Average Error: 0.1 → 0.0
Time: 42.2s
Precision: 64
\[\left(\left(x \cdot y + \frac{z \cdot t}{16.0}\right) - \frac{a \cdot b}{4.0}\right) + c\]
↓
\[\mathsf{fma}\left(\frac{t}{16.0}, z, \mathsf{fma}\left(y, x, c\right) - \frac{b}{4.0} \cdot a\right)\]
\left(\left(x \cdot y + \frac{z \cdot t}{16.0}\right) - \frac{a \cdot b}{4.0}\right) + c
↓
\mathsf{fma}\left(\frac{t}{16.0}, z, \mathsf{fma}\left(y, x, c\right) - \frac{b}{4.0} \cdot a\right)
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r10972024 = x;
        double r10972025 = y;
        double r10972026 = r10972024 * r10972025;
        double r10972027 = z;
        double r10972028 = t;
        double r10972029 = r10972027 * r10972028;
        double r10972030 = 16.0;
        double r10972031 = r10972029 / r10972030;
        double r10972032 = r10972026 + r10972031;
        double r10972033 = a;
        double r10972034 = b;
        double r10972035 = r10972033 * r10972034;
        double r10972036 = 4.0;
        double r10972037 = r10972035 / r10972036;
        double r10972038 = r10972032 - r10972037;
        double r10972039 = c;
        double r10972040 = r10972038 + r10972039;
        return r10972040;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r10972041 = t;
        double r10972042 = 16.0;
        double r10972043 = r10972041 / r10972042;
        double r10972044 = z;
        double r10972045 = y;
        double r10972046 = x;
        double r10972047 = c;
        double r10972048 = fma(r10972045, r10972046, r10972047);
        double r10972049 = b;
        double r10972050 = 4.0;
        double r10972051 = r10972049 / r10972050;
        double r10972052 = a;
        double r10972053 = r10972051 * r10972052;
        double r10972054 = r10972048 - r10972053;
        double r10972055 = fma(r10972043, r10972044, r10972054);
        return r10972055;
}

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 0.1

    \[\left(\left(x \cdot y + \frac{z \cdot t}{16.0}\right) - \frac{a \cdot b}{4.0}\right) + c\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{16.0}, z, \mathsf{fma}\left(y, x, c\right) - \frac{a \cdot b}{4.0}\right)}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.0

    \[\leadsto \mathsf{fma}\left(\frac{t}{16.0}, z, \mathsf{fma}\left(y, x, c\right) - \frac{a \cdot b}{\color{blue}{1 \cdot 4.0}}\right)\]

  5. Applied times-frac0.0

    \[\leadsto \mathsf{fma}\left(\frac{t}{16.0}, z, \mathsf{fma}\left(y, x, c\right) - \color{blue}{\frac{a}{1} \cdot \frac{b}{4.0}}\right)\]

  6. Simplified0.0

    \[\leadsto \mathsf{fma}\left(\frac{t}{16.0}, z, \mathsf{fma}\left(y, x, c\right) - \color{blue}{a} \cdot \frac{b}{4.0}\right)\]

  7. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(\frac{t}{16.0}, z, \mathsf{fma}\left(y, x, c\right) - \frac{b}{4.0} \cdot a\right)\]

Reproduce

herbie shell --seed 2019168 +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))