Average Error: 12.3 → 11.8
Time: 8.4s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -2.936819268186205650838255402135178279408 \cdot 10^{-268} \lor \neg \left(a \le 1.284975181144422199319550481563476668578 \cdot 10^{121}\right):\\ \;\;\;\;\left(\left(\left(\sqrt[3]{\left(x \cdot y\right) \cdot z} \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z}\right) \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z} + {\left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)}^{1}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]
\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}\;a \le -2.936819268186205650838255402135178279408 \cdot 10^{-268} \lor \neg \left(a \le 1.284975181144422199319550481563476668578 \cdot 10^{121}\right):\\
\;\;\;\;\left(\left(\left(\sqrt[3]{\left(x \cdot y\right) \cdot z} \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z}\right) \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z} + {\left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)}^{1}\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(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\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)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r788159 = x;
        double r788160 = y;
        double r788161 = z;
        double r788162 = r788160 * r788161;
        double r788163 = t;
        double r788164 = a;
        double r788165 = r788163 * r788164;
        double r788166 = r788162 - r788165;
        double r788167 = r788159 * r788166;
        double r788168 = b;
        double r788169 = c;
        double r788170 = r788169 * r788161;
        double r788171 = i;
        double r788172 = r788163 * r788171;
        double r788173 = r788170 - r788172;
        double r788174 = r788168 * r788173;
        double r788175 = r788167 - r788174;
        double r788176 = j;
        double r788177 = r788169 * r788164;
        double r788178 = r788160 * r788171;
        double r788179 = r788177 - r788178;
        double r788180 = r788176 * r788179;
        double r788181 = r788175 + r788180;
        return r788181;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r788182 = a;
        double r788183 = -2.9368192681862057e-268;
        bool r788184 = r788182 <= r788183;
        double r788185 = 1.2849751811444222e+121;
        bool r788186 = r788182 <= r788185;
        double r788187 = !r788186;
        bool r788188 = r788184 || r788187;
        double r788189 = x;
        double r788190 = y;
        double r788191 = r788189 * r788190;
        double r788192 = z;
        double r788193 = r788191 * r788192;
        double r788194 = cbrt(r788193);
        double r788195 = r788194 * r788194;
        double r788196 = r788195 * r788194;
        double r788197 = -1.0;
        double r788198 = t;
        double r788199 = r788189 * r788198;
        double r788200 = r788182 * r788199;
        double r788201 = r788197 * r788200;
        double r788202 = 1.0;
        double r788203 = pow(r788201, r788202);
        double r788204 = r788196 + r788203;
        double r788205 = b;
        double r788206 = c;
        double r788207 = r788206 * r788192;
        double r788208 = i;
        double r788209 = r788198 * r788208;
        double r788210 = r788207 - r788209;
        double r788211 = r788205 * r788210;
        double r788212 = r788204 - r788211;
        double r788213 = j;
        double r788214 = r788206 * r788182;
        double r788215 = r788190 * r788208;
        double r788216 = r788214 - r788215;
        double r788217 = r788213 * r788216;
        double r788218 = r788212 + r788217;
        double r788219 = cbrt(r788189);
        double r788220 = r788219 * r788219;
        double r788221 = r788190 * r788192;
        double r788222 = r788219 * r788221;
        double r788223 = r788220 * r788222;
        double r788224 = r788198 * r788182;
        double r788225 = -r788224;
        double r788226 = r788189 * r788225;
        double r788227 = r788223 + r788226;
        double r788228 = r788227 - r788211;
        double r788229 = r788228 + r788217;
        double r788230 = r788188 ? r788218 : r788229;
        return r788230;
}

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
Target20.3
Herbie11.8
\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -2.9368192681862057e-268 or 1.2849751811444222e+121 < a

    1. Initial program 14.0

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

    3. Applied sub-neg14.0

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

    4. Applied distribute-lft-in14.0

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

    6. Applied associate-*r*14.1

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

    8. Applied pow114.1

      \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z + x \cdot \color{blue}{{\left(-t \cdot a\right)}^{1}}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]

    9. Applied pow114.1

      \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z + \color{blue}{{x}^{1}} \cdot {\left(-t \cdot a\right)}^{1}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]

    10. Applied pow-prod-down14.1

      \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z + \color{blue}{{\left(x \cdot \left(-t \cdot a\right)\right)}^{1}}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]

    11. Simplified12.9

      \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z + {\color{blue}{\left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)}}^{1}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
    12. Using strategy rm

    13. Applied add-cube-cbrt13.0

      \[\leadsto \left(\left(\color{blue}{\left(\sqrt[3]{\left(x \cdot y\right) \cdot z} \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z}\right) \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z}} + {\left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)}^{1}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]

    if -2.9368192681862057e-268 < a < 1.2849751811444222e+121

    1. Initial program 10.1

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

    3. Applied sub-neg10.1

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

    4. Applied distribute-lft-in10.1

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

    6. Applied add-cube-cbrt10.3

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

    7. Applied associate-*l*10.3

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

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.936819268186205650838255402135178279408 \cdot 10^{-268} \lor \neg \left(a \le 1.284975181144422199319550481563476668578 \cdot 10^{121}\right):\\ \;\;\;\;\left(\left(\left(\sqrt[3]{\left(x \cdot y\right) \cdot z} \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z}\right) \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z} + {\left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)}^{1}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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 -1.469694296777705e-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)))))