\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

↓

\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\]

\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)

↓

\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r489566 = x;
        double r489567 = y;
        double r489568 = z;
        double r489569 = r489567 * r489568;
        double r489570 = t;
        double r489571 = a;
        double r489572 = r489570 * r489571;
        double r489573 = r489569 - r489572;
        double r489574 = r489566 * r489573;
        double r489575 = b;
        double r489576 = c;
        double r489577 = r489576 * r489568;
        double r489578 = i;
        double r489579 = r489578 * r489571;
        double r489580 = r489577 - r489579;
        double r489581 = r489575 * r489580;
        double r489582 = r489574 - r489581;
        double r489583 = j;
        double r489584 = r489576 * r489570;
        double r489585 = r489578 * r489567;
        double r489586 = r489584 - r489585;
        double r489587 = r489583 * r489586;
        double r489588 = r489582 + r489587;
        return r489588;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r489589 = x;
        double r489590 = y;
        double r489591 = z;
        double r489592 = r489590 * r489591;
        double r489593 = t;
        double r489594 = a;
        double r489595 = r489593 * r489594;
        double r489596 = r489592 - r489595;
        double r489597 = r489589 * r489596;
        double r489598 = b;
        double r489599 = c;
        double r489600 = r489599 * r489591;
        double r489601 = i;
        double r489602 = r489601 * r489594;
        double r489603 = r489600 - r489602;
        double r489604 = r489598 * r489603;
        double r489605 = -r489594;
        double r489606 = r489594 * r489601;
        double r489607 = fma(r489605, r489601, r489606);
        double r489608 = r489598 * r489607;
        double r489609 = r489604 + r489608;
        double r489610 = r489597 - r489609;
        double r489611 = j;
        double r489612 = r489590 * r489601;
        double r489613 = -r489612;
        double r489614 = fma(r489599, r489593, r489613);
        double r489615 = r489611 * r489614;
        double r489616 = -r489590;
        double r489617 = fma(r489616, r489601, r489612);
        double r489618 = r489611 * r489617;
        double r489619 = r489615 + r489618;
        double r489620 = r489610 + r489619;
        return r489620;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	12.5
Target	16.3
Herbie	12.5

\[\begin{array}{l} \mathbf{if}\;t \lt -8.1209789191959122 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.7125538182184851 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.63353334603158369 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

Initial program 12.5
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
Using strategy rm
Applied prod-diff12.5
\[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \color{blue}{\left(\mathsf{fma}\left(c, t, -y \cdot i\right) + \mathsf{fma}\left(-y, i, y \cdot i\right)\right)}\]
Applied distribute-lft-in12.5
\[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)}\]
Using strategy rm
Applied prod-diff12.5
\[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(\mathsf{fma}\left(c, z, -a \cdot i\right) + \mathsf{fma}\left(-a, i, a \cdot i\right)\right)}\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\]
Applied distribute-lft-in12.5
\[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\left(b \cdot \mathsf{fma}\left(c, z, -a \cdot i\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)}\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\]
Simplified12.5
\[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\color{blue}{b \cdot \left(c \cdot z - i \cdot a\right)} + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\]
Final simplification12.5
\[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -8.120978919195912e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.712553818218485e-169) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -7.633533346031584e-308) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 1.0535888557455487e-139) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))