Average Error: 11.8 → 12.2
Time: 12.9s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -2.38118333135362335 \cdot 10^{92} \lor \neg \left(a \le 3.66852259421464815 \cdot 10^{132}\right):\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\sqrt[3]{\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}} \cdot \sqrt[3]{\sqrt[3]{c \cdot t - i \cdot y}}\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \end{array}\]
\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}\;a \le -2.38118333135362335 \cdot 10^{92} \lor \neg \left(a \le 3.66852259421464815 \cdot 10^{132}\right):\\
\;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\left(\left(\sqrt[3]{\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}} \cdot \sqrt[3]{\sqrt[3]{c \cdot t - i \cdot y}}\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}, j, x \cdot \left(y \cdot z - t \cdot a\right) - 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 r550087 = x;
        double r550088 = y;
        double r550089 = z;
        double r550090 = r550088 * r550089;
        double r550091 = t;
        double r550092 = a;
        double r550093 = r550091 * r550092;
        double r550094 = r550090 - r550093;
        double r550095 = r550087 * r550094;
        double r550096 = b;
        double r550097 = c;
        double r550098 = r550097 * r550089;
        double r550099 = i;
        double r550100 = r550099 * r550092;
        double r550101 = r550098 - r550100;
        double r550102 = r550096 * r550101;
        double r550103 = r550095 - r550102;
        double r550104 = j;
        double r550105 = r550097 * r550091;
        double r550106 = r550099 * r550088;
        double r550107 = r550105 - r550106;
        double r550108 = r550104 * r550107;
        double r550109 = r550103 + r550108;
        return r550109;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r550110 = a;
        double r550111 = -2.3811833313536233e+92;
        bool r550112 = r550110 <= r550111;
        double r550113 = 3.668522594214648e+132;
        bool r550114 = r550110 <= r550113;
        double r550115 = !r550114;
        bool r550116 = r550112 || r550115;
        double r550117 = i;
        double r550118 = b;
        double r550119 = r550117 * r550118;
        double r550120 = z;
        double r550121 = c;
        double r550122 = r550118 * r550121;
        double r550123 = x;
        double r550124 = t;
        double r550125 = r550123 * r550124;
        double r550126 = r550110 * r550125;
        double r550127 = fma(r550120, r550122, r550126);
        double r550128 = -r550127;
        double r550129 = fma(r550110, r550119, r550128);
        double r550130 = r550121 * r550124;
        double r550131 = y;
        double r550132 = r550117 * r550131;
        double r550133 = r550130 - r550132;
        double r550134 = cbrt(r550133);
        double r550135 = r550134 * r550134;
        double r550136 = cbrt(r550135);
        double r550137 = cbrt(r550134);
        double r550138 = r550136 * r550137;
        double r550139 = r550138 * r550134;
        double r550140 = r550139 * r550134;
        double r550141 = j;
        double r550142 = r550131 * r550120;
        double r550143 = r550124 * r550110;
        double r550144 = r550142 - r550143;
        double r550145 = r550123 * r550144;
        double r550146 = r550121 * r550120;
        double r550147 = r550117 * r550110;
        double r550148 = r550146 - r550147;
        double r550149 = r550118 * r550148;
        double r550150 = r550145 - r550149;
        double r550151 = fma(r550140, r550141, r550150);
        double r550152 = r550116 ? r550129 : r550151;
        return r550152;
}

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

Target

Original11.8
Target15.7
Herbie12.2
\[\begin{array}{l} \mathbf{if}\;t \lt -8.1209789191959122 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.7125538182184851 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.63353334603158369 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -2.3811833313536233e+92 or 3.668522594214648e+132 < a

    1. Initial program 21.0

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

    2. Simplified21.0

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

    3. Taylor expanded around inf 21.3

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

    4. Simplified21.3

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

    if -2.3811833313536233e+92 < a < 3.668522594214648e+132

    1. Initial program 9.4

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

    2. Simplified9.4

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

    4. Applied add-cube-cbrt9.7

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

    6. Applied add-cube-cbrt9.7

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

    7. Applied cbrt-prod9.8

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(\sqrt[3]{\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}} \cdot \sqrt[3]{\sqrt[3]{c \cdot t - i \cdot y}}\right)} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.38118333135362335 \cdot 10^{92} \lor \neg \left(a \le 3.66852259421464815 \cdot 10^{132}\right):\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\sqrt[3]{\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}} \cdot \sqrt[3]{\sqrt[3]{c \cdot t - i \cdot y}}\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(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)))))