Average Error: 12.3 → 12.2
Time: 34.1s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.35838280615037666349340971153519687763 \cdot 10^{-188}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(t \cdot c - i \cdot y\right) \cdot j\\ \mathbf{elif}\;b \le 9.95588474272539046952932324509302559692 \cdot 10^{-263}:\\ \;\;\;\;\left(t \cdot c - i \cdot y\right) \cdot j + \left(z \cdot y - t \cdot a\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot y - t \cdot a\right) \cdot x - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(z \cdot c - i \cdot a\right)\right)\right) + \left(t \cdot c - i \cdot y\right) \cdot j\\ \end{array}\]
\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}\;b \le -3.35838280615037666349340971153519687763 \cdot 10^{-188}:\\
\;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(t \cdot c - i \cdot y\right) \cdot j\\

\mathbf{elif}\;b \le 9.95588474272539046952932324509302559692 \cdot 10^{-263}:\\
\;\;\;\;\left(t \cdot c - i \cdot y\right) \cdot j + \left(z \cdot y - t \cdot a\right) \cdot x\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r11595024 = x;
        double r11595025 = y;
        double r11595026 = z;
        double r11595027 = r11595025 * r11595026;
        double r11595028 = t;
        double r11595029 = a;
        double r11595030 = r11595028 * r11595029;
        double r11595031 = r11595027 - r11595030;
        double r11595032 = r11595024 * r11595031;
        double r11595033 = b;
        double r11595034 = c;
        double r11595035 = r11595034 * r11595026;
        double r11595036 = i;
        double r11595037 = r11595036 * r11595029;
        double r11595038 = r11595035 - r11595037;
        double r11595039 = r11595033 * r11595038;
        double r11595040 = r11595032 - r11595039;
        double r11595041 = j;
        double r11595042 = r11595034 * r11595028;
        double r11595043 = r11595036 * r11595025;
        double r11595044 = r11595042 - r11595043;
        double r11595045 = r11595041 * r11595044;
        double r11595046 = r11595040 + r11595045;
        return r11595046;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r11595047 = b;
        double r11595048 = -3.3583828061503767e-188;
        bool r11595049 = r11595047 <= r11595048;
        double r11595050 = x;
        double r11595051 = z;
        double r11595052 = y;
        double r11595053 = r11595051 * r11595052;
        double r11595054 = r11595050 * r11595053;
        double r11595055 = a;
        double r11595056 = t;
        double r11595057 = r11595050 * r11595056;
        double r11595058 = r11595055 * r11595057;
        double r11595059 = r11595054 - r11595058;
        double r11595060 = c;
        double r11595061 = r11595051 * r11595060;
        double r11595062 = i;
        double r11595063 = r11595062 * r11595055;
        double r11595064 = r11595061 - r11595063;
        double r11595065 = r11595047 * r11595064;
        double r11595066 = r11595059 - r11595065;
        double r11595067 = r11595056 * r11595060;
        double r11595068 = r11595062 * r11595052;
        double r11595069 = r11595067 - r11595068;
        double r11595070 = j;
        double r11595071 = r11595069 * r11595070;
        double r11595072 = r11595066 + r11595071;
        double r11595073 = 9.95588474272539e-263;
        bool r11595074 = r11595047 <= r11595073;
        double r11595075 = r11595056 * r11595055;
        double r11595076 = r11595053 - r11595075;
        double r11595077 = r11595076 * r11595050;
        double r11595078 = r11595071 + r11595077;
        double r11595079 = sqrt(r11595047);
        double r11595080 = r11595079 * r11595064;
        double r11595081 = r11595079 * r11595080;
        double r11595082 = r11595077 - r11595081;
        double r11595083 = r11595082 + r11595071;
        double r11595084 = r11595074 ? r11595078 : r11595083;
        double r11595085 = r11595049 ? r11595072 : r11595084;
        return r11595085;
}

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
Target16.1
Herbie12.2
\[\begin{array}{l} \mathbf{if}\;t \lt -8.12097891919591218149793027759825150959 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.712553818218485141757938537793350881052 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.633533346031583686060259351057142920433 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.053588855745548710002760210539645467715 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.3583828061503767e-188

    1. Initial program 11.1

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

    2. Taylor expanded around inf 11.6

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

    if -3.3583828061503767e-188 < b < 9.95588474272539e-263

    1. Initial program 17.4

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

    2. Taylor expanded around 0 15.9

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

    if 9.95588474272539e-263 < b

    1. Initial program 11.2

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

    3. Applied add-sqr-sqrt11.3

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

    4. Applied associate-*l*11.3

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

  4. Final simplification12.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.35838280615037666349340971153519687763 \cdot 10^{-188}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - b \cdot \left(z \cdot c - i \cdot a\right)\right) + \left(t \cdot c - i \cdot y\right) \cdot j\\ \mathbf{elif}\;b \le 9.95588474272539046952932324509302559692 \cdot 10^{-263}:\\ \;\;\;\;\left(t \cdot c - i \cdot y\right) \cdot j + \left(z \cdot y - t \cdot a\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot y - t \cdot a\right) \cdot x - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(z \cdot c - i \cdot a\right)\right)\right) + \left(t \cdot c - i \cdot y\right) \cdot j\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"

  :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) 2.0) (pow (* i y) 2.0))) (+ (* 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) 2.0) (pow (* i y) 2.0))) (+ (* 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)))))