Average Error: 7.7 → 7.7
Time: 18.5s
Precision: 64
\[\frac{x \cdot y - z \cdot t}{a}\]
↓
\[\frac{\mathsf{fma}\left(x, y, -t \cdot z\right) + z \cdot \left(\left(-t\right) + t\right)}{a}\]
\frac{x \cdot y - z \cdot t}{a}
↓
\frac{\mathsf{fma}\left(x, y, -t \cdot z\right) + z \cdot \left(\left(-t\right) + t\right)}{a}
double f(double x, double y, double z, double t, double a) {
        double r554621 = x;
        double r554622 = y;
        double r554623 = r554621 * r554622;
        double r554624 = z;
        double r554625 = t;
        double r554626 = r554624 * r554625;
        double r554627 = r554623 - r554626;
        double r554628 = a;
        double r554629 = r554627 / r554628;
        return r554629;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r554630 = x;
        double r554631 = y;
        double r554632 = t;
        double r554633 = z;
        double r554634 = r554632 * r554633;
        double r554635 = -r554634;
        double r554636 = fma(r554630, r554631, r554635);
        double r554637 = -r554632;
        double r554638 = r554637 + r554632;
        double r554639 = r554633 * r554638;
        double r554640 = r554636 + r554639;
        double r554641 = a;
        double r554642 = r554640 / r554641;
        return r554642;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.7
Target6.2
Herbie7.7
\[\begin{array}{l} \mathbf{if}\;z \lt -2.46868496869954822 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.30983112197837121 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

  1. Initial program 7.7

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

  3. Applied prod-diff7.7

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

  4. Simplified7.7

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

  5. Final simplification7.7

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

Reproduce

herbie shell --seed 2019198 +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))