Average Error: 12.0 → 12.6
Time: 9.1s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.3735883346778377 \cdot 10^{-126}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right)\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\\ \mathbf{elif}\;x \le 1.34472083159762248 \cdot 10^{-129}:\\ \;\;\;\;\left(0 - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]
\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.3735883346778377 \cdot 10^{-126}:\\
\;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right)\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\\

\mathbf{elif}\;x \le 1.34472083159762248 \cdot 10^{-129}:\\
\;\;\;\;\left(0 - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r103424 = x;
        double r103425 = y;
        double r103426 = z;
        double r103427 = r103425 * r103426;
        double r103428 = t;
        double r103429 = a;
        double r103430 = r103428 * r103429;
        double r103431 = r103427 - r103430;
        double r103432 = r103424 * r103431;
        double r103433 = b;
        double r103434 = c;
        double r103435 = r103434 * r103426;
        double r103436 = i;
        double r103437 = r103436 * r103429;
        double r103438 = r103435 - r103437;
        double r103439 = r103433 * r103438;
        double r103440 = r103432 - r103439;
        double r103441 = j;
        double r103442 = r103434 * r103428;
        double r103443 = r103436 * r103425;
        double r103444 = r103442 - r103443;
        double r103445 = r103441 * r103444;
        double r103446 = r103440 + r103445;
        return r103446;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r103447 = x;
        double r103448 = -3.373588334677838e-126;
        bool r103449 = r103447 <= r103448;
        double r103450 = y;
        double r103451 = z;
        double r103452 = r103450 * r103451;
        double r103453 = t;
        double r103454 = a;
        double r103455 = r103453 * r103454;
        double r103456 = r103452 - r103455;
        double r103457 = r103447 * r103456;
        double r103458 = b;
        double r103459 = c;
        double r103460 = r103459 * r103451;
        double r103461 = i;
        double r103462 = r103461 * r103454;
        double r103463 = r103460 - r103462;
        double r103464 = r103458 * r103463;
        double r103465 = r103457 - r103464;
        double r103466 = j;
        double r103467 = r103459 * r103453;
        double r103468 = r103461 * r103450;
        double r103469 = r103467 - r103468;
        double r103470 = cbrt(r103469);
        double r103471 = r103470 * r103470;
        double r103472 = r103466 * r103471;
        double r103473 = r103472 * r103470;
        double r103474 = r103465 + r103473;
        double r103475 = 1.3447208315976225e-129;
        bool r103476 = r103447 <= r103475;
        double r103477 = 0.0;
        double r103478 = r103477 - r103464;
        double r103479 = r103466 * r103469;
        double r103480 = r103478 + r103479;
        double r103481 = sqrt(r103447);
        double r103482 = r103481 * r103456;
        double r103483 = r103481 * r103482;
        double r103484 = r103483 - r103464;
        double r103485 = r103484 + r103479;
        double r103486 = r103476 ? r103480 : r103485;
        double r103487 = r103449 ? r103474 : r103486;
        return r103487;
}

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

Derivation

  1. Split input into 3 regimes

  2. if x < -3.373588334677838e-126

    1. Initial program 9.8

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

    3. Applied add-cube-cbrt10.1

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

    4. Applied associate-*r*10.0

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

    if -3.373588334677838e-126 < x < 1.3447208315976225e-129

    1. Initial program 16.3

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

    2. Taylor expanded around 0 17.5

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

    if 1.3447208315976225e-129 < x

    1. Initial program 9.3

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

    3. Applied add-sqr-sqrt9.4

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

    4. Applied associate-*l*9.4

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

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.3735883346778377 \cdot 10^{-126}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right)\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\\ \mathbf{elif}\;x \le 1.34472083159762248 \cdot 10^{-129}:\\ \;\;\;\;\left(0 - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))