\[\frac{x \cdot y - z \cdot t}{a}\]

↓

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

\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 r27996467 = x;
        double r27996468 = y;
        double r27996469 = r27996467 * r27996468;
        double r27996470 = z;
        double r27996471 = t;
        double r27996472 = r27996470 * r27996471;
        double r27996473 = r27996469 - r27996472;
        double r27996474 = a;
        double r27996475 = r27996473 / r27996474;
        return r27996475;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r27996476 = x;
        double r27996477 = y;
        double r27996478 = z;
        double r27996479 = t;
        double r27996480 = r27996478 * r27996479;
        double r27996481 = -r27996480;
        double r27996482 = fma(r27996476, r27996477, r27996481);
        double r27996483 = a;
        double r27996484 = r27996482 / r27996483;
        return r27996484;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	7.1
Target	5.5
Herbie	7.1

\[\begin{array}{l} \mathbf{if}\;z \lt -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

Initial program 7.1
\[\frac{x \cdot y - z \cdot t}{a}\]
Using strategy rm
Applied fma-neg7.1
\[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y, -z \cdot t\right)}}{a}\]
Final simplification7.1
\[\leadsto \frac{\mathsf{fma}\left(x, y, -z \cdot t\right)}{a}\]

Reproduce

herbie shell --seed 2019162 +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))