Average Error: 12.0 → 9.1
Time: 23.3s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.252223170063660426996319611027382605263 \cdot 10^{-58} \lor \neg \left(b \le 4.140273813512067091775729699038869223041 \cdot 10^{-36}\right):\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]
\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 -4.252223170063660426996319611027382605263 \cdot 10^{-58} \lor \neg \left(b \le 4.140273813512067091775729699038869223041 \cdot 10^{-36}\right):\\
\;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r726065 = x;
        double r726066 = y;
        double r726067 = z;
        double r726068 = r726066 * r726067;
        double r726069 = t;
        double r726070 = a;
        double r726071 = r726069 * r726070;
        double r726072 = r726068 - r726071;
        double r726073 = r726065 * r726072;
        double r726074 = b;
        double r726075 = c;
        double r726076 = r726075 * r726067;
        double r726077 = i;
        double r726078 = r726069 * r726077;
        double r726079 = r726076 - r726078;
        double r726080 = r726074 * r726079;
        double r726081 = r726073 - r726080;
        double r726082 = j;
        double r726083 = r726075 * r726070;
        double r726084 = r726066 * r726077;
        double r726085 = r726083 - r726084;
        double r726086 = r726082 * r726085;
        double r726087 = r726081 + r726086;
        return r726087;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r726088 = b;
        double r726089 = -4.2522231700636604e-58;
        bool r726090 = r726088 <= r726089;
        double r726091 = 4.140273813512067e-36;
        bool r726092 = r726088 <= r726091;
        double r726093 = !r726092;
        bool r726094 = r726090 || r726093;
        double r726095 = y;
        double r726096 = z;
        double r726097 = r726095 * r726096;
        double r726098 = x;
        double r726099 = r726097 * r726098;
        double r726100 = a;
        double r726101 = t;
        double r726102 = r726098 * r726101;
        double r726103 = r726100 * r726102;
        double r726104 = -r726103;
        double r726105 = r726099 + r726104;
        double r726106 = c;
        double r726107 = r726106 * r726096;
        double r726108 = i;
        double r726109 = r726101 * r726108;
        double r726110 = r726107 - r726109;
        double r726111 = r726088 * r726110;
        double r726112 = r726105 - r726111;
        double r726113 = j;
        double r726114 = r726106 * r726100;
        double r726115 = r726095 * r726108;
        double r726116 = r726114 - r726115;
        double r726117 = r726113 * r726116;
        double r726118 = r726112 + r726117;
        double r726119 = r726101 * r726100;
        double r726120 = r726097 - r726119;
        double r726121 = r726098 * r726120;
        double r726122 = r726088 * r726106;
        double r726123 = r726096 * r726122;
        double r726124 = r726108 * r726088;
        double r726125 = r726101 * r726124;
        double r726126 = -r726125;
        double r726127 = r726123 + r726126;
        double r726128 = r726121 - r726127;
        double r726129 = r726128 + r726117;
        double r726130 = r726094 ? r726118 : r726129;
        return r726130;
}

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

Original12.0
Target19.0
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -4.2522231700636604e-58 or 4.140273813512067e-36 < b

    1. Initial program 7.7

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

    3. Applied sub-neg7.7

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

    4. Applied distribute-lft-in7.7

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

    5. Simplified7.7

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

    6. Simplified8.1

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

    if -4.2522231700636604e-58 < b < 4.140273813512067e-36

    1. Initial program 15.8

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

    3. Applied sub-neg15.8

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

    4. Applied distribute-lft-in15.8

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

    5. Simplified13.1

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

    6. Simplified10.1

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.252223170063660426996319611027382605263 \cdot 10^{-58} \lor \neg \left(b \le 4.140273813512067091775729699038869223041 \cdot 10^{-36}\right):\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (if (< x -1.46969429677770502e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* 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) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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