Data.Colour.Matrix:determinant from colour-2.3.3, A

Average Error: 12.4 → 12.7

Time: 38.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r103721098 = x;
        double r103721099 = y;
        double r103721100 = z;
        double r103721101 = r103721099 * r103721100;
        double r103721102 = t;
        double r103721103 = a;
        double r103721104 = r103721102 * r103721103;
        double r103721105 = r103721101 - r103721104;
        double r103721106 = r103721098 * r103721105;
        double r103721107 = b;
        double r103721108 = c;
        double r103721109 = r103721108 * r103721100;
        double r103721110 = i;
        double r103721111 = r103721102 * r103721110;
        double r103721112 = r103721109 - r103721111;
        double r103721113 = r103721107 * r103721112;
        double r103721114 = r103721106 - r103721113;
        double r103721115 = j;
        double r103721116 = r103721108 * r103721103;
        double r103721117 = r103721099 * r103721110;
        double r103721118 = r103721116 - r103721117;
        double r103721119 = r103721115 * r103721118;
        double r103721120 = r103721114 + r103721119;
        return r103721120;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r103721121 = y;
        double r103721122 = z;
        double r103721123 = r103721121 * r103721122;
        double r103721124 = t;
        double r103721125 = a;
        double r103721126 = r103721124 * r103721125;
        double r103721127 = r103721123 - r103721126;
        double r103721128 = x;
        double r103721129 = b;
        double r103721130 = i;
        double r103721131 = r103721124 * r103721130;
        double r103721132 = c;
        double r103721133 = r103721132 * r103721122;
        double r103721134 = r103721131 - r103721133;
        double r103721135 = j;
        double r103721136 = cbrt(r103721135);
        double r103721137 = r103721132 * r103721125;
        double r103721138 = r103721121 * r103721130;
        double r103721139 = r103721137 - r103721138;
        double r103721140 = cbrt(r103721139);
        double r103721141 = r103721136 * r103721140;
        double r103721142 = r103721135 * r103721139;
        double r103721143 = cbrt(r103721142);
        double r103721144 = r103721141 * r103721143;
        double r103721145 = r103721143 * r103721143;
        double r103721146 = r103721145 * r103721143;
        double r103721147 = cbrt(r103721146);
        double r103721148 = r103721144 * r103721147;
        double r103721149 = fma(r103721129, r103721134, r103721148);
        double r103721150 = fma(r103721127, r103721128, r103721149);
        return r103721150;
}

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

Original	12.4
Target	19.9
Herbie	12.7

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

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. Simplified 12.4

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

4. Applied add-cube-cbrt 12.7

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

6. Applied cbrt-prod 12.6

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

8. Applied add-cbrt-cube 12.7

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

9. Final simplification 12.7

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

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(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)))))