Average Error: 1.9 → 1.0
Time: 25.3s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\sqrt{{\left(\frac{1}{a}\right)}^{1}} \cdot \frac{x}{\frac{y}{\frac{\sqrt{{\left(\frac{1}{a}\right)}^{1}}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\sqrt{{\left(\frac{1}{a}\right)}^{1}} \cdot \frac{x}{\frac{y}{\frac{\sqrt{{\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)}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r126520 = x;
        double r126521 = y;
        double r126522 = z;
        double r126523 = log(r126522);
        double r126524 = r126521 * r126523;
        double r126525 = t;
        double r126526 = 1.0;
        double r126527 = r126525 - r126526;
        double r126528 = a;
        double r126529 = log(r126528);
        double r126530 = r126527 * r126529;
        double r126531 = r126524 + r126530;
        double r126532 = b;
        double r126533 = r126531 - r126532;
        double r126534 = exp(r126533);
        double r126535 = r126520 * r126534;
        double r126536 = r126535 / r126521;
        return r126536;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r126537 = 1.0;
        double r126538 = a;
        double r126539 = r126537 / r126538;
        double r126540 = 1.0;
        double r126541 = pow(r126539, r126540);
        double r126542 = sqrt(r126541);
        double r126543 = x;
        double r126544 = y;
        double r126545 = z;
        double r126546 = r126537 / r126545;
        double r126547 = log(r126546);
        double r126548 = log(r126539);
        double r126549 = t;
        double r126550 = b;
        double r126551 = fma(r126548, r126549, r126550);
        double r126552 = fma(r126544, r126547, r126551);
        double r126553 = exp(r126552);
        double r126554 = r126542 / r126553;
        double r126555 = r126544 / r126554;
        double r126556 = r126543 / r126555;
        double r126557 = r126542 * r126556;
        return r126557;
}

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

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. Simplified1.2

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

  5. Applied associate-/l*1.2

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

  7. Applied *-un-lft-identity1.2

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

  8. Applied add-sqr-sqrt1.3

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

  9. Applied times-frac1.3

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

  10. Applied *-un-lft-identity1.3

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

  11. Applied times-frac1.3

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

  12. Applied *-un-lft-identity1.3

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

  13. Applied times-frac1.0

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

  14. Simplified1.0

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

  15. Final simplification1.0

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

Reproduce

herbie shell --seed 2020002 +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))