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

Average Error: 5.5 → 3.9

Time: 16.0s

Precision: 64

Math

\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.995746752057705666616031777662888381911 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{elif}\;z \le 31.71288006302372508571352227590978145599:\\ \;\;\;\;t \cdot \left({\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r791088 = x;
        double r791089 = 18.0;
        double r791090 = r791088 * r791089;
        double r791091 = y;
        double r791092 = r791090 * r791091;
        double r791093 = z;
        double r791094 = r791092 * r791093;
        double r791095 = t;
        double r791096 = r791094 * r791095;
        double r791097 = a;
        double r791098 = 4.0;
        double r791099 = r791097 * r791098;
        double r791100 = r791099 * r791095;
        double r791101 = r791096 - r791100;
        double r791102 = b;
        double r791103 = c;
        double r791104 = r791102 * r791103;
        double r791105 = r791101 + r791104;
        double r791106 = r791088 * r791098;
        double r791107 = i;
        double r791108 = r791106 * r791107;
        double r791109 = r791105 - r791108;
        double r791110 = j;
        double r791111 = 27.0;
        double r791112 = r791110 * r791111;
        double r791113 = k;
        double r791114 = r791112 * r791113;
        double r791115 = r791109 - r791114;
        return r791115;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r791116 = z;
        double r791117 = -4.995746752057706e-77;
        bool r791118 = r791116 <= r791117;
        double r791119 = t;
        double r791120 = x;
        double r791121 = 18.0;
        double r791122 = y;
        double r791123 = r791121 * r791122;
        double r791124 = r791120 * r791123;
        double r791125 = r791124 * r791116;
        double r791126 = a;
        double r791127 = 4.0;
        double r791128 = r791126 * r791127;
        double r791129 = r791125 - r791128;
        double r791130 = b;
        double r791131 = c;
        double r791132 = r791130 * r791131;
        double r791133 = i;
        double r791134 = r791127 * r791133;
        double r791135 = j;
        double r791136 = 27.0;
        double r791137 = k;
        double r791138 = r791136 * r791137;
        double r791139 = r791135 * r791138;
        double r791140 = fma(r791120, r791134, r791139);
        double r791141 = r791132 - r791140;
        double r791142 = fma(r791119, r791129, r791141);
        double r791143 = 31.712880063023725;
        bool r791144 = r791116 <= r791143;
        double r791145 = r791116 * r791122;
        double r791146 = r791120 * r791145;
        double r791147 = r791121 * r791146;
        double r791148 = 1.0;
        double r791149 = pow(r791147, r791148);
        double r791150 = r791149 - r791128;
        double r791151 = r791119 * r791150;
        double r791152 = r791151 + r791141;
        double r791153 = r791120 * r791121;
        double r791154 = r791153 * r791122;
        double r791155 = r791154 * r791116;
        double r791156 = r791155 - r791128;
        double r791157 = r791135 * r791136;
        double r791158 = cbrt(r791137);
        double r791159 = r791158 * r791158;
        double r791160 = r791157 * r791159;
        double r791161 = r791160 * r791158;
        double r791162 = fma(r791120, r791134, r791161);
        double r791163 = r791132 - r791162;
        double r791164 = fma(r791119, r791156, r791163);
        double r791165 = r791144 ? r791152 : r791164;
        double r791166 = r791118 ? r791142 : r791165;
        return r791166;
}

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	3.9

\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -4.995746752057706e-77

   1. Initial program 5.7

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   2. Simplified 5.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]
   3. Using strategy rm

   4. Applied associate-*l* 5.8

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \color{blue}{j \cdot \left(27 \cdot k\right)}\right)\right)\]
   5. Using strategy rm

   6. Applied associate-*l* 5.8

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(x \cdot \left(18 \cdot y\right)\right)} \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   if -4.995746752057706e-77 < z < 31.712880063023725

   1. Initial program 4.6

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   2. Simplified 4.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]
   3. Using strategy rm

   4. Applied associate-*l* 4.6

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \color{blue}{j \cdot \left(27 \cdot k\right)}\right)\right)\]
   5. Using strategy rm

   6. Applied fma-udef 4.6

      \[\leadsto \color{blue}{t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)}\]
   7. Using strategy rm

   8. Applied pow1 4.6

      \[\leadsto t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \color{blue}{{z}^{1}} - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   9. Applied pow1 4.6

      \[\leadsto t \cdot \left(\left(\left(x \cdot 18\right) \cdot \color{blue}{{y}^{1}}\right) \cdot {z}^{1} - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   10. Applied pow1 4.6

       \[\leadsto t \cdot \left(\left(\left(x \cdot \color{blue}{{18}^{1}}\right) \cdot {y}^{1}\right) \cdot {z}^{1} - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   11. Applied pow1 4.6

       \[\leadsto t \cdot \left(\left(\left(\color{blue}{{x}^{1}} \cdot {18}^{1}\right) \cdot {y}^{1}\right) \cdot {z}^{1} - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   12. Applied pow-prod-down 4.6

       \[\leadsto t \cdot \left(\left(\color{blue}{{\left(x \cdot 18\right)}^{1}} \cdot {y}^{1}\right) \cdot {z}^{1} - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   13. Applied pow-prod-down 4.6

       \[\leadsto t \cdot \left(\color{blue}{{\left(\left(x \cdot 18\right) \cdot y\right)}^{1}} \cdot {z}^{1} - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   14. Applied pow-prod-down 4.6

       \[\leadsto t \cdot \left(\color{blue}{{\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right)}^{1}} - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   15. Simplified 1.2

       \[\leadsto t \cdot \left({\color{blue}{\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}}^{1} - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\]

   if 31.712880063023725 < z

   1. Initial program 7.0

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   2. Simplified 7.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 7.3

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot \color{blue}{\left(\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right) \cdot \sqrt[3]{k}\right)}\right)\right)\]

   5. Applied associate-*r* 7.3

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \color{blue}{\left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}}\right)\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 3.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.995746752057705666616031777662888381911 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{elif}\;z \le 31.71288006302372508571352227590978145599:\\ \;\;\;\;t \cdot \left({\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))