Average Error: 1.9 → 0.3
Time: 10.2s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot y + \mathsf{fma}\left(\log \left(\sqrt[3]{z}\right) - t, y, a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot y + \mathsf{fma}\left(\log \left(\sqrt[3]{z}\right) - t, y, a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)\right)}
double f(double x, double y, double z, double t, double a, double b) {
        double r112631 = x;
        double r112632 = y;
        double r112633 = z;
        double r112634 = log(r112633);
        double r112635 = t;
        double r112636 = r112634 - r112635;
        double r112637 = r112632 * r112636;
        double r112638 = a;
        double r112639 = 1.0;
        double r112640 = r112639 - r112633;
        double r112641 = log(r112640);
        double r112642 = b;
        double r112643 = r112641 - r112642;
        double r112644 = r112638 * r112643;
        double r112645 = r112637 + r112644;
        double r112646 = exp(r112645);
        double r112647 = r112631 * r112646;
        return r112647;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r112648 = x;
        double r112649 = z;
        double r112650 = cbrt(r112649);
        double r112651 = r112650 * r112650;
        double r112652 = log(r112651);
        double r112653 = y;
        double r112654 = r112652 * r112653;
        double r112655 = log(r112650);
        double r112656 = t;
        double r112657 = r112655 - r112656;
        double r112658 = a;
        double r112659 = 1.0;
        double r112660 = log(r112659);
        double r112661 = 0.5;
        double r112662 = 2.0;
        double r112663 = pow(r112649, r112662);
        double r112664 = pow(r112659, r112662);
        double r112665 = r112663 / r112664;
        double r112666 = r112661 * r112665;
        double r112667 = r112659 * r112649;
        double r112668 = r112666 + r112667;
        double r112669 = r112660 - r112668;
        double r112670 = b;
        double r112671 = r112669 - r112670;
        double r112672 = r112658 * r112671;
        double r112673 = fma(r112657, r112653, r112672);
        double r112674 = r112654 + r112673;
        double r112675 = exp(r112674);
        double r112676 = r112648 * r112675;
        return r112676;
}

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

    \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.9

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

  4. Applied log-prod1.9

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

  5. Applied associate--l+1.9

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

  6. Applied distribute-rgt-in1.9

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

  7. Applied associate-+l+1.9

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

  8. Simplified1.7

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

  9. Taylor expanded around 0 0.3

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

  10. Final simplification0.3

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

Reproduce

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