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

Average Error: 3.6 → 2.7

Time: 30.0s

Precision: 64

Math

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \le 1.934650635555646046239049173909307910435 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\frac{-1}{z}}{3}, y, \frac{\frac{1}{z}}{3} \cdot y\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\right) + \mathsf{fma}\left(1, x, y \cdot \frac{\frac{-1}{z}}{3}\right)\\ \mathbf{elif}\;z \cdot 3 \le 2.296793905184516690316228557659555390868 \cdot 10^{221}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{1}{z}}{y} \cdot \frac{t}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-\frac{\frac{y}{z}}{\sqrt[3]{3}}, \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{\frac{y}{z}}{\sqrt[3]{3}} \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\right) + \mathsf{fma}\left(1, x, \frac{-1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\frac{y}{z}}{\sqrt[3]{3}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r492106 = x;
        double r492107 = y;
        double r492108 = z;
        double r492109 = 3.0;
        double r492110 = r492108 * r492109;
        double r492111 = r492107 / r492110;
        double r492112 = r492106 - r492111;
        double r492113 = t;
        double r492114 = r492110 * r492107;
        double r492115 = r492113 / r492114;
        double r492116 = r492112 + r492115;
        return r492116;
}

↓

double f(double x, double y, double z, double t) {
        double r492117 = z;
        double r492118 = 3.0;
        double r492119 = r492117 * r492118;
        double r492120 = 1.934650635555646e-06;
        bool r492121 = r492119 <= r492120;
        double r492122 = -1.0;
        double r492123 = r492122 / r492117;
        double r492124 = r492123 / r492118;
        double r492125 = y;
        double r492126 = 1.0;
        double r492127 = r492126 / r492117;
        double r492128 = r492127 / r492118;
        double r492129 = r492128 * r492125;
        double r492130 = fma(r492124, r492125, r492129);
        double r492131 = t;
        double r492132 = r492131 / r492118;
        double r492133 = r492125 / r492132;
        double r492134 = r492127 / r492133;
        double r492135 = r492130 + r492134;
        double r492136 = x;
        double r492137 = r492125 * r492124;
        double r492138 = fma(r492126, r492136, r492137);
        double r492139 = r492135 + r492138;
        double r492140 = 2.2967939051845167e+221;
        bool r492141 = r492119 <= r492140;
        double r492142 = r492125 / r492119;
        double r492143 = r492136 - r492142;
        double r492144 = r492127 / r492125;
        double r492145 = r492144 * r492132;
        double r492146 = r492143 + r492145;
        double r492147 = r492125 / r492117;
        double r492148 = cbrt(r492118);
        double r492149 = r492147 / r492148;
        double r492150 = -r492149;
        double r492151 = r492148 * r492148;
        double r492152 = r492126 / r492151;
        double r492153 = r492149 * r492152;
        double r492154 = fma(r492150, r492152, r492153);
        double r492155 = r492154 + r492134;
        double r492156 = r492122 / r492151;
        double r492157 = r492156 * r492149;
        double r492158 = fma(r492126, r492136, r492157);
        double r492159 = r492155 + r492158;
        double r492160 = r492141 ? r492146 : r492159;
        double r492161 = r492121 ? r492139 : r492160;
        return r492161;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	3.6
Target	1.7
Herbie	2.7

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\]

Derivation

1. Split input into 3 regimes

2. if (* z 3.0) < 1.934650635555646e-06

   1. Initial program 5.2

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
   2. Using strategy rm

   3. Applied associate-/r* 2.2

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 2.2

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{\color{blue}{1 \cdot t}}{z \cdot 3}}{y}\]

   6. Applied times-frac 2.2

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{\frac{1}{z} \cdot \frac{t}{3}}}{y}\]

   7. Applied associate-/l* 2.9

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}}\]
   8. Using strategy rm

   9. Applied associate-/r* 2.9

      \[\leadsto \left(x - \color{blue}{\frac{\frac{y}{z}}{3}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]
   10. Using strategy rm

   11. Applied *-un-lft-identity 2.9

       \[\leadsto \left(x - \frac{\frac{y}{z}}{\color{blue}{1 \cdot 3}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]

   12. Applied div-inv 2.9

       \[\leadsto \left(x - \frac{\color{blue}{y \cdot \frac{1}{z}}}{1 \cdot 3}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]

   13. Applied times-frac 2.9

       \[\leadsto \left(x - \color{blue}{\frac{y}{1} \cdot \frac{\frac{1}{z}}{3}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]

   14. Applied *-un-lft-identity 2.9

       \[\leadsto \left(\color{blue}{1 \cdot x} - \frac{y}{1} \cdot \frac{\frac{1}{z}}{3}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]

   15. Applied prod-diff 2.9

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(1, x, -\frac{\frac{1}{z}}{3} \cdot \frac{y}{1}\right) + \mathsf{fma}\left(-\frac{\frac{1}{z}}{3}, \frac{y}{1}, \frac{\frac{1}{z}}{3} \cdot \frac{y}{1}\right)\right)} + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]

