Average Error: 12.6 → 12.8
Time: 9.1s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\left(\left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}\right) \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} - \left(b \cdot \left(c \cdot z - t \cdot i\right) + b \cdot \mathsf{fma}\left(-i, t, i \cdot t\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
\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)
↓
\left(\left(\sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)}\right) \cdot \sqrt[3]{x \cdot \left(y \cdot z - t \cdot a\right)} - \left(b \cdot \left(c \cdot z - t \cdot i\right) + b \cdot \mathsf{fma}\left(-i, t, i \cdot t\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1016244 = x;
        double r1016245 = y;
        double r1016246 = z;
        double r1016247 = r1016245 * r1016246;
        double r1016248 = t;
        double r1016249 = a;
        double r1016250 = r1016248 * r1016249;
        double r1016251 = r1016247 - r1016250;
        double r1016252 = r1016244 * r1016251;
        double r1016253 = b;
        double r1016254 = c;
        double r1016255 = r1016254 * r1016246;
        double r1016256 = i;
        double r1016257 = r1016248 * r1016256;
        double r1016258 = r1016255 - r1016257;
        double r1016259 = r1016253 * r1016258;
        double r1016260 = r1016252 - r1016259;
        double r1016261 = j;
        double r1016262 = r1016254 * r1016249;
        double r1016263 = r1016245 * r1016256;
        double r1016264 = r1016262 - r1016263;
        double r1016265 = r1016261 * r1016264;
        double r1016266 = r1016260 + r1016265;
        return r1016266;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1016267 = x;
        double r1016268 = y;
        double r1016269 = z;
        double r1016270 = r1016268 * r1016269;
        double r1016271 = t;
        double r1016272 = a;
        double r1016273 = r1016271 * r1016272;
        double r1016274 = r1016270 - r1016273;
        double r1016275 = r1016267 * r1016274;
        double r1016276 = cbrt(r1016275);
        double r1016277 = r1016276 * r1016276;
        double r1016278 = r1016277 * r1016276;
        double r1016279 = b;
        double r1016280 = c;
        double r1016281 = r1016280 * r1016269;
        double r1016282 = i;
        double r1016283 = r1016271 * r1016282;
        double r1016284 = r1016281 - r1016283;
        double r1016285 = r1016279 * r1016284;
        double r1016286 = -r1016282;
        double r1016287 = r1016282 * r1016271;
        double r1016288 = fma(r1016286, r1016271, r1016287);
        double r1016289 = r1016279 * r1016288;
        double r1016290 = r1016285 + r1016289;
        double r1016291 = r1016278 - r1016290;
        double r1016292 = j;
        double r1016293 = r1016280 * r1016272;
        double r1016294 = r1016268 * r1016282;
        double r1016295 = r1016293 - r1016294;
        double r1016296 = r1016292 * r1016295;
        double r1016297 = r1016291 + r1016296;
        return r1016297;
}

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

Original12.6
Target20.3
Herbie12.8
\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Initial program 12.6

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

  3. Applied prod-diff12.6

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

  4. Applied distribute-lft-in12.6

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

  5. Simplified12.6

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

  7. Applied add-cube-cbrt12.8

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

  8. Final simplification12.8

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

Reproduce

herbie shell --seed 2019362 +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)))))