Average Error: 7.1 → 7.1
Time: 9.1s
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 r14499493 = x;
        double r14499494 = y;
        double r14499495 = r14499493 * r14499494;
        double r14499496 = z;
        double r14499497 = t;
        double r14499498 = r14499496 * r14499497;
        double r14499499 = r14499495 - r14499498;
        double r14499500 = a;
        double r14499501 = r14499499 / r14499500;
        return r14499501;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r14499502 = x;
        double r14499503 = y;
        double r14499504 = z;
        double r14499505 = t;
        double r14499506 = r14499504 * r14499505;
        double r14499507 = -r14499506;
        double r14499508 = fma(r14499502, r14499503, r14499507);
        double r14499509 = a;
        double r14499510 = r14499508 / r14499509;
        return r14499510;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original7.1
Target5.8
Herbie7.1
\[\begin{array}{l} \mathbf{if}\;z \lt -2.468684968699548 \cdot 10^{+170}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \mathbf{elif}\;z \lt 6.309831121978371 \cdot 10^{-71}:\\ \;\;\;\;\frac{x \cdot y - z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot x - \frac{t}{a} \cdot z\\ \end{array}\]

Derivation

  1. Initial program 7.1

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

  3. Applied fma-neg7.1

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

  4. Final simplification7.1

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

Reproduce

herbie shell --seed 2019156 +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))