Average Error: 12.3 → 12.6
Time: 31.4s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\left(\left(z \cdot y - t \cdot a\right) \cdot x - \left(b \cdot \mathsf{fma}\left(-i, t, i \cdot t\right) + b \cdot \left(z \cdot c - i \cdot t\right)\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot a - i \cdot y} \cdot \sqrt[3]{c \cdot a - i \cdot y}\right)\right) \cdot \sqrt[3]{c \cdot a - i \cdot y}\]
\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)
↓
\left(\left(z \cdot y - t \cdot a\right) \cdot x - \left(b \cdot \mathsf{fma}\left(-i, t, i \cdot t\right) + b \cdot \left(z \cdot c - i \cdot t\right)\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot a - i \cdot y} \cdot \sqrt[3]{c \cdot a - i \cdot y}\right)\right) \cdot \sqrt[3]{c \cdot a - i \cdot y}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r32993193 = x;
        double r32993194 = y;
        double r32993195 = z;
        double r32993196 = r32993194 * r32993195;
        double r32993197 = t;
        double r32993198 = a;
        double r32993199 = r32993197 * r32993198;
        double r32993200 = r32993196 - r32993199;
        double r32993201 = r32993193 * r32993200;
        double r32993202 = b;
        double r32993203 = c;
        double r32993204 = r32993203 * r32993195;
        double r32993205 = i;
        double r32993206 = r32993197 * r32993205;
        double r32993207 = r32993204 - r32993206;
        double r32993208 = r32993202 * r32993207;
        double r32993209 = r32993201 - r32993208;
        double r32993210 = j;
        double r32993211 = r32993203 * r32993198;
        double r32993212 = r32993194 * r32993205;
        double r32993213 = r32993211 - r32993212;
        double r32993214 = r32993210 * r32993213;
        double r32993215 = r32993209 + r32993214;
        return r32993215;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r32993216 = z;
        double r32993217 = y;
        double r32993218 = r32993216 * r32993217;
        double r32993219 = t;
        double r32993220 = a;
        double r32993221 = r32993219 * r32993220;
        double r32993222 = r32993218 - r32993221;
        double r32993223 = x;
        double r32993224 = r32993222 * r32993223;
        double r32993225 = b;
        double r32993226 = i;
        double r32993227 = -r32993226;
        double r32993228 = r32993226 * r32993219;
        double r32993229 = fma(r32993227, r32993219, r32993228);
        double r32993230 = r32993225 * r32993229;
        double r32993231 = c;
        double r32993232 = r32993216 * r32993231;
        double r32993233 = r32993232 - r32993228;
        double r32993234 = r32993225 * r32993233;
        double r32993235 = r32993230 + r32993234;
        double r32993236 = r32993224 - r32993235;
        double r32993237 = j;
        double r32993238 = r32993231 * r32993220;
        double r32993239 = r32993226 * r32993217;
        double r32993240 = r32993238 - r32993239;
        double r32993241 = cbrt(r32993240);
        double r32993242 = r32993241 * r32993241;
        double r32993243 = r32993237 * r32993242;
        double r32993244 = r32993243 * r32993241;
        double r32993245 = r32993236 + r32993244;
        return r32993245;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Bits error versus j

Target

Original12.3
Target20.1
Herbie12.6
\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Initial program 12.3

    \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
  2. Using strategy rm

  3. Applied prod-diff12.3

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

  4. Applied distribute-lft-in12.3

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

  5. Simplified12.3

    \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\color{blue}{\left(c \cdot z - i \cdot t\right) \cdot b} + b \cdot \mathsf{fma}\left(-i, t, i \cdot t\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
  6. Using strategy rm

  7. Applied add-cube-cbrt12.6

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

  8. Applied associate-*r*12.6

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

  9. Final simplification12.6

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0))) (+ (* 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) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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