   16. Applied associate-+l+ 2.9

       \[\leadsto \color{blue}{\mathsf{fma}\left(1, x, -\frac{\frac{1}{z}}{3} \cdot \frac{y}{1}\right) + \left(\mathsf{fma}\left(-\frac{\frac{1}{z}}{3}, \frac{y}{1}, \frac{\frac{1}{z}}{3} \cdot \frac{y}{1}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\right)}\]

   17. Simplified 2.9

       \[\leadsto \mathsf{fma}\left(1, x, -\frac{\frac{1}{z}}{3} \cdot \frac{y}{1}\right) + \color{blue}{\left(\mathsf{fma}\left(-\frac{\frac{1}{z}}{3}, y, \frac{\frac{1}{z}}{3} \cdot y\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\right)}\]

   if 1.934650635555646e-06 < (* z 3.0) < 2.2967939051845167e+221

   1. Initial program 0.2

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
   2. Using strategy rm

   3. Applied associate-/r* 0.9

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 0.9

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{\color{blue}{1 \cdot t}}{z \cdot 3}}{y}\]

   6. Applied times-frac 0.9

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{\frac{1}{z} \cdot \frac{t}{3}}}{y}\]

   7. Applied associate-/l* 4.3

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}}\]
   8. Using strategy rm

   9. Applied associate-/r/ 0.3

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{1}{z}}{y} \cdot \frac{t}{3}}\]

   if 2.2967939051845167e+221 < (* z 3.0)

   1. Initial program 0.8

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
   2. Using strategy rm

   3. Applied associate-/r* 1.1

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 1.1

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{\color{blue}{1 \cdot t}}{z \cdot 3}}{y}\]

   6. Applied times-frac 1.0

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{\frac{1}{z} \cdot \frac{t}{3}}}{y}\]

   7. Applied associate-/l* 7.1

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}}\]
   8. Using strategy rm

   9. Applied associate-/r* 7.2

      \[\leadsto \left(x - \color{blue}{\frac{\frac{y}{z}}{3}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 7.2

       \[\leadsto \left(x - \frac{\frac{y}{z}}{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]

   12. Applied *-un-lft-identity 7.2

       \[\leadsto \left(x - \frac{\frac{y}{\color{blue}{1 \cdot z}}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]

   13. Applied *-un-lft-identity 7.2

       \[\leadsto \left(x - \frac{\frac{\color{blue}{1 \cdot y}}{1 \cdot z}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]

   14. Applied times-frac 7.2

       \[\leadsto \left(x - \frac{\color{blue}{\frac{1}{1} \cdot \frac{y}{z}}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]

   15. Applied times-frac 7.2

       \[\leadsto \left(x - \color{blue}{\frac{\frac{1}{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\frac{y}{z}}{\sqrt[3]{3}}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]

   16. Applied *-un-lft-identity 7.2

       \[\leadsto \left(\color{blue}{1 \cdot x} - \frac{\frac{1}{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\frac{y}{z}}{\sqrt[3]{3}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]

   17. Applied prod-diff 7.2

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(1, x, -\frac{\frac{y}{z}}{\sqrt[3]{3}} \cdot \frac{\frac{1}{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right) + \mathsf{fma}\left(-\frac{\frac{y}{z}}{\sqrt[3]{3}}, \frac{\frac{1}{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{\frac{y}{z}}{\sqrt[3]{3}} \cdot \frac{\frac{1}{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right)} + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\]

   18. Applied associate-+l+ 7.2

       \[\leadsto \color{blue}{\mathsf{fma}\left(1, x, -\frac{\frac{y}{z}}{\sqrt[3]{3}} \cdot \frac{\frac{1}{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right) + \left(\mathsf{fma}\left(-\frac{\frac{y}{z}}{\sqrt[3]{3}}, \frac{\frac{1}{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{\frac{y}{z}}{\sqrt[3]{3}} \cdot \frac{\frac{1}{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\right)}\]

   19. Simplified 7.2

       \[\leadsto \mathsf{fma}\left(1, x, -\frac{\frac{y}{z}}{\sqrt[3]{3}} \cdot \frac{\frac{1}{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right) + \color{blue}{\left(\mathsf{fma}\left(-\frac{\frac{y}{z}}{\sqrt[3]{3}}, \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{\frac{y}{z}}{\sqrt[3]{3}} \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \le 1.934650635555646046239049173909307910435 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\frac{-1}{z}}{3}, y, \frac{\frac{1}{z}}{3} \cdot y\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\right) + \mathsf{fma}\left(1, x, y \cdot \frac{\frac{-1}{z}}{3}\right)\\ \mathbf{elif}\;z \cdot 3 \le 2.296793905184516690316228557659555390868 \cdot 10^{221}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{1}{z}}{y} \cdot \frac{t}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-\frac{\frac{y}{z}}{\sqrt[3]{3}}, \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{\frac{y}{z}}{\sqrt[3]{3}} \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\right) + \mathsf{fma}\left(1, x, \frac{-1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\frac{y}{z}}{\sqrt[3]{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3))) (/ (/ t (* z 3)) y))

  (+ (- x (/ y (* z 3))) (/ t (* (* z 3) y))))