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

Average Error: 19.6 → 3.1

Time: 21.5s

Precision: 64

Math

\[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9.0\right) - 4.0 \cdot \left(t \cdot \frac{a}{c}\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -1.5906234503561492 \cdot 10^{-156}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 5.0596648813598755 \cdot 10^{+87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9.0, \frac{y \cdot x}{z}, \frac{b}{z}\right) - 4.0 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 7.263326730960966 \cdot 10^{+307}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + 9.0 \cdot \frac{x}{\frac{c}{\sqrt[3]{y}} \cdot \frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) - \frac{t}{\frac{c}{a}} \cdot 4.0\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r21095733 = x;
        double r21095734 = 9.0;
        double r21095735 = r21095733 * r21095734;
        double r21095736 = y;
        double r21095737 = r21095735 * r21095736;
        double r21095738 = z;
        double r21095739 = 4.0;
        double r21095740 = r21095738 * r21095739;
        double r21095741 = t;
        double r21095742 = r21095740 * r21095741;
        double r21095743 = a;
        double r21095744 = r21095742 * r21095743;
        double r21095745 = r21095737 - r21095744;
        double r21095746 = b;
        double r21095747 = r21095745 + r21095746;
        double r21095748 = c;
        double r21095749 = r21095738 * r21095748;
        double r21095750 = r21095747 / r21095749;
        return r21095750;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r21095751 = x;
        double r21095752 = 9.0;
        double r21095753 = r21095751 * r21095752;
        double r21095754 = y;
        double r21095755 = r21095753 * r21095754;
        double r21095756 = z;
        double r21095757 = 4.0;
        double r21095758 = r21095756 * r21095757;
        double r21095759 = t;
        double r21095760 = r21095758 * r21095759;
        double r21095761 = a;
        double r21095762 = r21095760 * r21095761;
        double r21095763 = r21095755 - r21095762;
        double r21095764 = b;
        double r21095765 = r21095763 + r21095764;
        double r21095766 = c;
        double r21095767 = r21095766 * r21095756;
        double r21095768 = r21095765 / r21095767;
        double r21095769 = -inf.0;
        bool r21095770 = r21095768 <= r21095769;
        double r21095771 = r21095764 / r21095767;
        double r21095772 = r21095767 / r21095754;
        double r21095773 = r21095751 / r21095772;
        double r21095774 = r21095773 * r21095752;
        double r21095775 = r21095771 + r21095774;
        double r21095776 = r21095761 / r21095766;
        double r21095777 = r21095759 * r21095776;
        double r21095778 = r21095757 * r21095777;
        double r21095779 = r21095775 - r21095778;
        double r21095780 = -1.5906234503561492e-156;
        bool r21095781 = r21095768 <= r21095780;
        double r21095782 = 5.0596648813598755e+87;
        bool r21095783 = r21095768 <= r21095782;
        double r21095784 = r21095754 * r21095751;
        double r21095785 = r21095784 / r21095756;
        double r21095786 = r21095764 / r21095756;
        double r21095787 = fma(r21095752, r21095785, r21095786);
        double r21095788 = r21095759 * r21095761;
        double r21095789 = r21095757 * r21095788;
        double r21095790 = r21095787 - r21095789;
        double r21095791 = r21095790 / r21095766;
        double r21095792 = 7.263326730960966e+307;
        bool r21095793 = r21095768 <= r21095792;
        double r21095794 = cbrt(r21095754);
        double r21095795 = r21095766 / r21095794;
        double r21095796 = r21095794 * r21095794;
        double r21095797 = r21095756 / r21095796;
        double r21095798 = r21095795 * r21095797;
        double r21095799 = r21095751 / r21095798;
        double r21095800 = r21095752 * r21095799;
        double r21095801 = r21095771 + r21095800;
        double r21095802 = r21095766 / r21095761;
        double r21095803 = r21095759 / r21095802;
        double r21095804 = r21095803 * r21095757;
        double r21095805 = r21095801 - r21095804;
        double r21095806 = r21095793 ? r21095768 : r21095805;
        double r21095807 = r21095783 ? r21095791 : r21095806;
        double r21095808 = r21095781 ? r21095768 : r21095807;
        double r21095809 = r21095770 ? r21095779 : r21095808;
        return r21095809;
}

