Average Error: 12.4 → 10.5
Time: 22.8s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -8722537516109230520438041346048 \lor \neg \left(c \le 4016321628929563854216906355507200\right):\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, c \cdot \left(t \cdot j - z \cdot b\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \left(b \cdot \left(\sqrt[3]{i \cdot a - c \cdot z} \cdot \sqrt[3]{i \cdot a - c \cdot z}\right)\right) \cdot \sqrt[3]{i \cdot a - c \cdot z} + j \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \end{array}\]
\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}\;c \le -8722537516109230520438041346048 \lor \neg \left(c \le 4016321628929563854216906355507200\right):\\
\;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, c \cdot \left(t \cdot j - z \cdot b\right) - i \cdot \left(j \cdot y\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \left(b \cdot \left(\sqrt[3]{i \cdot a - c \cdot z} \cdot \sqrt[3]{i \cdot a - c \cdot z}\right)\right) \cdot \sqrt[3]{i \cdot a - c \cdot z} + j \cdot \left(c \cdot t - i \cdot y\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 r390206 = x;
        double r390207 = y;
        double r390208 = z;
        double r390209 = r390207 * r390208;
        double r390210 = t;
        double r390211 = a;
        double r390212 = r390210 * r390211;
        double r390213 = r390209 - r390212;
        double r390214 = r390206 * r390213;
        double r390215 = b;
        double r390216 = c;
        double r390217 = r390216 * r390208;
        double r390218 = i;
        double r390219 = r390218 * r390211;
        double r390220 = r390217 - r390219;
        double r390221 = r390215 * r390220;
        double r390222 = r390214 - r390221;
        double r390223 = j;
        double r390224 = r390216 * r390210;
        double r390225 = r390218 * r390207;
        double r390226 = r390224 - r390225;
        double r390227 = r390223 * r390226;
        double r390228 = r390222 + r390227;
        return r390228;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r390229 = c;
        double r390230 = -8.72253751610923e+30;
        bool r390231 = r390229 <= r390230;
        double r390232 = 4.016321628929564e+33;
        bool r390233 = r390229 <= r390232;
        double r390234 = !r390233;
        bool r390235 = r390231 || r390234;
        double r390236 = x;
        double r390237 = y;
        double r390238 = z;
        double r390239 = r390237 * r390238;
        double r390240 = t;
        double r390241 = a;
        double r390242 = r390240 * r390241;
        double r390243 = r390239 - r390242;
        double r390244 = j;
        double r390245 = r390240 * r390244;
        double r390246 = b;
        double r390247 = r390238 * r390246;
        double r390248 = r390245 - r390247;
        double r390249 = r390229 * r390248;
        double r390250 = i;
        double r390251 = r390244 * r390237;
        double r390252 = r390250 * r390251;
        double r390253 = r390249 - r390252;
        double r390254 = fma(r390236, r390243, r390253);
        double r390255 = r390250 * r390241;
        double r390256 = r390229 * r390238;
        double r390257 = r390255 - r390256;
        double r390258 = cbrt(r390257);
        double r390259 = r390258 * r390258;
        double r390260 = r390246 * r390259;
        double r390261 = r390260 * r390258;
        double r390262 = r390229 * r390240;
        double r390263 = r390250 * r390237;
        double r390264 = r390262 - r390263;
        double r390265 = r390244 * r390264;
        double r390266 = r390261 + r390265;
        double r390267 = fma(r390236, r390243, r390266);
        double r390268 = r390235 ? r390254 : r390267;
        return r390268;
}

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

Original12.4
Target16.2
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;t \lt -8.12097891919591218149793027759825150959 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.712553818218485141757938537793350881052 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.633533346031583686060259351057142920433 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.053588855745548710002760210539645467715 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if c < -8.72253751610923e+30 or 4.016321628929564e+33 < c

    1. Initial program 18.5

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

    2. Simplified18.5

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

    3. Taylor expanded around inf 22.0

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

    4. Simplified12.7

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

    if -8.72253751610923e+30 < c < 4.016321628929564e+33

    1. Initial program 9.1

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

    2. Simplified9.1

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

    4. Applied fma-udef9.1

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

    6. Applied add-cube-cbrt9.4

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

    7. Applied associate-*r*9.4

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -8722537516109230520438041346048 \lor \neg \left(c \le 4016321628929563854216906355507200\right):\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, c \cdot \left(t \cdot j - z \cdot b\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \left(b \cdot \left(\sqrt[3]{i \cdot a - c \cdot z} \cdot \sqrt[3]{i \cdot a - c \cdot z}\right)\right) \cdot \sqrt[3]{i \cdot a - c \cdot z} + j \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +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 -8.1209789191959122e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.7125538182184851e-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 -7.63353334603158369e-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)))))