Average Error: 2.0 → 1.4
Time: 16.6s
Precision: 64
\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
↓
\[\frac{\left({\left(\frac{1}{{a}^{1}}\right)}^{0.5} \cdot \left(x \cdot {\left(\frac{1}{{\left(e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}\right)}^{2}}\right)}^{\frac{1}{3}}\right)\right) \cdot \frac{{\left(\frac{\sqrt[3]{1}}{\sqrt{a}}\right)}^{1}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]
\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}
↓
\frac{\left({\left(\frac{1}{{a}^{1}}\right)}^{0.5} \cdot \left(x \cdot {\left(\frac{1}{{\left(e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}\right)}^{2}}\right)}^{\frac{1}{3}}\right)\right) \cdot \frac{{\left(\frac{\sqrt[3]{1}}{\sqrt{a}}\right)}^{1}}{\sqrt[3]{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}
double f(double x, double y, double z, double t, double a, double b) {
        double r106572 = x;
        double r106573 = y;
        double r106574 = z;
        double r106575 = log(r106574);
        double r106576 = r106573 * r106575;
        double r106577 = t;
        double r106578 = 1.0;
        double r106579 = r106577 - r106578;
        double r106580 = a;
        double r106581 = log(r106580);
        double r106582 = r106579 * r106581;
        double r106583 = r106576 + r106582;
        double r106584 = b;
        double r106585 = r106583 - r106584;
        double r106586 = exp(r106585);
        double r106587 = r106572 * r106586;
        double r106588 = r106587 / r106573;
        return r106588;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r106589 = 1.0;
        double r106590 = a;
        double r106591 = 1.0;
        double r106592 = pow(r106590, r106591);
        double r106593 = r106589 / r106592;
        double r106594 = 0.5;
        double r106595 = pow(r106593, r106594);
        double r106596 = x;
        double r106597 = z;
        double r106598 = r106589 / r106597;
        double r106599 = log(r106598);
        double r106600 = y;
        double r106601 = r106599 * r106600;
        double r106602 = r106589 / r106590;
        double r106603 = log(r106602);
        double r106604 = t;
        double r106605 = r106603 * r106604;
        double r106606 = b;
        double r106607 = r106605 + r106606;
        double r106608 = r106601 + r106607;
        double r106609 = exp(r106608);
        double r106610 = 2.0;
        double r106611 = pow(r106609, r106610);
        double r106612 = r106589 / r106611;
        double r106613 = 0.3333333333333333;
        double r106614 = pow(r106612, r106613);
        double r106615 = r106596 * r106614;
        double r106616 = r106595 * r106615;
        double r106617 = cbrt(r106589);
        double r106618 = sqrt(r106590);
        double r106619 = r106617 / r106618;
        double r106620 = pow(r106619, r106591);
        double r106621 = fma(r106603, r106604, r106606);
        double r106622 = fma(r106600, r106599, r106621);
        double r106623 = exp(r106622);
        double r106624 = cbrt(r106623);
        double r106625 = r106620 / r106624;
        double r106626 = r106616 * r106625;
        double r106627 = r106626 / r106600;
        return r106627;
}

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

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

  3. Simplified1.3

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

  5. Applied add-cube-cbrt1.4

    \[\leadsto \frac{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{\color{blue}{\left(\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}}{y}\]

  6. Applied add-sqr-sqrt1.4

    \[\leadsto \frac{x \cdot \frac{{\left(\frac{1}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}\right)}^{1}}{\left(\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]

  7. Applied add-cube-cbrt1.4

    \[\leadsto \frac{x \cdot \frac{{\left(\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{a} \cdot \sqrt{a}}\right)}^{1}}{\left(\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]

  8. Applied times-frac1.4

    \[\leadsto \frac{x \cdot \frac{{\color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{a}} \cdot \frac{\sqrt[3]{1}}{\sqrt{a}}\right)}}^{1}}{\left(\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]

  9. Applied unpow-prod-down1.4

    \[\leadsto \frac{x \cdot \frac{\color{blue}{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{a}}\right)}^{1} \cdot {\left(\frac{\sqrt[3]{1}}{\sqrt{a}}\right)}^{1}}}{\left(\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]

  10. Applied times-frac1.4

    \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{a}}\right)}^{1}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot \frac{{\left(\frac{\sqrt[3]{1}}{\sqrt{a}}\right)}^{1}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}\right)}}{y}\]

  11. Applied associate-*r*1.4

    \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{a}}\right)}^{1}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}\right) \cdot \frac{{\left(\frac{\sqrt[3]{1}}{\sqrt{a}}\right)}^{1}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}}{y}\]

  12. Taylor expanded around inf 1.4

    \[\leadsto \frac{\color{blue}{\left({\left(\frac{1}{{a}^{1}}\right)}^{0.5} \cdot \left(x \cdot {\left(\frac{1}{{\left(e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}\right)}^{2}}\right)}^{\frac{1}{3}}\right)\right)} \cdot \frac{{\left(\frac{\sqrt[3]{1}}{\sqrt{a}}\right)}^{1}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]

  13. Final simplification1.4

    \[\leadsto \frac{\left({\left(\frac{1}{{a}^{1}}\right)}^{0.5} \cdot \left(x \cdot {\left(\frac{1}{{\left(e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}\right)}^{2}}\right)}^{\frac{1}{3}}\right)\right) \cdot \frac{{\left(\frac{\sqrt[3]{1}}{\sqrt{a}}\right)}^{1}}{\sqrt[3]{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]

Reproduce

herbie shell --seed 2020047 +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))