Average Error: 7.3 → 7.3
Time: 1.2m
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\frac{\mathsf{fma}\left(x, y, \left(-t\right) \cdot z\right)}{a}\]
\frac{x \cdot y - z \cdot t}{a}
↓
\frac{\mathsf{fma}\left(x, y, \left(-t\right) \cdot z\right)}{a}
double f(double x, double y, double z, double t, double a) {
        double r38991635 = x;
        double r38991636 = y;
        double r38991637 = r38991635 * r38991636;
        double r38991638 = z;
        double r38991639 = t;
        double r38991640 = r38991638 * r38991639;
        double r38991641 = r38991637 - r38991640;
        double r38991642 = a;
        double r38991643 = r38991641 / r38991642;
        return r38991643;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r38991644 = x;
        double r38991645 = y;
        double r38991646 = t;
        double r38991647 = -r38991646;
        double r38991648 = z;
        double r38991649 = r38991647 * r38991648;
        double r38991650 = fma(r38991644, r38991645, r38991649);
        double r38991651 = a;
        double r38991652 = r38991650 / r38991651;
        return r38991652;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.3
Target5.9
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

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

Reproduce

herbie shell --seed 2019200 +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))