Average Error: 12.2 → 11.8
Time: 25.0s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;j \le -9.929536798422278605563067575704178711758 \cdot 10^{96}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(-b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;j \le -1.098607656333811782069347702956620830271 \cdot 10^{-146}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;j \le -6.651508483195953646948775651706394971033 \cdot 10^{-258}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(c \cdot \left(t \cdot j\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;j \le 2.649066615389731199752472489106042554466 \cdot 10^{253}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right) + \left(c \cdot \left(t \cdot j\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(-b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \end{array}\]
\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}\;j \le -9.929536798422278605563067575704178711758 \cdot 10^{96}:\\
\;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(-b \cdot \left(c \cdot z - i \cdot a\right)\right)\\

\mathbf{elif}\;j \le -1.098607656333811782069347702956620830271 \cdot 10^{-146}:\\
\;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\

\mathbf{elif}\;j \le -6.651508483195953646948775651706394971033 \cdot 10^{-258}:\\
\;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(c \cdot \left(t \cdot j\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\

\mathbf{elif}\;j \le 2.649066615389731199752472489106042554466 \cdot 10^{253}:\\
\;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right) + \left(c \cdot \left(t \cdot j\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(-b \cdot \left(c \cdot z - i \cdot a\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 r377003 = x;
        double r377004 = y;
        double r377005 = z;
        double r377006 = r377004 * r377005;
        double r377007 = t;
        double r377008 = a;
        double r377009 = r377007 * r377008;
        double r377010 = r377006 - r377009;
        double r377011 = r377003 * r377010;
        double r377012 = b;
        double r377013 = c;
        double r377014 = r377013 * r377005;
        double r377015 = i;
        double r377016 = r377015 * r377008;
        double r377017 = r377014 - r377016;
        double r377018 = r377012 * r377017;
        double r377019 = r377011 - r377018;
        double r377020 = j;
        double r377021 = r377013 * r377007;
        double r377022 = r377015 * r377004;
        double r377023 = r377021 - r377022;
        double r377024 = r377020 * r377023;
        double r377025 = r377019 + r377024;
        return r377025;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r377026 = j;
        double r377027 = -9.929536798422279e+96;
        bool r377028 = r377026 <= r377027;
        double r377029 = c;
        double r377030 = t;
        double r377031 = r377029 * r377030;
        double r377032 = i;
        double r377033 = y;
        double r377034 = r377032 * r377033;
        double r377035 = r377031 - r377034;
        double r377036 = r377026 * r377035;
        double r377037 = b;
        double r377038 = z;
        double r377039 = r377029 * r377038;
        double r377040 = a;
        double r377041 = r377032 * r377040;
        double r377042 = r377039 - r377041;
        double r377043 = r377037 * r377042;
        double r377044 = -r377043;
        double r377045 = r377036 + r377044;
        double r377046 = -1.0986076563338118e-146;
        bool r377047 = r377026 <= r377046;
        double r377048 = x;
        double r377049 = r377033 * r377038;
        double r377050 = r377030 * r377040;
        double r377051 = r377049 - r377050;
        double r377052 = r377048 * r377051;
        double r377053 = r377052 - r377043;
        double r377054 = r377026 * r377029;
        double r377055 = r377030 * r377054;
        double r377056 = r377026 * r377033;
        double r377057 = r377032 * r377056;
        double r377058 = -r377057;
        double r377059 = r377055 + r377058;
        double r377060 = r377053 + r377059;
        double r377061 = -6.651508483195954e-258;
        bool r377062 = r377026 <= r377061;
        double r377063 = r377037 * r377029;
        double r377064 = r377038 * r377063;
        double r377065 = -r377041;
        double r377066 = r377065 * r377037;
        double r377067 = r377064 + r377066;
        double r377068 = r377052 - r377067;
        double r377069 = r377030 * r377026;
        double r377070 = r377029 * r377069;
        double r377071 = r377070 + r377058;
        double r377072 = r377068 + r377071;
        double r377073 = 2.6490666153897312e+253;
        bool r377074 = r377026 <= r377073;
        double r377075 = cbrt(r377042);
        double r377076 = r377075 * r377075;
        double r377077 = r377037 * r377076;
        double r377078 = r377077 * r377075;
        double r377079 = r377052 - r377078;
        double r377080 = r377079 + r377071;
        double r377081 = r377074 ? r377080 : r377045;
        double r377082 = r377062 ? r377072 : r377081;
        double r377083 = r377047 ? r377060 : r377082;
        double r377084 = r377028 ? r377045 : r377083;
        return r377084;
}

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
Target16.1
Herbie11.8
\[\begin{array}{l} \mathbf{if}\;t \lt -8.12097891919591218149793027759825150959 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.712553818218485141757938537793350881052 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.633533346031583686060259351057142920433 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.053588855745548710002760210539645467715 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if j < -9.929536798422279e+96 or 2.6490666153897312e+253 < j

    1. Initial program 7.3

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

    2. Taylor expanded around 0 15.8

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

    if -9.929536798422279e+96 < j < -1.0986076563338118e-146

    1. Initial program 11.3

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

    3. Applied sub-neg11.3

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

    4. Applied distribute-lft-in11.3

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

    5. Simplified11.3

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

    6. Simplified10.4

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

    7. Taylor expanded around inf 9.1

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

    if -1.0986076563338118e-146 < j < -6.651508483195954e-258

    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

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

    4. Applied distribute-lft-in17.1

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

    5. Simplified17.1

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

    6. Simplified14.0

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

    8. Applied associate-*l*10.9

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

    10. Applied sub-neg10.9

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

    11. Applied distribute-lft-in10.9

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

    12. Simplified11.5

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

    13. Simplified11.5

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

    if -6.651508483195954e-258 < j < 2.6490666153897312e+253

    1. Initial program 12.9

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

    3. Applied sub-neg12.9

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

    4. Applied distribute-lft-in12.9

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

    5. Simplified12.9

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

    6. Simplified12.4

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

    8. Applied associate-*l*11.6

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

    10. Applied add-cube-cbrt11.9

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

    11. Applied associate-*r*11.9

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}}\right) + \left(c \cdot \left(t \cdot j\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -9.929536798422278605563067575704178711758 \cdot 10^{96}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(-b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;j \le -1.098607656333811782069347702956620830271 \cdot 10^{-146}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;j \le -6.651508483195953646948775651706394971033 \cdot 10^{-258}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(c \cdot \left(t \cdot j\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;j \le 2.649066615389731199752472489106042554466 \cdot 10^{253}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right) + \left(c \cdot \left(t \cdot j\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(-b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :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) (pow (* i y) 2))) (+ (* 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) (pow (* i y) 2))) (+ (* 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)))))