Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, E

Average Error: 5.5 → 4.3

Time: 59.1s

Precision: 64

Math

\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -8.760666895464302 \cdot 10^{-244}:\\ \;\;\;\;\left(b \cdot c - \left(\left(i \cdot x\right) \cdot 4.0 + \left(j \cdot k\right) \cdot 27.0\right)\right) + t \cdot \left(\left(x \cdot \left(z \cdot y\right)\right) \cdot 18.0 - 4.0 \cdot a\right)\\ \mathbf{elif}\;t \le 2.4482662989943874 \cdot 10^{+44}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot \left(y \cdot 18.0\right)\right) \cdot \left(t \cdot z\right) - t \cdot \left(4.0 \cdot a\right)\right) + b \cdot c\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c - \left(\left(i \cdot x\right) \cdot 4.0 + \left(j \cdot k\right) \cdot 27.0\right)\right) + t \cdot \left(\left(x \cdot \left(z \cdot y\right)\right) \cdot 18.0 - 4.0 \cdot a\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r34674027 = x;
        double r34674028 = 18.0;
        double r34674029 = r34674027 * r34674028;
        double r34674030 = y;
        double r34674031 = r34674029 * r34674030;
        double r34674032 = z;
        double r34674033 = r34674031 * r34674032;
        double r34674034 = t;
        double r34674035 = r34674033 * r34674034;
        double r34674036 = a;
        double r34674037 = 4.0;
        double r34674038 = r34674036 * r34674037;
        double r34674039 = r34674038 * r34674034;
        double r34674040 = r34674035 - r34674039;
        double r34674041 = b;
        double r34674042 = c;
        double r34674043 = r34674041 * r34674042;
        double r34674044 = r34674040 + r34674043;
        double r34674045 = r34674027 * r34674037;
        double r34674046 = i;
        double r34674047 = r34674045 * r34674046;
        double r34674048 = r34674044 - r34674047;
        double r34674049 = j;
        double r34674050 = 27.0;
        double r34674051 = r34674049 * r34674050;
        double r34674052 = k;
        double r34674053 = r34674051 * r34674052;
        double r34674054 = r34674048 - r34674053;
        return r34674054;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r34674055 = t;
        double r34674056 = -8.760666895464302e-244;
        bool r34674057 = r34674055 <= r34674056;
        double r34674058 = b;
        double r34674059 = c;
        double r34674060 = r34674058 * r34674059;
        double r34674061 = i;
        double r34674062 = x;
        double r34674063 = r34674061 * r34674062;
        double r34674064 = 4.0;
        double r34674065 = r34674063 * r34674064;
        double r34674066 = j;
        double r34674067 = k;
        double r34674068 = r34674066 * r34674067;
        double r34674069 = 27.0;
        double r34674070 = r34674068 * r34674069;
        double r34674071 = r34674065 + r34674070;
        double r34674072 = r34674060 - r34674071;
        double r34674073 = z;
        double r34674074 = y;
        double r34674075 = r34674073 * r34674074;
        double r34674076 = r34674062 * r34674075;
        double r34674077 = 18.0;
        double r34674078 = r34674076 * r34674077;
        double r34674079 = a;
        double r34674080 = r34674064 * r34674079;
        double r34674081 = r34674078 - r34674080;
        double r34674082 = r34674055 * r34674081;
        double r34674083 = r34674072 + r34674082;
        double r34674084 = 2.4482662989943874e+44;
        bool r34674085 = r34674055 <= r34674084;
        double r34674086 = r34674074 * r34674077;
        double r34674087 = r34674062 * r34674086;
        double r34674088 = r34674055 * r34674073;
        double r34674089 = r34674087 * r34674088;
        double r34674090 = r34674055 * r34674080;
        double r34674091 = r34674089 - r34674090;
        double r34674092 = r34674091 + r34674060;
        double r34674093 = r34674064 * r34674062;
        double r34674094 = r34674093 * r34674061;
        double r34674095 = r34674092 - r34674094;
        double r34674096 = r34674066 * r34674069;
        double r34674097 = r34674096 * r34674067;
        double r34674098 = r34674095 - r34674097;
        double r34674099 = r34674085 ? r34674098 : r34674083;
        double r34674100 = r34674057 ? r34674083 : r34674099;
        return r34674100;
}

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

Bits error versus k

Target

Original	5.5
Target	1.6
Herbie	4.3

\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18.0 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) - \left(\left(k \cdot j\right) \cdot 27.0 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.68027943805222:\\ \;\;\;\;\left(\left(18.0 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) + \left(c \cdot b - 27.0 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18.0 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) - \left(\left(k \cdot j\right) \cdot 27.0 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < -8.760666895464302e-244 or 2.4482662989943874e+44 < t

   1. Initial program 4.1

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

   2. Simplified 4.1

      \[\leadsto \color{blue}{\left(b \cdot c - \left(k \cdot \left(j \cdot 27.0\right) + \left(i \cdot x\right) \cdot 4.0\right)\right) + \left(\left(y \cdot x\right) \cdot \left(z \cdot 18.0\right) - a \cdot 4.0\right) \cdot t}\]

   3. Taylor expanded around inf 4.0

      \[\leadsto \color{blue}{\left(b \cdot c - \left(27.0 \cdot \left(j \cdot k\right) + 4.0 \cdot \left(x \cdot i\right)\right)\right)} + \left(\left(y \cdot x\right) \cdot \left(z \cdot 18.0\right) - a \cdot 4.0\right) \cdot t\]

   4. Taylor expanded around inf 4.4

      \[\leadsto \left(b \cdot c - \left(27.0 \cdot \left(j \cdot k\right) + 4.0 \cdot \left(x \cdot i\right)\right)\right) + \left(\color{blue}{18.0 \cdot \left(x \cdot \left(z \cdot y\right)\right)} - a \cdot 4.0\right) \cdot t\]

   if -8.760666895464302e-244 < t < 2.4482662989943874e+44

   1. Initial program 7.4

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
   2. Using strategy rm

   3. Applied associate-*l* 7.4

      \[\leadsto \left(\left(\left(\left(\color{blue}{\left(x \cdot \left(18.0 \cdot y\right)\right)} \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
   4. Using strategy rm

   5. Applied associate-*l* 4.1

      \[\leadsto \left(\left(\left(\color{blue}{\left(x \cdot \left(18.0 \cdot y\right)\right) \cdot \left(z \cdot t\right)} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.760666895464302 \cdot 10^{-244}:\\ \;\;\;\;\left(b \cdot c - \left(\left(i \cdot x\right) \cdot 4.0 + \left(j \cdot k\right) \cdot 27.0\right)\right) + t \cdot \left(\left(x \cdot \left(z \cdot y\right)\right) \cdot 18.0 - 4.0 \cdot a\right)\\ \mathbf{elif}\;t \le 2.4482662989943874 \cdot 10^{+44}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot \left(y \cdot 18.0\right)\right) \cdot \left(t \cdot z\right) - t \cdot \left(4.0 \cdot a\right)\right) + b \cdot c\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c - \left(\left(i \cdot x\right) \cdot 4.0 + \left(j \cdot k\right) \cdot 27.0\right)\right) + t \cdot \left(\left(x \cdot \left(z \cdot y\right)\right) \cdot 18.0 - 4.0 \cdot a\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))