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

Target

Original	19.6
Target	13.7
Herbie	3.1

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804105 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(z \cdot 4.0\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(z \cdot 4.0\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9.0 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4.0 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(z \cdot 4.0\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9.0 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4.0 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0

   1. Initial program 60.2

      \[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 21.9

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot 9.0, x, b\right)}{z} - \left(a \cdot t\right) \cdot 4.0}{c}}\]

   3. Taylor expanded around 0 26.2

      \[\leadsto \color{blue}{\left(9.0 \cdot \frac{x \cdot y}{z \cdot c} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{t \cdot a}{c}}\]
   4. Using strategy rm

   5. Applied associate-/l* 13.0

      \[\leadsto \left(9.0 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{t \cdot a}{c}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 13.0

      \[\leadsto \left(9.0 \cdot \frac{x}{\frac{z \cdot c}{y}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{t \cdot a}{\color{blue}{1 \cdot c}}\]

   8. Applied times-frac 6.0

      \[\leadsto \left(9.0 \cdot \frac{x}{\frac{z \cdot c}{y}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \color{blue}{\left(\frac{t}{1} \cdot \frac{a}{c}\right)}\]

   9. Simplified 6.0

      \[\leadsto \left(9.0 \cdot \frac{x}{\frac{z \cdot c}{y}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \left(\color{blue}{t} \cdot \frac{a}{c}\right)\]

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -1.5906234503561492e-156 or 5.0596648813598755e+87 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 7.263326730960966e+307

   1. Initial program 0.6

      \[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   if -1.5906234503561492e-156 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 5.0596648813598755e+87

   1. Initial program 13.7

      \[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 2.2

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot 9.0, x, b\right)}{z} - \left(a \cdot t\right) \cdot 4.0}{c}}\]

   3. Taylor expanded around 0 2.1

      \[\leadsto \frac{\color{blue}{\left(9.0 \cdot \frac{x \cdot y}{z} + \frac{b}{z}\right)} - \left(a \cdot t\right) \cdot 4.0}{c}\]

   4. Simplified 2.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(9.0, \frac{x \cdot y}{z}, \frac{b}{z}\right)} - \left(a \cdot t\right) \cdot 4.0}{c}\]

   if 7.263326730960966e+307 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 61.5

      \[\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 26.7

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y \cdot 9.0, x, b\right)}{z} - \left(a \cdot t\right) \cdot 4.0}{c}}\]

   3. Taylor expanded around 0 30.4

      \[\leadsto \color{blue}{\left(9.0 \cdot \frac{x \cdot y}{z \cdot c} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{t \cdot a}{c}}\]
   4. Using strategy rm

   5. Applied associate-/l* 21.8

      \[\leadsto \left(9.0 \cdot \color{blue}{\frac{x}{\frac{z \cdot c}{y}}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{t \cdot a}{c}\]
   6. Using strategy rm

   7. Applied associate-/l* 14.9

      \[\leadsto \left(9.0 \cdot \frac{x}{\frac{z \cdot c}{y}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \color{blue}{\frac{t}{\frac{c}{a}}}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 15.0

      \[\leadsto \left(9.0 \cdot \frac{x}{\frac{z \cdot c}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{t}{\frac{c}{a}}\]

   10. Applied times-frac 10.0

       \[\leadsto \left(9.0 \cdot \frac{x}{\color{blue}{\frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{c}{\sqrt[3]{y}}}} + \frac{b}{z \cdot c}\right) - 4.0 \cdot \frac{t}{\frac{c}{a}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9.0\right) - 4.0 \cdot \left(t \cdot \frac{a}{c}\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -1.5906234503561492 \cdot 10^{-156}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 5.0596648813598755 \cdot 10^{+87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9.0, \frac{y \cdot x}{z}, \frac{b}{z}\right) - 4.0 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 7.263326730960966 \cdot 10^{+307}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + 9.0 \cdot \frac{x}{\frac{c}{\sqrt[3]{y}} \cdot \frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) - \frac{t}{\frac{c}{a}} \cdot 4.0\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"

  :herbie-target
  (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -1.100156740804105e-171) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9.0 (/ y c)) (/ x z)) (/ b (* c z))) (* 4.0 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (- (+ (* 9.0 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4.0 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))