Average Error: 7.9 → 7.9
Time: 10.1s
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 r815629 = x;
        double r815630 = y;
        double r815631 = r815629 * r815630;
        double r815632 = z;
        double r815633 = t;
        double r815634 = r815632 * r815633;
        double r815635 = r815631 - r815634;
        double r815636 = a;
        double r815637 = r815635 / r815636;
        return r815637;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r815638 = x;
        double r815639 = y;
        double r815640 = z;
        double r815641 = t;
        double r815642 = r815640 * r815641;
        double r815643 = -r815642;
        double r815644 = fma(r815638, r815639, r815643);
        double r815645 = a;
        double r815646 = r815644 / r815645;
        return r815646;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.9
Target6.0
Herbie7.9
\[\begin{array}{l} \mathbf{if}\;z \lt -2.46868496869954822 \cdot 10^{170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.30983112197837121 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

  1. Initial program 7.9

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

  3. Applied fma-neg7.9

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

  4. Final simplification7.9

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

Reproduce

herbie shell --seed 2020047 +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))