Average Error: 7.3 → 7.3
Time: 15.4s
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\frac{\mathsf{fma}\left(x, y, -z \cdot t\right)}{a}\]
\frac{x \cdot y - z \cdot t}{a}
↓
\frac{\mathsf{fma}\left(x, y, -z \cdot t\right)}{a}
double f(double x, double y, double z, double t, double a) {
        double r36777585 = x;
        double r36777586 = y;
        double r36777587 = r36777585 * r36777586;
        double r36777588 = z;
        double r36777589 = t;
        double r36777590 = r36777588 * r36777589;
        double r36777591 = r36777587 - r36777590;
        double r36777592 = a;
        double r36777593 = r36777591 / r36777592;
        return r36777593;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r36777594 = x;
        double r36777595 = y;
        double r36777596 = z;
        double r36777597 = t;
        double r36777598 = r36777596 * r36777597;
        double r36777599 = -r36777598;
        double r36777600 = fma(r36777594, r36777595, r36777599);
        double r36777601 = a;
        double r36777602 = r36777600 / r36777601;
        return r36777602;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.3
Target6.1
Herbie7.3
\[\begin{array}{l} \mathbf{if}\;z \lt -2.468684968699548224247694913169778644284 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.309831121978371209578784129518242708809 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

  1. Initial program 7.3

    \[\frac{x \cdot y - z \cdot t}{a}\]
  2. Using strategy rm

  3. Applied fma-neg7.3

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

  4. Final simplification7.3

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"

  :herbie-target
  (if (< z -2.468684968699548e+170) (- (* (/ y a) x) (* (/ t a) z)) (if (< z 6.309831121978371e-71) (/ (- (* x y) (* z t)) a) (- (* (/ y a) x) (* (/ t a) z))))

  (/ (- (* x y) (* z t)) a))