Average Error: 11.3 → 12.1
Time: 27.4s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.2743838771657712 \cdot 10^{-169}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(\left(j \cdot t\right) \cdot c - i \cdot \left(y \cdot j\right)\right) + \mathsf{fma}\left(-y, i, i \cdot y\right) \cdot j\right)\\ \mathbf{elif}\;b \le 4.999298010517163 \cdot 10^{-281}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(y \cdot z - t \cdot a\right) \cdot x\\ \mathbf{elif}\;b \le 8.631973115792552 \cdot 10^{-67}:\\ \;\;\;\;\left(\mathsf{fma}\left(-y, i, i \cdot y\right) \cdot j + \left(\left(j \cdot t\right) \cdot c - j \cdot \left(i \cdot y\right)\right)\right) + \left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(\left(c \cdot j\right) \cdot t - j \cdot \left(i \cdot y\right)\right) + \mathsf{fma}\left(-y, i, i \cdot y\right) \cdot j\right)\\ \end{array}\]
\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}\;b \le -1.2743838771657712 \cdot 10^{-169}:\\
\;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(\left(j \cdot t\right) \cdot c - i \cdot \left(y \cdot j\right)\right) + \mathsf{fma}\left(-y, i, i \cdot y\right) \cdot j\right)\\

\mathbf{elif}\;b \le 4.999298010517163 \cdot 10^{-281}:\\
\;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(y \cdot z - t \cdot a\right) \cdot x\\

\mathbf{elif}\;b \le 8.631973115792552 \cdot 10^{-67}:\\
\;\;\;\;\left(\mathsf{fma}\left(-y, i, i \cdot y\right) \cdot j + \left(\left(j \cdot t\right) \cdot c - j \cdot \left(i \cdot y\right)\right)\right) + \left(\left(y \cdot z - t \cdot a\right) \cdot x - \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)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1820060 = x;
        double r1820061 = y;
        double r1820062 = z;
        double r1820063 = r1820061 * r1820062;
        double r1820064 = t;
        double r1820065 = a;
        double r1820066 = r1820064 * r1820065;
        double r1820067 = r1820063 - r1820066;
        double r1820068 = r1820060 * r1820067;
        double r1820069 = b;
        double r1820070 = c;
        double r1820071 = r1820070 * r1820062;
        double r1820072 = i;
        double r1820073 = r1820072 * r1820065;
        double r1820074 = r1820071 - r1820073;
        double r1820075 = r1820069 * r1820074;
        double r1820076 = r1820068 - r1820075;
        double r1820077 = j;
        double r1820078 = r1820070 * r1820064;
        double r1820079 = r1820072 * r1820061;
        double r1820080 = r1820078 - r1820079;
        double r1820081 = r1820077 * r1820080;
        double r1820082 = r1820076 + r1820081;
        return r1820082;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1820083 = b;
        double r1820084 = -1.2743838771657712e-169;
        bool r1820085 = r1820083 <= r1820084;
        double r1820086 = y;
        double r1820087 = z;
        double r1820088 = r1820086 * r1820087;
        double r1820089 = t;
        double r1820090 = a;
        double r1820091 = r1820089 * r1820090;
        double r1820092 = r1820088 - r1820091;
        double r1820093 = x;
        double r1820094 = r1820092 * r1820093;
        double r1820095 = c;
        double r1820096 = r1820095 * r1820087;
        double r1820097 = i;
        double r1820098 = r1820097 * r1820090;
        double r1820099 = r1820096 - r1820098;
        double r1820100 = r1820083 * r1820099;
        double r1820101 = r1820094 - r1820100;
        double r1820102 = j;
        double r1820103 = r1820102 * r1820089;
        double r1820104 = r1820103 * r1820095;
        double r1820105 = r1820086 * r1820102;
        double r1820106 = r1820097 * r1820105;
        double r1820107 = r1820104 - r1820106;
        double r1820108 = -r1820086;
        double r1820109 = r1820097 * r1820086;
        double r1820110 = fma(r1820108, r1820097, r1820109);
        double r1820111 = r1820110 * r1820102;
        double r1820112 = r1820107 + r1820111;
        double r1820113 = r1820101 + r1820112;
        double r1820114 = 4.999298010517163e-281;
        bool r1820115 = r1820083 <= r1820114;
        double r1820116 = r1820095 * r1820089;
        double r1820117 = r1820116 - r1820109;
        double r1820118 = r1820102 * r1820117;
        double r1820119 = r1820118 + r1820094;
        double r1820120 = 8.631973115792552e-67;
        bool r1820121 = r1820083 <= r1820120;
        double r1820122 = r1820102 * r1820109;
        double r1820123 = r1820104 - r1820122;
        double r1820124 = r1820111 + r1820123;
        double r1820125 = cbrt(r1820099);
        double r1820126 = r1820125 * r1820125;
        double r1820127 = r1820083 * r1820126;
        double r1820128 = r1820127 * r1820125;
        double r1820129 = r1820094 - r1820128;
        double r1820130 = r1820124 + r1820129;
        double r1820131 = r1820095 * r1820102;
        double r1820132 = r1820131 * r1820089;
        double r1820133 = r1820132 - r1820122;
        double r1820134 = r1820133 + r1820111;
        double r1820135 = r1820101 + r1820134;
        double r1820136 = r1820121 ? r1820130 : r1820135;
        double r1820137 = r1820115 ? r1820119 : r1820136;
        double r1820138 = r1820085 ? r1820113 : r1820137;
        return r1820138;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -1.2743838771657712e-169

    1. Initial program 9.5

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

    3. Applied prod-diff9.5

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

    4. Applied distribute-rgt-in9.5

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

    5. Taylor expanded around inf 11.8

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

    6. Simplified9.5

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

    8. Applied associate-*r*11.3

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

    if -1.2743838771657712e-169 < b < 4.999298010517163e-281

    1. Initial program 16.8

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

    2. Taylor expanded around 0 17.1

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

    if 4.999298010517163e-281 < b < 8.631973115792552e-67

    1. Initial program 14.7

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

    3. Applied prod-diff14.7

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

    4. Applied distribute-rgt-in14.7

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

    5. Taylor expanded around inf 16.4

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

    6. Simplified14.5

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

    8. Applied add-cube-cbrt14.6

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

    9. Applied associate-*r*14.6

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

    if 8.631973115792552e-67 < b

    1. Initial program 7.4

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

    3. Applied prod-diff7.4

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

    4. Applied distribute-rgt-in7.4

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

    5. Taylor expanded around inf 9.9

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

    6. Simplified8.1

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

    7. Taylor expanded around inf 7.9

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

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.2743838771657712 \cdot 10^{-169}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(\left(j \cdot t\right) \cdot c - i \cdot \left(y \cdot j\right)\right) + \mathsf{fma}\left(-y, i, i \cdot y\right) \cdot j\right)\\ \mathbf{elif}\;b \le 4.999298010517163 \cdot 10^{-281}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(y \cdot z - t \cdot a\right) \cdot x\\ \mathbf{elif}\;b \le 8.631973115792552 \cdot 10^{-67}:\\ \;\;\;\;\left(\mathsf{fma}\left(-y, i, i \cdot y\right) \cdot j + \left(\left(j \cdot t\right) \cdot c - j \cdot \left(i \cdot y\right)\right)\right) + \left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(b \cdot \left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(\left(c \cdot j\right) \cdot t - j \cdot \left(i \cdot y\right)\right) + \mathsf{fma}\left(-y, i, i \cdot y\right) \cdot j\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))