Average Error: 12.3 → 12.3
Time: 31.9s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.66438809751155875077165123717638653921 \cdot 10^{-165}:\\ \;\;\;\;\left(c \cdot a - y \cdot i\right) \cdot j + \left(\left(\sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x} \cdot \sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x}\right) \cdot \sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x} - b \cdot \left(c \cdot z - i \cdot t\right)\right)\\ \mathbf{elif}\;b \le 1.032827567662806685902054411305148385494 \cdot 10^{-306}:\\ \;\;\;\;\left(y \cdot z - t \cdot a\right) \cdot x + \left(c \cdot a - y \cdot i\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot a - y \cdot i\right) \cdot j + \left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(c \cdot z - i \cdot t\right) \cdot \sqrt{b}\right) \cdot \sqrt{b}\right)\\ \end{array}\]
\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)
↓
\begin{array}{l}
\mathbf{if}\;b \le -8.66438809751155875077165123717638653921 \cdot 10^{-165}:\\
\;\;\;\;\left(c \cdot a - y \cdot i\right) \cdot j + \left(\left(\sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x} \cdot \sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x}\right) \cdot \sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x} - b \cdot \left(c \cdot z - i \cdot t\right)\right)\\

\mathbf{elif}\;b \le 1.032827567662806685902054411305148385494 \cdot 10^{-306}:\\
\;\;\;\;\left(y \cdot z - t \cdot a\right) \cdot x + \left(c \cdot a - y \cdot i\right) \cdot j\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r37915927 = x;
        double r37915928 = y;
        double r37915929 = z;
        double r37915930 = r37915928 * r37915929;
        double r37915931 = t;
        double r37915932 = a;
        double r37915933 = r37915931 * r37915932;
        double r37915934 = r37915930 - r37915933;
        double r37915935 = r37915927 * r37915934;
        double r37915936 = b;
        double r37915937 = c;
        double r37915938 = r37915937 * r37915929;
        double r37915939 = i;
        double r37915940 = r37915931 * r37915939;
        double r37915941 = r37915938 - r37915940;
        double r37915942 = r37915936 * r37915941;
        double r37915943 = r37915935 - r37915942;
        double r37915944 = j;
        double r37915945 = r37915937 * r37915932;
        double r37915946 = r37915928 * r37915939;
        double r37915947 = r37915945 - r37915946;
        double r37915948 = r37915944 * r37915947;
        double r37915949 = r37915943 + r37915948;
        return r37915949;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r37915950 = b;
        double r37915951 = -8.664388097511559e-165;
        bool r37915952 = r37915950 <= r37915951;
        double r37915953 = c;
        double r37915954 = a;
        double r37915955 = r37915953 * r37915954;
        double r37915956 = y;
        double r37915957 = i;
        double r37915958 = r37915956 * r37915957;
        double r37915959 = r37915955 - r37915958;
        double r37915960 = j;
        double r37915961 = r37915959 * r37915960;
        double r37915962 = z;
        double r37915963 = r37915956 * r37915962;
        double r37915964 = t;
        double r37915965 = r37915964 * r37915954;
        double r37915966 = r37915963 - r37915965;
        double r37915967 = x;
        double r37915968 = r37915966 * r37915967;
        double r37915969 = cbrt(r37915968);
        double r37915970 = r37915969 * r37915969;
        double r37915971 = r37915970 * r37915969;
        double r37915972 = r37915953 * r37915962;
        double r37915973 = r37915957 * r37915964;
        double r37915974 = r37915972 - r37915973;
        double r37915975 = r37915950 * r37915974;
        double r37915976 = r37915971 - r37915975;
        double r37915977 = r37915961 + r37915976;
        double r37915978 = 1.0328275676628067e-306;
        bool r37915979 = r37915950 <= r37915978;
        double r37915980 = r37915968 + r37915961;
        double r37915981 = sqrt(r37915950);
        double r37915982 = r37915974 * r37915981;
        double r37915983 = r37915982 * r37915981;
        double r37915984 = r37915968 - r37915983;
        double r37915985 = r37915961 + r37915984;
        double r37915986 = r37915979 ? r37915980 : r37915985;
        double r37915987 = r37915952 ? r37915977 : r37915986;
        return r37915987;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.3
Target19.2
Herbie12.3
\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.664388097511559e-165

    1. Initial program 10.4

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

    3. Applied add-cube-cbrt10.6

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

    if -8.664388097511559e-165 < b < 1.0328275676628067e-306

    1. Initial program 18.2

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

    2. Taylor expanded around 0 17.7

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

    if 1.0328275676628067e-306 < b

    1. Initial program 11.9

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

    3. Applied add-sqr-sqrt12.0

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

    4. Applied associate-*l*12.0

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

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.66438809751155875077165123717638653921 \cdot 10^{-165}:\\ \;\;\;\;\left(c \cdot a - y \cdot i\right) \cdot j + \left(\left(\sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x} \cdot \sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x}\right) \cdot \sqrt[3]{\left(y \cdot z - t \cdot a\right) \cdot x} - b \cdot \left(c \cdot z - i \cdot t\right)\right)\\ \mathbf{elif}\;b \le 1.032827567662806685902054411305148385494 \cdot 10^{-306}:\\ \;\;\;\;\left(y \cdot z - t \cdot a\right) \cdot x + \left(c \cdot a - y \cdot i\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot a - y \cdot i\right) \cdot j + \left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(c \cdot z - i \cdot t\right) \cdot \sqrt{b}\right) \cdot \sqrt{b}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 3.2113527362226803e-147) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))