Average Error: 7.3 → 7.3
Time: 15.0s
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\frac{\mathsf{fma}\left(x, -y, t \cdot z\right)}{-a}\]
\frac{x \cdot y - z \cdot t}{a}
↓
\frac{\mathsf{fma}\left(x, -y, t \cdot z\right)}{-a}
double f(double x, double y, double z, double t, double a) {
        double r553872 = x;
        double r553873 = y;
        double r553874 = r553872 * r553873;
        double r553875 = z;
        double r553876 = t;
        double r553877 = r553875 * r553876;
        double r553878 = r553874 - r553877;
        double r553879 = a;
        double r553880 = r553878 / r553879;
        return r553880;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r553881 = x;
        double r553882 = y;
        double r553883 = -r553882;
        double r553884 = t;
        double r553885 = z;
        double r553886 = r553884 * r553885;
        double r553887 = fma(r553881, r553883, r553886);
        double r553888 = a;
        double r553889 = -r553888;
        double r553890 = r553887 / r553889;
        return r553890;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.3
Target5.7
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 frac-2neg7.3

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

  4. Simplified7.3

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

  5. Final simplification7.3

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z t a)
  :name "Data.Colour.Matrix:inverse from colour-2.3.3, B"
  :precision binary64

  :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))