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

Average Error: 12.2 → 11.2

Time: 26.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.502651937255943249008686150206970880095 \cdot 10^{49}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot i\right) \cdot b\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \le 5.444011925251250727376291305043069110217 \cdot 10^{-18}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + \left(a \cdot \left(j \cdot c\right) + j \cdot \left(-y \cdot i\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r502042 = x;
        double r502043 = y;
        double r502044 = z;
        double r502045 = r502043 * r502044;
        double r502046 = t;
        double r502047 = a;
        double r502048 = r502046 * r502047;
        double r502049 = r502045 - r502048;
        double r502050 = r502042 * r502049;
        double r502051 = b;
        double r502052 = c;
        double r502053 = r502052 * r502044;
        double r502054 = i;
        double r502055 = r502046 * r502054;
        double r502056 = r502053 - r502055;
        double r502057 = r502051 * r502056;
        double r502058 = r502050 - r502057;
        double r502059 = j;
        double r502060 = r502052 * r502047;
        double r502061 = r502043 * r502054;
        double r502062 = r502060 - r502061;
        double r502063 = r502059 * r502062;
        double r502064 = r502058 + r502063;
        return r502064;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r502065 = x;
        double r502066 = -1.5026519372559432e+49;
        bool r502067 = r502065 <= r502066;
        double r502068 = y;
        double r502069 = z;
        double r502070 = r502068 * r502069;
        double r502071 = t;
        double r502072 = a;
        double r502073 = r502071 * r502072;
        double r502074 = r502070 - r502073;
        double r502075 = r502065 * r502074;
        double r502076 = b;
        double r502077 = c;
        double r502078 = r502076 * r502077;
        double r502079 = r502069 * r502078;
        double r502080 = i;
        double r502081 = r502071 * r502080;
        double r502082 = -r502081;
        double r502083 = r502082 * r502076;
        double r502084 = r502079 + r502083;
        double r502085 = r502075 - r502084;
        double r502086 = j;
        double r502087 = r502077 * r502072;
        double r502088 = r502068 * r502080;
        double r502089 = r502087 - r502088;
        double r502090 = r502086 * r502089;
        double r502091 = r502085 + r502090;
        double r502092 = 5.444011925251251e-18;
        bool r502093 = r502065 <= r502092;
        double r502094 = r502065 * r502070;
        double r502095 = r502065 * r502071;
        double r502096 = r502072 * r502095;
        double r502097 = -r502096;
        double r502098 = r502094 + r502097;
        double r502099 = r502077 * r502069;
        double r502100 = r502099 - r502081;
        double r502101 = r502076 * r502100;
        double r502102 = r502098 - r502101;
        double r502103 = r502102 + r502090;
        double r502104 = cbrt(r502101);
        double r502105 = r502104 * r502104;
        double r502106 = r502105 * r502104;
        double r502107 = r502075 - r502106;
        double r502108 = r502086 * r502077;
        double r502109 = r502072 * r502108;
        double r502110 = -r502088;
        double r502111 = r502086 * r502110;
        double r502112 = r502109 + r502111;
        double r502113 = r502107 + r502112;
        double r502114 = r502093 ? r502103 : r502113;
        double r502115 = r502067 ? r502091 : r502114;
        return r502115;
}

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	12.2
Target	19.1
Herbie	11.2

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

Derivation

1. Split input into 3 regimes

2. if x < -1.5026519372559432e+49

   1. Initial program 6.9

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

   3. Applied sub-neg 6.9

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

   4. Applied distribute-lft-in 6.9

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

   5. Simplified 8.2

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

   6. Simplified 8.2

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

   if -1.5026519372559432e+49 < x < 5.444011925251251e-18

   1. Initial program 15.0

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

   3. Applied sub-neg 15.0

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

   4. Applied distribute-lft-in 15.0

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

   5. Simplified 12.8

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

   if 5.444011925251251e-18 < x

   1. Initial program 7.4

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

   3. Applied add-cube-cbrt 7.6

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

   5. Applied sub-neg 7.6

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

   6. Applied distribute-lft-in 7.6

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

   7. Simplified 8.8

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

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.502651937255943249008686150206970880095 \cdot 10^{49}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot i\right) \cdot b\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \le 5.444011925251250727376291305043069110217 \cdot 10^{-18}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + \left(a \cdot \left(j \cdot c\right) + j \cdot \left(-y \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))))