Average Error: 1.9 → 0.9
Time: 20.3s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\sqrt{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot \frac{x}{\frac{y}{\sqrt{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\sqrt{\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)}}} \cdot \frac{x}{\frac{y}{\sqrt{\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)}}}}}
double f(double x, double y, double z, double t, double a, double b) {
        double r103577 = x;
        double r103578 = y;
        double r103579 = z;
        double r103580 = log(r103579);
        double r103581 = r103578 * r103580;
        double r103582 = t;
        double r103583 = 1.0;
        double r103584 = r103582 - r103583;
        double r103585 = a;
        double r103586 = log(r103585);
        double r103587 = r103584 * r103586;
        double r103588 = r103581 + r103587;
        double r103589 = b;
        double r103590 = r103588 - r103589;
        double r103591 = exp(r103590);
        double r103592 = r103577 * r103591;
        double r103593 = r103592 / r103578;
        return r103593;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r103594 = 1.0;
        double r103595 = a;
        double r103596 = r103594 / r103595;
        double r103597 = 1.0;
        double r103598 = pow(r103596, r103597);
        double r103599 = y;
        double r103600 = z;
        double r103601 = r103594 / r103600;
        double r103602 = log(r103601);
        double r103603 = log(r103596);
        double r103604 = t;
        double r103605 = b;
        double r103606 = fma(r103603, r103604, r103605);
        double r103607 = fma(r103599, r103602, r103606);
        double r103608 = exp(r103607);
        double r103609 = r103598 / r103608;
        double r103610 = sqrt(r103609);
        double r103611 = x;
        double r103612 = r103599 / r103610;
        double r103613 = r103611 / r103612;
        double r103614 = r103610 * r103613;
        return r103614;
}

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*1.3

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

  7. Applied add-sqr-sqrt1.3

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

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

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

  9. Applied times-frac1.3

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

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

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

  11. Applied times-frac0.9

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

  12. Simplified0.9

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

  13. Final simplification0.9

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

Reproduce

herbie shell --seed 2019354 +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))