508 = r67506 - r67507;
        double r67509 = 5.0;
        double r67510 = 6.0;
        double r67511 = r67509 / r67510;
        double r67512 = r67501 + r67511;
        double r67513 = 3.0;
        double r67514 = r67500 * r67513;
        double r67515 = r67498 / r67514;
        double r67516 = r67512 - r67515;
        double r67517 = r67508 * r67516;
        double r67518 = r67505 - r67517;
        double r67519 = r67498 * r67518;
        double r67520 = exp(r67519);
        double r67521 = r67497 * r67520;
        double r67522 = r67496 + r67521;
        double r67523 = r67496 / r67522;
        return r67523;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r67524 = x;
        double r67525 = y;
        double r67526 = 2.0;
        double r67527 = exp(r67526);
        double r67528 = t;
        double r67529 = r67526 / r67528;
        double r67530 = 3.0;
        double r67531 = r67529 / r67530;
        double r67532 = a;
        double r67533 = 5.0;
        double r67534 = 6.0;
        double r67535 = r67533 / r67534;
        double r67536 = r67532 + r67535;
        double r67537 = r67531 - r67536;
        double r67538 = b;
        double r67539 = c;
        double r67540 = r67538 - r67539;
        double r67541 = z;
        double r67542 = r67528 + r67532;
        double r67543 = sqrt(r67542);
        double r67544 = r67543 / r67528;
        double r67545 = r67541 * r67544;
        double r67546 = fma(r67537, r67540, r67545);
        double r67547 = pow(r67527, r67546);
        double r67548 = fma(r67525, r67547, r67524);
        double r67549 = r67524 / r67548;
        return r67549;
}

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)))))))))))