Average Error: 12.3 → 12.3
Time: 31.9s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;j \le -1.769364613621733552383620262384135612998 \cdot 10^{-286}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot t\right)\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;j \le 4.199087590050386664363720491477147491239 \cdot 10^{-242}:\\ \;\;\;\;\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right)\right) + \left(\sqrt{j} \cdot \left(c \cdot a - y \cdot i\right)\right) \cdot \sqrt{j}\\ \end{array}\]
\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}\;j \le -1.769364613621733552383620262384135612998 \cdot 10^{-286}:\\
\;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot t\right)\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\

\mathbf{elif}\;j \le 4.199087590050386664363720491477147491239 \cdot 10^{-242}:\\
\;\;\;\;\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r37675382 = x;
        double r37675383 = y;
        double r37675384 = z;
        double r37675385 = r37675383 * r37675384;
        double r37675386 = t;
        double r37675387 = a;
        double r37675388 = r37675386 * r37675387;
        double r37675389 = r37675385 - r37675388;
        double r37675390 = r37675382 * r37675389;
        double r37675391 = b;
        double r37675392 = c;
        double r37675393 = r37675392 * r37675384;
        double r37675394 = i;
        double r37675395 = r37675386 * r37675394;
        double r37675396 = r37675393 - r37675395;
        double r37675397 = r37675391 * r37675396;
        double r37675398 = r37675390 - r37675397;
        double r37675399 = j;
        double r37675400 = r37675392 * r37675387;
        double r37675401 = r37675383 * r37675394;
        double r37675402 = r37675400 - r37675401;
        double r37675403 = r37675399 * r37675402;
        double r37675404 = r37675398 + r37675403;
        return r37675404;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r37675405 = j;
        double r37675406 = -1.7693646136217336e-286;
        bool r37675407 = r37675405 <= r37675406;
        double r37675408 = y;
        double r37675409 = z;
        double r37675410 = r37675408 * r37675409;
        double r37675411 = t;
        double r37675412 = a;
        double r37675413 = r37675411 * r37675412;
        double r37675414 = r37675410 - r37675413;
        double r37675415 = x;
        double r37675416 = r37675414 * r37675415;
        double r37675417 = b;
        double r37675418 = cbrt(r37675417);
        double r37675419 = c;
        double r37675420 = r37675419 * r37675409;
        double r37675421 = i;
        double r37675422 = r37675421 * r37675411;
        double r37675423 = r37675420 - r37675422;
        double r37675424 = r37675418 * r37675423;
        double r37675425 = r37675418 * r37675418;
        double r37675426 = r37675424 * r37675425;
        double r37675427 = r37675416 - r37675426;
        double r37675428 = r37675419 * r37675412;
        double r37675429 = r37675408 * r37675421;
        double r37675430 = r37675428 - r37675429;
        double r37675431 = r37675405 * r37675430;
        double r37675432 = r37675427 + r37675431;
        double r37675433 = 4.199087590050387e-242;
        bool r37675434 = r37675405 <= r37675433;
        double r37675435 = r37675417 * r37675423;
        double r37675436 = r37675416 - r37675435;
        double r37675437 = sqrt(r37675405);
        double r37675438 = r37675437 * r37675430;
        double r37675439 = r37675438 * r37675437;
        double r37675440 = r37675436 + r37675439;
        double r37675441 = r37675434 ? r37675436 : r37675440;
        double r37675442 = r37675407 ? r37675432 : r37675441;
        return r37675442;
}

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

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

Derivation

  1. Split input into 3 regimes

  2. if j < -1.7693646136217336e-286

    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. Using strategy rm

    3. Applied add-cube-cbrt12.7

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

    4. Applied associate-*l*12.7

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

    if -1.7693646136217336e-286 < j < 4.199087590050387e-242

    1. Initial program 18.6

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

    2. Taylor expanded around 0 16.9

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \color{blue}{0}\]

    if 4.199087590050387e-242 < j

    1. Initial program 10.9

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

    3. Applied add-sqr-sqrt11.0

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

    4. Applied associate-*l*11.0

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

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -1.769364613621733552383620262384135612998 \cdot 10^{-286}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot t\right)\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;j \le 4.199087590050386664363720491477147491239 \cdot 10^{-242}:\\ \;\;\;\;\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right)\right) + \left(\sqrt{j} \cdot \left(c \cdot a - y \cdot i\right)\right) \cdot \sqrt{j}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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)))))