Average Error: 0.1 → 0.0
Time: 21.0s
Precision: 64
\[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\]
↓
\[\frac{t}{16} \cdot z + \mathsf{fma}\left(y, x, c - \left(a \cdot 0.25\right) \cdot b\right)\]
\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c
↓
\frac{t}{16} \cdot z + \mathsf{fma}\left(y, x, c - \left(a \cdot 0.25\right) \cdot b\right)
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r11307091 = x;
        double r11307092 = y;
        double r11307093 = r11307091 * r11307092;
        double r11307094 = z;
        double r11307095 = t;
        double r11307096 = r11307094 * r11307095;
        double r11307097 = 16.0;
        double r11307098 = r11307096 / r11307097;
        double r11307099 = r11307093 + r11307098;
        double r11307100 = a;
        double r11307101 = b;
        double r11307102 = r11307100 * r11307101;
        double r11307103 = 4.0;
        double r11307104 = r11307102 / r11307103;
        double r11307105 = r11307099 - r11307104;
        double r11307106 = c;
        double r11307107 = r11307105 + r11307106;
        return r11307107;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r11307108 = t;
        double r11307109 = 16.0;
        double r11307110 = r11307108 / r11307109;
        double r11307111 = z;
        double r11307112 = r11307110 * r11307111;
        double r11307113 = y;
        double r11307114 = x;
        double r11307115 = c;
        double r11307116 = a;
        double r11307117 = 0.25;
        double r11307118 = r11307116 * r11307117;
        double r11307119 = b;
        double r11307120 = r11307118 * r11307119;
        double r11307121 = r11307115 - r11307120;
        double r11307122 = fma(r11307113, r11307114, r11307121);
        double r11307123 = r11307112 + r11307122;
        return r11307123;
}

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

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

  2. Simplified0.0

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

  3. Taylor expanded around inf 0.0

    \[\leadsto \mathsf{fma}\left(\frac{t}{16}, z, \color{blue}{\left(c + x \cdot y\right) - 0.25 \cdot \left(a \cdot b\right)}\right)\]

  4. Simplified0.0

    \[\leadsto \mathsf{fma}\left(\frac{t}{16}, z, \color{blue}{\mathsf{fma}\left(y, x, c - b \cdot \left(a \cdot 0.25\right)\right)}\right)\]
  5. Using strategy rm

  6. Applied fma-udef0.0

    \[\leadsto \color{blue}{\frac{t}{16} \cdot z + \mathsf{fma}\left(y, x, c - b \cdot \left(a \cdot 0.25\right)\right)}\]

  7. Final simplification0.0

    \[\leadsto \frac{t}{16} \cdot z + \mathsf{fma}\left(y, x, c - \left(a \cdot 0.25\right) \cdot b\right)\]

Reproduce

herbie shell --seed 2019170 +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))