Average Error: 1.9 → 1.3
Time: 16.6s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[{\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
{\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}
double f(double x, double y, double z, double t, double a, double b) {
        double r105447 = x;
        double r105448 = y;
        double r105449 = z;
        double r105450 = log(r105449);
        double r105451 = r105448 * r105450;
        double r105452 = t;
        double r105453 = 1.0;
        double r105454 = r105452 - r105453;
        double r105455 = a;
        double r105456 = log(r105455);
        double r105457 = r105454 * r105456;
        double r105458 = r105451 + r105457;
        double r105459 = b;
        double r105460 = r105458 - r105459;
        double r105461 = exp(r105460);
        double r105462 = r105447 * r105461;
        double r105463 = r105462 / r105448;
        return r105463;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r105464 = 1.0;
        double r105465 = a;
        double r105466 = 1.0;
        double r105467 = pow(r105465, r105466);
        double r105468 = r105464 / r105467;
        double r105469 = pow(r105468, r105466);
        double r105470 = x;
        double r105471 = z;
        double r105472 = r105464 / r105471;
        double r105473 = log(r105472);
        double r105474 = y;
        double r105475 = r105473 * r105474;
        double r105476 = r105464 / r105465;
        double r105477 = log(r105476);
        double r105478 = t;
        double r105479 = r105477 * r105478;
        double r105480 = b;
        double r105481 = r105479 + r105480;
        double r105482 = r105475 + r105481;
        double r105483 = exp(r105482);
        double r105484 = r105483 * r105474;
        double r105485 = r105470 / r105484;
        double r105486 = r105469 * r105485;
        return r105486;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.9

    \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

  2. Taylor expanded around inf 1.9

    \[\leadsto \frac{x \cdot \color{blue}{e^{1 \cdot \log \left(\frac{1}{a}\right) - \left(y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)\right)}}}{y}\]

  3. Simplified1.1

    \[\leadsto \frac{x \cdot \color{blue}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]
  4. Using strategy rm

  5. Applied associate-/l*1.3

    \[\leadsto \color{blue}{\frac{x}{\frac{y}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}}\]

  6. Taylor expanded around inf 1.3

    \[\leadsto \color{blue}{{\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}}\]

  7. Final simplification1.3

    \[\leadsto {\left(\frac{1}{{a}^{1}}\right)}^{1} \cdot \frac{x}{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))