Average Error: 0.2 → 0.3
Time: 1.2m
Precision: 64
\[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\]
↓
\[\mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, c\right)\right) - \left(a \cdot \sqrt[3]{b}\right) \cdot \frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{4}\]
\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c
↓
\mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, c\right)\right) - \left(a \cdot \sqrt[3]{b}\right) \cdot \frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{4}
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r13532773 = x;
        double r13532774 = y;
        double r13532775 = r13532773 * r13532774;
        double r13532776 = z;
        double r13532777 = t;
        double r13532778 = r13532776 * r13532777;
        double r13532779 = 16.0;
        double r13532780 = r13532778 / r13532779;
        double r13532781 = r13532775 + r13532780;
        double r13532782 = a;
        double r13532783 = b;
        double r13532784 = r13532782 * r13532783;
        double r13532785 = 4.0;
        double r13532786 = r13532784 / r13532785;
        double r13532787 = r13532781 - r13532786;
        double r13532788 = c;
        double r13532789 = r13532787 + r13532788;
        return r13532789;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r13532790 = z;
        double r13532791 = 16.0;
        double r13532792 = r13532790 / r13532791;
        double r13532793 = t;
        double r13532794 = x;
        double r13532795 = y;
        double r13532796 = c;
        double r13532797 = fma(r13532794, r13532795, r13532796);
        double r13532798 = fma(r13532792, r13532793, r13532797);
        double r13532799 = a;
        double r13532800 = b;
        double r13532801 = cbrt(r13532800);
        double r13532802 = r13532799 * r13532801;
        double r13532803 = r13532801 * r13532801;
        double r13532804 = 4.0;
        double r13532805 = r13532803 / r13532804;
        double r13532806 = r13532802 * r13532805;
        double r13532807 = r13532798 - r13532806;
        return r13532807;
}

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

Derivation

  1. Initial program 0.2

    \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, c\right)\right) - \frac{b \cdot a}{4}}\]
  3. Using strategy rm

  4. Applied associate-/l*0.1

    \[\leadsto \mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, c\right)\right) - \color{blue}{\frac{b}{\frac{4}{a}}}\]
  5. Using strategy rm

  6. Applied div-inv0.1

    \[\leadsto \mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, c\right)\right) - \frac{b}{\color{blue}{4 \cdot \frac{1}{a}}}\]

  7. Applied add-cube-cbrt0.3

    \[\leadsto \mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, c\right)\right) - \frac{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}{4 \cdot \frac{1}{a}}\]

  8. Applied times-frac0.3

    \[\leadsto \mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, c\right)\right) - \color{blue}{\frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{4} \cdot \frac{\sqrt[3]{b}}{\frac{1}{a}}}\]

  9. Simplified0.3

    \[\leadsto \mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, c\right)\right) - \frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{4} \cdot \color{blue}{\left(\sqrt[3]{b} \cdot a\right)}\]

  10. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, c\right)\right) - \left(a \cdot \sqrt[3]{b}\right) \cdot \frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{4}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))