Average Error: 0.1 → 0.0
Time: 11.8s
Precision: 64
\[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\]
↓
\[\mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, c\right) - 0.25 \cdot \left(a \cdot b\right)\right)\]
\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) - 0.25 \cdot \left(a \cdot b\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r231084 = x;
        double r231085 = y;
        double r231086 = r231084 * r231085;
        double r231087 = z;
        double r231088 = t;
        double r231089 = r231087 * r231088;
        double r231090 = 16.0;
        double r231091 = r231089 / r231090;
        double r231092 = r231086 + r231091;
        double r231093 = a;
        double r231094 = b;
        double r231095 = r231093 * r231094;
        double r231096 = 4.0;
        double r231097 = r231095 / r231096;
        double r231098 = r231092 - r231097;
        double r231099 = c;
        double r231100 = r231098 + r231099;
        return r231100;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r231101 = z;
        double r231102 = 16.0;
        double r231103 = r231101 / r231102;
        double r231104 = t;
        double r231105 = x;
        double r231106 = y;
        double r231107 = c;
        double r231108 = fma(r231105, r231106, r231107);
        double r231109 = 0.25;
        double r231110 = a;
        double r231111 = b;
        double r231112 = r231110 * r231111;
        double r231113 = r231109 * r231112;
        double r231114 = r231108 - r231113;
        double r231115 = fma(r231103, r231104, r231114);
        return r231115;
}

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

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

  3. Taylor expanded around inf 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  :precision binary64
  (+ (- (+ (* x y) (/ (* z t) 16)) (/ (* a b) 4)) c))