Average Error: 0.1 → 0.1
Time: 13.8s
Precision: 64
\[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
↓
\[\left(\left(\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), 0 + x \cdot \log \left({y}^{\frac{1}{3}}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]
\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(\sqrt[3]{y}\right), 0 + x \cdot \log \left({y}^{\frac{1}{3}}\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 r68509 = x;
        double r68510 = y;
        double r68511 = log(r68510);
        double r68512 = r68509 * r68511;
        double r68513 = z;
        double r68514 = r68512 + r68513;
        double r68515 = t;
        double r68516 = r68514 + r68515;
        double r68517 = a;
        double r68518 = r68516 + r68517;
        double r68519 = b;
        double r68520 = 0.5;
        double r68521 = r68519 - r68520;
        double r68522 = c;
        double r68523 = log(r68522);
        double r68524 = r68521 * r68523;
        double r68525 = r68518 + r68524;
        double r68526 = i;
        double r68527 = r68510 * r68526;
        double r68528 = r68525 + r68527;
        return r68528;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r68529 = x;
        double r68530 = 2.0;
        double r68531 = y;
        double r68532 = cbrt(r68531);
        double r68533 = log(r68532);
        double r68534 = r68530 * r68533;
        double r68535 = 0.0;
        double r68536 = 0.3333333333333333;
        double r68537 = pow(r68531, r68536);
        double r68538 = log(r68537);
        double r68539 = r68529 * r68538;
        double r68540 = r68535 + r68539;
        double r68541 = fma(r68529, r68534, r68540);
        double r68542 = z;
        double r68543 = r68541 + r68542;
        double r68544 = t;
        double r68545 = r68543 + r68544;
        double r68546 = a;
        double r68547 = r68545 + r68546;
        double r68548 = b;
        double r68549 = 0.5;
        double r68550 = r68548 - r68549;
        double r68551 = c;
        double r68552 = log(r68551);
        double r68553 = r68550 * r68552;
        double r68554 = r68547 + r68553;
        double r68555 = i;
        double r68556 = r68531 * r68555;
        double r68557 = r68554 + r68556;
        return r68557;
}

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. Using strategy rm

  8. Applied fma-def0.1

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

  10. Applied *-un-lft-identity0.1

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

  11. Applied cbrt-prod0.1

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

  12. Applied log-prod0.1

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

  13. Applied distribute-lft-in0.1

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

  14. Simplified0.1

    \[\leadsto \left(\left(\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), \color{blue}{0} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

  15. Simplified0.1

    \[\leadsto \left(\left(\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), 0 + \color{blue}{x \cdot \log \left({y}^{\frac{1}{3}}\right)}\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

  16. Final simplification0.1

    \[\leadsto \left(\left(\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), 0 + x \cdot \log \left({y}^{\frac{1}{3}}\right)\right) + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\]

Reproduce

herbie shell --seed 2020001 +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)))