Average Error: 12.4 → 12.7
Time: 12.7s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\mathsf{fma}\left(c \cdot a - y \cdot i, j, \left(\left(x \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)}\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)} + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)\]
\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)
↓
\mathsf{fma}\left(c \cdot a - y \cdot i, j, \left(\left(x \cdot \left(\sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)}\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)} + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r967076 = x;
        double r967077 = y;
        double r967078 = z;
        double r967079 = r967077 * r967078;
        double r967080 = t;
        double r967081 = a;
        double r967082 = r967080 * r967081;
        double r967083 = r967079 - r967082;
        double r967084 = r967076 * r967083;
        double r967085 = b;
        double r967086 = c;
        double r967087 = r967086 * r967078;
        double r967088 = i;
        double r967089 = r967080 * r967088;
        double r967090 = r967087 - r967089;
        double r967091 = r967085 * r967090;
        double r967092 = r967084 - r967091;
        double r967093 = j;
        double r967094 = r967086 * r967081;
        double r967095 = r967077 * r967088;
        double r967096 = r967094 - r967095;
        double r967097 = r967093 * r967096;
        double r967098 = r967092 + r967097;
        return r967098;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r967099 = c;
        double r967100 = a;
        double r967101 = r967099 * r967100;
        double r967102 = y;
        double r967103 = i;
        double r967104 = r967102 * r967103;
        double r967105 = r967101 - r967104;
        double r967106 = j;
        double r967107 = x;
        double r967108 = z;
        double r967109 = t;
        double r967110 = r967100 * r967109;
        double r967111 = -r967110;
        double r967112 = fma(r967102, r967108, r967111);
        double r967113 = cbrt(r967112);
        double r967114 = r967113 * r967113;
        double r967115 = r967107 * r967114;
        double r967116 = r967115 * r967113;
        double r967117 = -r967100;
        double r967118 = fma(r967117, r967109, r967110);
        double r967119 = r967107 * r967118;
        double r967120 = r967116 + r967119;
        double r967121 = b;
        double r967122 = r967099 * r967108;
        double r967123 = r967109 * r967103;
        double r967124 = r967122 - r967123;
        double r967125 = r967121 * r967124;
        double r967126 = r967120 - r967125;
        double r967127 = fma(r967105, r967106, r967126);
        return r967127;
}

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
Target20.4
Herbie12.7
\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Initial program 12.4

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

  2. Simplified12.4

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

  4. Applied prod-diff12.4

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

  5. Applied distribute-lft-in12.4

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

  7. Applied add-cube-cbrt12.7

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

  8. Applied associate-*r*12.7

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

  9. Final simplification12.7

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

Reproduce

herbie shell --seed 2019353 +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)))))