Average Error: 12.2 → 9.3
Time: 16.2s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.44311144531657368 \cdot 10^{-44} \lor \neg \left(x \le 1.08484586376827762 \cdot 10^{49}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(j \cdot y\right) \cdot \left(-i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(j \cdot y\right) \cdot \left(-i\right)\right)\\ \end{array}\]
\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)
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.44311144531657368 \cdot 10^{-44} \lor \neg \left(x \le 1.08484586376827762 \cdot 10^{49}\right):\\
\;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(j \cdot y\right) \cdot \left(-i\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(j \cdot y\right) \cdot \left(-i\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r593208 = x;
        double r593209 = y;
        double r593210 = z;
        double r593211 = r593209 * r593210;
        double r593212 = t;
        double r593213 = a;
        double r593214 = r593212 * r593213;
        double r593215 = r593211 - r593214;
        double r593216 = r593208 * r593215;
        double r593217 = b;
        double r593218 = c;
        double r593219 = r593218 * r593210;
        double r593220 = i;
        double r593221 = r593212 * r593220;
        double r593222 = r593219 - r593221;
        double r593223 = r593217 * r593222;
        double r593224 = r593216 - r593223;
        double r593225 = j;
        double r593226 = r593218 * r593213;
        double r593227 = r593209 * r593220;
        double r593228 = r593226 - r593227;
        double r593229 = r593225 * r593228;
        double r593230 = r593224 + r593229;
        return r593230;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r593231 = x;
        double r593232 = -5.443111445316574e-44;
        bool r593233 = r593231 <= r593232;
        double r593234 = 1.0848458637682776e+49;
        bool r593235 = r593231 <= r593234;
        double r593236 = !r593235;
        bool r593237 = r593233 || r593236;
        double r593238 = y;
        double r593239 = z;
        double r593240 = r593238 * r593239;
        double r593241 = t;
        double r593242 = a;
        double r593243 = r593241 * r593242;
        double r593244 = r593240 - r593243;
        double r593245 = r593231 * r593244;
        double r593246 = b;
        double r593247 = c;
        double r593248 = r593247 * r593239;
        double r593249 = i;
        double r593250 = r593241 * r593249;
        double r593251 = r593248 - r593250;
        double r593252 = r593246 * r593251;
        double r593253 = r593245 - r593252;
        double r593254 = j;
        double r593255 = r593242 * r593254;
        double r593256 = r593255 * r593247;
        double r593257 = r593254 * r593238;
        double r593258 = -r593249;
        double r593259 = r593257 * r593258;
        double r593260 = r593256 + r593259;
        double r593261 = r593253 + r593260;
        double r593262 = r593239 * r593231;
        double r593263 = r593238 * r593262;
        double r593264 = r593231 * r593241;
        double r593265 = r593242 * r593264;
        double r593266 = -r593265;
        double r593267 = r593263 + r593266;
        double r593268 = r593267 - r593252;
        double r593269 = r593254 * r593247;
        double r593270 = r593242 * r593269;
        double r593271 = r593270 + r593259;
        double r593272 = r593268 + r593271;
        double r593273 = r593237 ? r593261 : r593272;
        return r593273;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.2
Target19.4
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;x \lt -1.46969429677770502 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.443111445316574e-44 or 1.0848458637682776e+49 < x

    1. Initial program 7.6

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

    3. Applied sub-neg7.6

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

    4. Applied distribute-lft-in7.6

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

    5. Simplified8.5

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

    7. Applied distribute-rgt-neg-in8.5

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

    8. Applied associate-*r*8.3

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

    10. Applied associate-*r*7.6

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

    if -5.443111445316574e-44 < x < 1.0848458637682776e+49

    1. Initial program 15.1

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

    3. Applied sub-neg15.1

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

    4. Applied distribute-lft-in15.1

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

    5. Simplified15.4

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

    7. Applied distribute-rgt-neg-in15.4

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

    8. Applied associate-*r*15.5

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

    10. Applied sub-neg15.5

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

    11. Applied distribute-lft-in15.5

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

    12. Simplified15.5

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

    13. Simplified12.8

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

    15. Applied associate-*l*10.3

      \[\leadsto \left(\left(\color{blue}{y \cdot \left(z \cdot x\right)} + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(j \cdot y\right) \cdot \left(-i\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.44311144531657368 \cdot 10^{-44} \lor \neg \left(x \le 1.08484586376827762 \cdot 10^{49}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(j \cdot y\right) \cdot \left(-i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(j \cdot y\right) \cdot \left(-i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +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)))))