Average Error: 12.4 → 12.7
Time: 12.3s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\mathsf{fma}\left(c \cdot a - y \cdot i, j, \left(\left(x \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)}\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)} + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\]
\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(\left(x \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)}\right)\right) \cdot \sqrt[3]{\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)
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r807281 = x;
        double r807282 = y;
        double r807283 = z;
        double r807284 = r807282 * r807283;
        double r807285 = t;
        double r807286 = a;
        double r807287 = r807285 * r807286;
        double r807288 = r807284 - r807287;
        double r807289 = r807281 * r807288;
        double r807290 = b;
        double r807291 = c;
        double r807292 = r807291 * r807283;
        double r807293 = i;
        double r807294 = r807285 * r807293;
        double r807295 = r807292 - r807294;
        double r807296 = r807290 * r807295;
        double r807297 = r807289 - r807296;
        double r807298 = j;
        double r807299 = r807291 * r807286;
        double r807300 = r807282 * r807293;
        double r807301 = r807299 - r807300;
        double r807302 = r807298 * r807301;
        double r807303 = r807297 + r807302;
        return r807303;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r807304 = c;
        double r807305 = a;
        double r807306 = r807304 * r807305;
        double r807307 = y;
        double r807308 = i;
        double r807309 = r807307 * r807308;
        double r807310 = r807306 - r807309;
        double r807311 = j;
        double r807312 = x;
        double r807313 = z;
        double r807314 = t;
        double r807315 = r807305 * r807314;
        double r807316 = -r807315;
        double r807317 = fma(r807307, r807313, r807316);
        double r807318 = cbrt(r807317);
        double r807319 = r807318 * r807318;
        double r807320 = r807312 * r807319;
        double r807321 = r807320 * r807318;
        double r807322 = -r807305;
        double r807323 = fma(r807322, r807314, r807315);
        double r807324 = r807312 * r807323;
        double r807325 = r807321 + r807324;
        double r807326 = b;
        double r807327 = r807304 * r807313;
        double r807328 = r807314 * r807308;
        double r807329 = r807327 - r807328;
        double r807330 = r807326 * r807329;
        double r807331 = r807325 - r807330;
        double r807332 = fma(r807310, r807311, r807331);
        return r807332;
}

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

Original12.4
Target20.4
Herbie12.7
\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Initial program 12.4

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

  2. Simplified12.4

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

  4. Applied prod-diff12.4

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

  5. Applied distribute-lft-in12.4

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

  7. Applied add-cube-cbrt12.7

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

  8. Applied associate-*r*12.7

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

  9. Final simplification12.7

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

Reproduce

herbie shell --seed 2019353 +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)))))