Average Error: 2.0 → 1.0
Time: 30.3s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y} \cdot \left(\frac{{\left(\frac{\sqrt{1}}{\sqrt{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot x\right)\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{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} \cdot \left(\frac{{\left(\frac{\sqrt{1}}{\sqrt{a}}\right)}^{1}}{\sqrt{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 x\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r164634 = x;
        double r164635 = y;
        double r164636 = z;
        double r164637 = log(r164636);
        double r164638 = r164635 * r164637;
        double r164639 = t;
        double r164640 = 1.0;
        double r164641 = r164639 - r164640;
        double r164642 = a;
        double r164643 = log(r164642);
        double r164644 = r164641 * r164643;
        double r164645 = r164638 + r164644;
        double r164646 = b;
        double r164647 = r164645 - r164646;
        double r164648 = exp(r164647);
        double r164649 = r164634 * r164648;
        double r164650 = r164649 / r164635;
        return r164650;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r164651 = 1.0;
        double r164652 = a;
        double r164653 = r164651 / r164652;
        double r164654 = sqrt(r164653);
        double r164655 = 1.0;
        double r164656 = pow(r164654, r164655);
        double r164657 = y;
        double r164658 = z;
        double r164659 = r164651 / r164658;
        double r164660 = log(r164659);
        double r164661 = log(r164653);
        double r164662 = t;
        double r164663 = b;
        double r164664 = fma(r164661, r164662, r164663);
        double r164665 = fma(r164657, r164660, r164664);
        double r164666 = exp(r164665);
        double r164667 = sqrt(r164666);
        double r164668 = r164656 / r164667;
        double r164669 = r164668 / r164657;
        double r164670 = sqrt(r164651);
        double r164671 = sqrt(r164652);
        double r164672 = r164670 / r164671;
        double r164673 = pow(r164672, r164655);
        double r164674 = r164673 / r164667;
        double r164675 = x;
        double r164676 = r164674 * r164675;
        double r164677 = r164669 * r164676;
        return r164677;
}

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 2.0

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

  2. Taylor expanded around inf 2.0

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

  3. Simplified6.2

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

  5. Applied div-inv6.2

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

  6. Applied add-sqr-sqrt6.2

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

  7. Applied add-sqr-sqrt6.2

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

  8. Applied add-sqr-sqrt6.2

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

  9. Applied times-frac6.2

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

  10. Applied unpow-prod-down6.2

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

  11. Applied times-frac6.2

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

  12. Applied times-frac1.0

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

  13. Simplified1.0

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

  15. Applied sqrt-undiv1.0

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

  16. Final simplification1.0

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

Reproduce

herbie shell --seed 2019353 +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))