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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.69001902444813421 \cdot 10^{149} \lor \neg \left(y \le 6.8519775127199077 \cdot 10^{134}\right):\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\right)\\ \end{array}\]

\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}\;y \le -2.69001902444813421 \cdot 10^{149} \lor \neg \left(y \le 6.8519775127199077 \cdot 10^{134}\right):\\
\;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)\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 r553825 = x;
        double r553826 = y;
        double r553827 = z;
        double r553828 = r553826 * r553827;
        double r553829 = t;
        double r553830 = a;
        double r553831 = r553829 * r553830;
        double r553832 = r553828 - r553831;
        double r553833 = r553825 * r553832;
        double r553834 = b;
        double r553835 = c;
        double r553836 = r553835 * r553827;
        double r553837 = i;
        double r553838 = r553837 * r553830;
        double r553839 = r553836 - r553838;
        double r553840 = r553834 * r553839;
        double r553841 = r553833 - r553840;
        double r553842 = j;
        double r553843 = r553835 * r553829;
        double r553844 = r553837 * r553826;
        double r553845 = r553843 - r553844;
        double r553846 = r553842 * r553845;
        double r553847 = r553841 + r553846;
        return r553847;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r553848 = y;
        double r553849 = -2.690019024448134e+149;
        bool r553850 = r553848 <= r553849;
        double r553851 = 6.851977512719908e+134;
        bool r553852 = r553848 <= r553851;
        double r553853 = !r553852;
        bool r553854 = r553850 || r553853;
        double r553855 = i;
        double r553856 = a;
        double r553857 = r553855 * r553856;
        double r553858 = c;
        double r553859 = z;
        double r553860 = r553858 * r553859;
        double r553861 = r553857 - r553860;
        double r553862 = b;
        double r553863 = x;
        double r553864 = r553863 * r553859;
        double r553865 = j;
        double r553866 = r553855 * r553865;
        double r553867 = r553864 - r553866;
        double r553868 = r553848 * r553867;
        double r553869 = t;
        double r553870 = r553863 * r553869;
        double r553871 = r553856 * r553870;
        double r553872 = r553868 - r553871;
        double r553873 = fma(r553861, r553862, r553872);
        double r553874 = r553858 * r553869;
        double r553875 = r553855 * r553848;
        double r553876 = r553874 - r553875;
        double r553877 = cbrt(r553863);
        double r553878 = r553877 * r553877;
        double r553879 = r553848 * r553859;
        double r553880 = r553869 * r553856;
        double r553881 = r553879 - r553880;
        double r553882 = r553877 * r553881;
        double r553883 = r553878 * r553882;
        double r553884 = fma(r553865, r553876, r553883);
        double r553885 = fma(r553861, r553862, r553884);
        double r553886 = r553854 ? r553873 : r553885;
        return r553886;
}

Target

Original	12.3
Target	16.3
Herbie	10.4

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

Derivation

Split input into 2 regimes
if y < -2.690019024448134e+149 or 6.851977512719908e+134 < y
1. Initial program 23.4
  \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
2. Simplified23.4
  \[\leadsto \color{blue}{\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(y \cdot z - t \cdot a\right)\right)\right)}\]
3. Using strategy rm
4. Applied add-cube-cbrt23.6
  \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\]
5. Applied associate-*l*23.6
  \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)}\right)\right)\]
6. Taylor expanded around inf 25.0
  \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \color{blue}{x \cdot \left(z \cdot y\right) - \left(i \cdot \left(j \cdot y\right) + a \cdot \left(x \cdot t\right)\right)}\right)\]
7. Simplified10.5
  \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \color{blue}{y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)}\right)\]
if -2.690019024448134e+149 < y < 6.851977512719908e+134
1. Initial program 10.1
  \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
2. Simplified10.1
  \[\leadsto \color{blue}{\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(y \cdot z - t \cdot a\right)\right)\right)}\]
3. Using strategy rm
4. Applied add-cube-cbrt10.4
  \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\]
5. Applied associate-*l*10.3
  \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)}\right)\right)\]
Recombined 2 regimes into one program.
Final simplification10.4
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.69001902444813421 \cdot 10^{149} \lor \neg \left(y \le 6.8519775127199077 \cdot 10^{134}\right):\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))))

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Target

Derivation

`if y < -2.690019024448134e+149 or 6.851977512719908e+134 < y`

`if -2.690019024448134e+149 < y < 6.851977512719908e+134`

Reproduce