Average Error: 12.4 → 14.3
Time: 11.1s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.13021924517858906 \cdot 10^{149} \lor \neg \left(z \le 3.5058906695120028 \cdot 10^{174}\right):\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(\left(b \cdot \left(\sqrt[3]{z \cdot c - a \cdot i} \cdot \sqrt[3]{z \cdot c - a \cdot i}\right)\right) \cdot \sqrt[3]{z \cdot c - a \cdot i} + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right)\\ \end{array}\]
\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}\;z \le -3.13021924517858906 \cdot 10^{149} \lor \neg \left(z \le 3.5058906695120028 \cdot 10^{174}\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(c \cdot t - i \cdot y, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(\left(b \cdot \left(\sqrt[3]{z \cdot c - a \cdot i} \cdot \sqrt[3]{z \cdot c - a \cdot i}\right)\right) \cdot \sqrt[3]{z \cdot c - a \cdot i} + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\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 r659721 = x;
        double r659722 = y;
        double r659723 = z;
        double r659724 = r659722 * r659723;
        double r659725 = t;
        double r659726 = a;
        double r659727 = r659725 * r659726;
        double r659728 = r659724 - r659727;
        double r659729 = r659721 * r659728;
        double r659730 = b;
        double r659731 = c;
        double r659732 = r659731 * r659723;
        double r659733 = i;
        double r659734 = r659733 * r659726;
        double r659735 = r659732 - r659734;
        double r659736 = r659730 * r659735;
        double r659737 = r659729 - r659736;
        double r659738 = j;
        double r659739 = r659731 * r659725;
        double r659740 = r659733 * r659722;
        double r659741 = r659739 - r659740;
        double r659742 = r659738 * r659741;
        double r659743 = r659737 + r659742;
        return r659743;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r659744 = z;
        double r659745 = -3.130219245178589e+149;
        bool r659746 = r659744 <= r659745;
        double r659747 = 3.505890669512003e+174;
        bool r659748 = r659744 <= r659747;
        double r659749 = !r659748;
        bool r659750 = r659746 || r659749;
        double r659751 = a;
        double r659752 = i;
        double r659753 = b;
        double r659754 = r659752 * r659753;
        double r659755 = c;
        double r659756 = r659753 * r659755;
        double r659757 = x;
        double r659758 = t;
        double r659759 = r659757 * r659758;
        double r659760 = r659751 * r659759;
        double r659761 = fma(r659744, r659756, r659760);
        double r659762 = -r659761;
        double r659763 = fma(r659751, r659754, r659762);
        double r659764 = r659755 * r659758;
        double r659765 = y;
        double r659766 = r659752 * r659765;
        double r659767 = r659764 - r659766;
        double r659768 = j;
        double r659769 = r659751 * r659758;
        double r659770 = -r659769;
        double r659771 = fma(r659765, r659744, r659770);
        double r659772 = r659757 * r659771;
        double r659773 = -r659751;
        double r659774 = fma(r659773, r659758, r659769);
        double r659775 = r659757 * r659774;
        double r659776 = r659772 + r659775;
        double r659777 = r659744 * r659755;
        double r659778 = r659751 * r659752;
        double r659779 = r659777 - r659778;
        double r659780 = cbrt(r659779);
        double r659781 = r659780 * r659780;
        double r659782 = r659753 * r659781;
        double r659783 = r659782 * r659780;
        double r659784 = fma(r659773, r659752, r659778);
        double r659785 = r659753 * r659784;
        double r659786 = r659783 + r659785;
        double r659787 = r659776 - r659786;
        double r659788 = fma(r659767, r659768, r659787);
        double r659789 = r659750 ? r659763 : r659788;
        return r659789;
}

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
Target16.1
Herbie14.3
\[\begin{array}{l} \mathbf{if}\;t \lt -8.1209789191959122 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.7125538182184851 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.63353334603158369 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.130219245178589e+149 or 3.505890669512003e+174 < z

    1. Initial program 23.0

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

    2. Simplified23.0

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

    3. Taylor expanded around inf 34.4

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

    4. Simplified34.4

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

    if -3.130219245178589e+149 < z < 3.505890669512003e+174

    1. Initial program 10.6

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

    2. Simplified10.6

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

    4. Applied prod-diff10.6

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

    5. Applied distribute-lft-in10.6

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

    6. Simplified10.6

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

    8. Applied prod-diff10.6

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

    9. Applied distribute-lft-in10.6

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

    11. Applied add-cube-cbrt10.9

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

    12. Applied associate-*r*10.9

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

  4. Final simplification14.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.13021924517858906 \cdot 10^{149} \lor \neg \left(z \le 3.5058906695120028 \cdot 10^{174}\right):\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(\left(b \cdot \left(\sqrt[3]{z \cdot c - a \cdot i} \cdot \sqrt[3]{z \cdot c - a \cdot i}\right)\right) \cdot \sqrt[3]{z \cdot c - a \cdot i} + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))))