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

Average Error: 20.3 → 5.4

Time: 19.4s

Precision: 64

Math

\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{z}}{\frac{c}{y}}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.880322514237006678222058407382437321799 \cdot 10^{-109}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 6.857809735758019969223840291451408280898 \cdot 10^{50}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{\mathsf{fma}\left(y, x \cdot 9, b\right)}} - \left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 6.524854797087028066381557610408788493249 \cdot 10^{304}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{z}}{\frac{c}{y}}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r569707 = x;
        double r569708 = 9.0;
        double r569709 = r569707 * r569708;
        double r569710 = y;
        double r569711 = r569709 * r569710;
        double r569712 = z;
        double r569713 = 4.0;
        double r569714 = r569712 * r569713;
        double r569715 = t;
        double r569716 = r569714 * r569715;
        double r569717 = a;
        double r569718 = r569716 * r569717;
        double r569719 = r569711 - r569718;
        double r569720 = b;
        double r569721 = r569719 + r569720;
        double r569722 = c;
        double r569723 = r569712 * r569722;
        double r569724 = r569721 / r569723;
        return r569724;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r569725 = x;
        double r569726 = 9.0;
        double r569727 = r569725 * r569726;
        double r569728 = y;
        double r569729 = r569727 * r569728;
        double r569730 = z;
        double r569731 = 4.0;
        double r569732 = r569730 * r569731;
        double r569733 = t;
        double r569734 = r569732 * r569733;
        double r569735 = a;
        double r569736 = r569734 * r569735;
        double r569737 = r569729 - r569736;
        double r569738 = b;
        double r569739 = r569737 + r569738;
        double r569740 = c;
        double r569741 = r569730 * r569740;
        double r569742 = r569739 / r569741;
        double r569743 = -inf.0;
        bool r569744 = r569742 <= r569743;
        double r569745 = r569725 / r569730;
        double r569746 = r569740 / r569728;
        double r569747 = r569745 / r569746;
        double r569748 = r569738 / r569741;
        double r569749 = fma(r569747, r569726, r569748);
        double r569750 = r569735 * r569731;
        double r569751 = r569750 * r569733;
        double r569752 = r569751 / r569740;
        double r569753 = r569749 - r569752;
        double r569754 = -2.8803225142370067e-109;
        bool r569755 = r569742 <= r569754;
        double r569756 = 6.85780973575802e+50;
        bool r569757 = r569742 <= r569756;
        double r569758 = 1.0;
        double r569759 = fma(r569728, r569727, r569738);
        double r569760 = r569730 / r569759;
        double r569761 = r569758 / r569760;
        double r569762 = r569761 - r569751;
        double r569763 = r569762 / r569740;
        double r569764 = 6.524854797087028e+304;
        bool r569765 = r569742 <= r569764;
        double r569766 = r569765 ? r569742 : r569753;
        double r569767 = r569757 ? r569763 : r569766;
        double r569768 = r569755 ? r569742 : r569767;
        double r569769 = r569744 ? r569753 : r569768;
        return r569769;
}

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	20.3
Target	14.1
Herbie	5.4

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.100156740804104887233830094663413900721 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.170887791174748819600820354912645756062 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.876823679546137226963937101710277849382 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.383851504245631860711731716196098366993 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -inf.0 or 6.524854797087028e+304 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 63.5

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 26.5

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t}{c}}\]

   3. Taylor expanded around 0 30.3

      \[\leadsto \color{blue}{\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{t \cdot a}{c}}\]

   4. Simplified 30.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot y}{z \cdot c}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}}\]
   5. Using strategy rm

   6. Applied associate-/l* 20.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\frac{z \cdot c}{y}}}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 20.2

      \[\leadsto \mathsf{fma}\left(\frac{x}{\frac{z \cdot c}{\color{blue}{1 \cdot y}}}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\]

   9. Applied times-frac 18.4

      \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\frac{z}{1} \cdot \frac{c}{y}}}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\]

   10. Applied associate-/r* 18.7

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{\frac{z}{1}}}{\frac{c}{y}}}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\]

   11. Simplified 18.7

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{z}}}{\frac{c}{y}}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\]

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -2.8803225142370067e-109 or 6.85780973575802e+50 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 6.524854797087028e+304

   1. Initial program 0.6

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   if -2.8803225142370067e-109 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 6.85780973575802e+50

   1. Initial program 14.2

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\]

   2. Simplified 1.2

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(y, x \cdot 9, b\right)}{z} - \left(a \cdot 4\right) \cdot t}{c}}\]
   3. Using strategy rm

   4. Applied clear-num 1.3

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{z}{\mathsf{fma}\left(y, x \cdot 9, b\right)}}} - \left(a \cdot 4\right) \cdot t}{c}\]
3. Recombined 3 regimes into one program.

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{z}}{\frac{c}{y}}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.880322514237006678222058407382437321799 \cdot 10^{-109}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 6.857809735758019969223840291451408280898 \cdot 10^{50}:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{\mathsf{fma}\left(y, x \cdot 9, b\right)}} - \left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 6.524854797087028066381557610408788493249 \cdot 10^{304}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x}{z}}{\frac{c}{y}}, 9, \frac{b}{z \cdot c}\right) - \frac{\left(a \cdot 4\right) \cdot t}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :herbie-target
  (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -1.1001567408041049e-171) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.17088779117474882e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.8768236795461372e130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e158) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (- (+ (* 9 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)))