Average Error: 0.1 → 0.1
Time: 12.4s
Precision: 64
\[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
↓
\[\left(\left(\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right), x \cdot \log \left(\sqrt[3]{y}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i
↓
\left(\left(\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right), x \cdot \log \left(\sqrt[3]{y}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r83603 = x;
        double r83604 = y;
        double r83605 = log(r83604);
        double r83606 = r83603 * r83605;
        double r83607 = z;
        double r83608 = r83606 + r83607;
        double r83609 = t;
        double r83610 = r83608 + r83609;
        double r83611 = a;
        double r83612 = r83610 + r83611;
        double r83613 = b;
        double r83614 = 0.5;
        double r83615 = r83613 - r83614;
        double r83616 = c;
        double r83617 = log(r83616);
        double r83618 = r83615 * r83617;
        double r83619 = r83612 + r83618;
        double r83620 = i;
        double r83621 = r83604 * r83620;
        double r83622 = r83619 + r83621;
        return r83622;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r83623 = x;
        double r83624 = 2.0;
        double r83625 = 1.0;
        double r83626 = y;
        double r83627 = r83625 / r83626;
        double r83628 = -0.3333333333333333;
        double r83629 = pow(r83627, r83628);
        double r83630 = log(r83629);
        double r83631 = r83624 * r83630;
        double r83632 = cbrt(r83626);
        double r83633 = log(r83632);
        double r83634 = r83623 * r83633;
        double r83635 = fma(r83623, r83631, r83634);
        double r83636 = z;
        double r83637 = r83635 + r83636;
        double r83638 = t;
        double r83639 = r83637 + r83638;
        double r83640 = a;
        double r83641 = r83639 + r83640;
        double r83642 = b;
        double r83643 = 0.5;
        double r83644 = r83642 - r83643;
        double r83645 = c;
        double r83646 = log(r83645);
        double r83647 = r83644 * r83646;
        double r83648 = r83641 + r83647;
        double r83649 = i;
        double r83650 = r83626 * r83649;
        double r83651 = r83648 + r83650;
        return r83651;
}

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

Bits error versus c

Bits error versus i

Derivation

  1. Initial program 0.1

    \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.1

    \[\leadsto \left(\left(\left(\left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

  4. Applied log-prod0.1

    \[\leadsto \left(\left(\left(\left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

  5. Applied distribute-lft-in0.1

    \[\leadsto \left(\left(\left(\left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

  6. Simplified0.1

    \[\leadsto \left(\left(\left(\left(\left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

  7. Taylor expanded around inf 0.1

    \[\leadsto \left(\left(\left(\left(\left(x \cdot \left(2 \cdot \log \color{blue}{\left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right)}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
  8. Using strategy rm

  9. Applied fma-def0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))