Average Error: 12.0 → 9.7
Time: 27.5s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.152794873699225302890935406008557975939 \cdot 10^{-102}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \sqrt[3]{y \cdot z - a \cdot t}\right)\right) \cdot x - b \cdot \left(z \cdot c - i \cdot a\right)\right)\\ \mathbf{elif}\;x \le 2.967077761808358008699176682955861081052 \cdot 10^{-95}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(\left(y \cdot x\right) \cdot z + \left(x \cdot t\right) \cdot \left(-a\right)\right) - \sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot t\right) \cdot \left(-x\right) + \left(\sqrt{x} \cdot \left(y \cdot z\right)\right) \cdot \sqrt{x}\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(c \cdot t - y \cdot i\right) \cdot j\\ \end{array}\]
\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 -6.152794873699225302890935406008557975939 \cdot 10^{-102}:\\
\;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \sqrt[3]{y \cdot z - a \cdot t}\right)\right) \cdot x - b \cdot \left(z \cdot c - i \cdot a\right)\right)\\

\mathbf{elif}\;x \le 2.967077761808358008699176682955861081052 \cdot 10^{-95}:\\
\;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(\left(y \cdot x\right) \cdot z + \left(x \cdot t\right) \cdot \left(-a\right)\right) - \sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)}\right)\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r27790577 = x;
        double r27790578 = y;
        double r27790579 = z;
        double r27790580 = r27790578 * r27790579;
        double r27790581 = t;
        double r27790582 = a;
        double r27790583 = r27790581 * r27790582;
        double r27790584 = r27790580 - r27790583;
        double r27790585 = r27790577 * r27790584;
        double r27790586 = b;
        double r27790587 = c;
        double r27790588 = r27790587 * r27790579;
        double r27790589 = i;
        double r27790590 = r27790589 * r27790582;
        double r27790591 = r27790588 - r27790590;
        double r27790592 = r27790586 * r27790591;
        double r27790593 = r27790585 - r27790592;
        double r27790594 = j;
        double r27790595 = r27790587 * r27790581;
        double r27790596 = r27790589 * r27790578;
        double r27790597 = r27790595 - r27790596;
        double r27790598 = r27790594 * r27790597;
        double r27790599 = r27790593 + r27790598;
        return r27790599;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r27790600 = x;
        double r27790601 = -6.152794873699225e-102;
        bool r27790602 = r27790600 <= r27790601;
        double r27790603 = c;
        double r27790604 = t;
        double r27790605 = r27790603 * r27790604;
        double r27790606 = y;
        double r27790607 = i;
        double r27790608 = r27790606 * r27790607;
        double r27790609 = r27790605 - r27790608;
        double r27790610 = j;
        double r27790611 = r27790609 * r27790610;
        double r27790612 = z;
        double r27790613 = r27790606 * r27790612;
        double r27790614 = a;
        double r27790615 = r27790614 * r27790604;
        double r27790616 = r27790613 - r27790615;
        double r27790617 = cbrt(r27790616);
        double r27790618 = r27790617 * r27790617;
        double r27790619 = r27790617 * r27790618;
        double r27790620 = r27790619 * r27790600;
        double r27790621 = b;
        double r27790622 = r27790612 * r27790603;
        double r27790623 = r27790607 * r27790614;
        double r27790624 = r27790622 - r27790623;
        double r27790625 = r27790621 * r27790624;
        double r27790626 = r27790620 - r27790625;
        double r27790627 = r27790611 + r27790626;
        double r27790628 = 2.967077761808358e-95;
        bool r27790629 = r27790600 <= r27790628;
        double r27790630 = r27790606 * r27790600;
        double r27790631 = r27790630 * r27790612;
        double r27790632 = r27790600 * r27790604;
        double r27790633 = -r27790614;
        double r27790634 = r27790632 * r27790633;
        double r27790635 = r27790631 + r27790634;
        double r27790636 = cbrt(r27790625);
        double r27790637 = r27790636 * r27790636;
        double r27790638 = r27790636 * r27790637;
        double r27790639 = r27790635 - r27790638;
        double r27790640 = r27790611 + r27790639;
        double r27790641 = -r27790600;
        double r27790642 = r27790615 * r27790641;
        double r27790643 = sqrt(r27790600);
        double r27790644 = r27790643 * r27790613;
        double r27790645 = r27790644 * r27790643;
        double r27790646 = r27790642 + r27790645;
        double r27790647 = r27790646 - r27790625;
        double r27790648 = r27790647 + r27790611;
        double r27790649 = r27790629 ? r27790640 : r27790648;
        double r27790650 = r27790602 ? r27790627 : r27790649;
        return r27790650;
}

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

Original12.0
Target15.7
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;t \lt -8.12097891919591218149793027759825150959 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.712553818218485141757938537793350881052 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.633533346031583686060259351057142920433 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.053588855745548710002760210539645467715 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -6.152794873699225e-102

    1. Initial program 8.5

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

    3. Applied add-cube-cbrt8.9

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

    if -6.152794873699225e-102 < x < 2.967077761808358e-95

    1. Initial program 17.1

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

    3. Applied sub-neg17.1

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

    4. Applied distribute-lft-in17.1

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

    6. Applied associate-*r*13.9

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

    8. Applied distribute-lft-neg-in13.9

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

    9. Applied associate-*r*10.9

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

    11. Applied add-cube-cbrt11.3

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

    if 2.967077761808358e-95 < x

    1. Initial program 8.0

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

    3. Applied sub-neg8.0

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

    4. Applied distribute-lft-in8.0

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

    6. Applied add-sqr-sqrt8.1

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

    7. Applied associate-*l*8.1

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.152794873699225302890935406008557975939 \cdot 10^{-102}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \sqrt[3]{y \cdot z - a \cdot t}\right)\right) \cdot x - b \cdot \left(z \cdot c - i \cdot a\right)\right)\\ \mathbf{elif}\;x \le 2.967077761808358008699176682955861081052 \cdot 10^{-95}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(\left(y \cdot x\right) \cdot z + \left(x \cdot t\right) \cdot \left(-a\right)\right) - \sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \left(\sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)} \cdot \sqrt[3]{b \cdot \left(z \cdot c - i \cdot a\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot t\right) \cdot \left(-x\right) + \left(\sqrt{x} \cdot \left(y \cdot z\right)\right) \cdot \sqrt{x}\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(c \cdot t - y \cdot i\right) \cdot j\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"

  :herbie-target
  (if (< t -8.120978919195912e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.712553818218485e-169) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0))) (+ (* c t) (* i y)))) (if (< t -7.633533346031584e-308) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 1.0535888557455487e-139) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))