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

↓

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

\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(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - b \cdot \left(c \cdot z - i \cdot a\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 r481701 = x;
        double r481702 = y;
        double r481703 = z;
        double r481704 = r481702 * r481703;
        double r481705 = t;
        double r481706 = a;
        double r481707 = r481705 * r481706;
        double r481708 = r481704 - r481707;
        double r481709 = r481701 * r481708;
        double r481710 = b;
        double r481711 = c;
        double r481712 = r481711 * r481703;
        double r481713 = i;
        double r481714 = r481713 * r481706;
        double r481715 = r481712 - r481714;
        double r481716 = r481710 * r481715;
        double r481717 = r481709 - r481716;
        double r481718 = j;
        double r481719 = r481711 * r481705;
        double r481720 = r481713 * r481702;
        double r481721 = r481719 - r481720;
        double r481722 = r481718 * r481721;
        double r481723 = r481717 + r481722;
        return r481723;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r481724 = x;
        double r481725 = y;
        double r481726 = z;
        double r481727 = a;
        double r481728 = t;
        double r481729 = r481727 * r481728;
        double r481730 = -r481729;
        double r481731 = fma(r481725, r481726, r481730);
        double r481732 = r481724 * r481731;
        double r481733 = -r481727;
        double r481734 = fma(r481733, r481728, r481729);
        double r481735 = r481724 * r481734;
        double r481736 = r481732 + r481735;
        double r481737 = b;
        double r481738 = c;
        double r481739 = r481738 * r481726;
        double r481740 = i;
        double r481741 = r481740 * r481727;
        double r481742 = r481739 - r481741;
        double r481743 = r481737 * r481742;
        double r481744 = r481736 - r481743;
        double r481745 = j;
        double r481746 = r481725 * r481740;
        double r481747 = -r481746;
        double r481748 = fma(r481738, r481728, r481747);
        double r481749 = r481745 * r481748;
        double r481750 = -r481725;
        double r481751 = fma(r481750, r481740, r481746);
        double r481752 = r481745 * r481751;
        double r481753 = r481749 + r481752;
        double r481754 = r481744 + r481753;
        return r481754;
}

Error

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

Target

Original	12.4
Target	16.2
Herbie	12.4

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

Derivation

Initial program 12.4
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
Using strategy rm
Applied prod-diff12.4
\[\leadsto \left(x \cdot \color{blue}{\left(\mathsf{fma}\left(y, z, -a \cdot t\right) + \mathsf{fma}\left(-a, t, a \cdot t\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
Applied distribute-lft-in12.4
\[\leadsto \left(\color{blue}{\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right)} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
Using strategy rm
Applied prod-diff12.4
\[\leadsto \left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \color{blue}{\left(\mathsf{fma}\left(c, t, -y \cdot i\right) + \mathsf{fma}\left(-y, i, y \cdot i\right)\right)}\]
Applied distribute-lft-in12.4
\[\leadsto \left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)}\]
Final simplification12.4
\[\leadsto \left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \mathsf{fma}\left(c, t, -y \cdot i\right) + j \cdot \mathsf{fma}\left(-y, i, y \cdot i\right)\right)\]

Reproduce

herbie shell --seed 2020039 +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)))))