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

↓

\[\mathsf{fma}\left(c \cdot a - y \cdot i, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right)\]

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

↓

\mathsf{fma}\left(c \cdot a - y \cdot i, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right)

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r949975 = x;
        double r949976 = y;
        double r949977 = z;
        double r949978 = r949976 * r949977;
        double r949979 = t;
        double r949980 = a;
        double r949981 = r949979 * r949980;
        double r949982 = r949978 - r949981;
        double r949983 = r949975 * r949982;
        double r949984 = b;
        double r949985 = c;
        double r949986 = r949985 * r949977;
        double r949987 = i;
        double r949988 = r949979 * r949987;
        double r949989 = r949986 - r949988;
        double r949990 = r949984 * r949989;
        double r949991 = r949983 - r949990;
        double r949992 = j;
        double r949993 = r949985 * r949980;
        double r949994 = r949976 * r949987;
        double r949995 = r949993 - r949994;
        double r949996 = r949992 * r949995;
        double r949997 = r949991 + r949996;
        return r949997;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r949998 = c;
        double r949999 = a;
        double r950000 = r949998 * r949999;
        double r950001 = y;
        double r950002 = i;
        double r950003 = r950001 * r950002;
        double r950004 = r950000 - r950003;
        double r950005 = j;
        double r950006 = x;
        double r950007 = z;
        double r950008 = t;
        double r950009 = r949999 * r950008;
        double r950010 = -r950009;
        double r950011 = fma(r950001, r950007, r950010);
        double r950012 = r950006 * r950011;
        double r950013 = -r949999;
        double r950014 = fma(r950013, r950008, r950009);
        double r950015 = r950006 * r950014;
        double r950016 = r950012 + r950015;
        double r950017 = b;
        double r950018 = cbrt(r950017);
        double r950019 = r950018 * r950018;
        double r950020 = r949998 * r950007;
        double r950021 = r950008 * r950002;
        double r950022 = r950020 - r950021;
        double r950023 = r950018 * r950022;
        double r950024 = r950019 * r950023;
        double r950025 = r950016 - r950024;
        double r950026 = fma(r950004, r950005, r950025);
        return r950026;
}

Error

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

Target

Original	11.6
Target	19.6
Herbie	11.9

\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

Initial program 11.6
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
Simplified11.6
\[\leadsto \color{blue}{\mathsf{fma}\left(c \cdot a - y \cdot i, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)}\]
Using strategy rm
Applied prod-diff11.6
\[\leadsto \mathsf{fma}\left(c \cdot a - y \cdot i, j, x \cdot \color{blue}{\left(\mathsf{fma}\left(y, z, -a \cdot t\right) + \mathsf{fma}\left(-a, t, a \cdot t\right)\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right)\]
Applied distribute-lft-in11.6
\[\leadsto \mathsf{fma}\left(c \cdot a - y \cdot i, j, \color{blue}{\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right)\]
Using strategy rm
Applied add-cube-cbrt11.9
\[\leadsto \mathsf{fma}\left(c \cdot a - y \cdot i, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \color{blue}{\left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}\right)} \cdot \left(c \cdot z - t \cdot i\right)\right)\]
Applied associate-*l*11.9
\[\leadsto \mathsf{fma}\left(c \cdot a - y \cdot i, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)\right)}\right)\]
Final simplification11.9
\[\leadsto \mathsf{fma}\left(c \cdot a - y \cdot i, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right)\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 3.2113527362226803e-147) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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