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

Average Error: 20.7 → 8.4

Time: 24.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.049750915246272776707763168040883621167 \cdot 10^{-94}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 9418504667764352876544:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{\mathsf{fma}\left(x \cdot y, 9, b\right)}} - \left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 1.262932583339488317194724427101820494475 \cdot 10^{299}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c}{\frac{\mathsf{fma}\left(x \cdot y, 9, b\right)}{z} - a \cdot \left(4 \cdot t\right)}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r434693 = x;
        double r434694 = 9.0;
        double r434695 = r434693 * r434694;
        double r434696 = y;
        double r434697 = r434695 * r434696;
        double r434698 = z;
        double r434699 = 4.0;
        double r434700 = r434698 * r434699;
        double r434701 = t;
        double r434702 = r434700 * r434701;
        double r434703 = a;
        double r434704 = r434702 * r434703;
        double r434705 = r434697 - r434704;
        double r434706 = b;
        double r434707 = r434705 + r434706;
        double r434708 = c;
        double r434709 = r434698 * r434708;
        double r434710 = r434707 / r434709;
        return r434710;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r434711 = x;
        double r434712 = 9.0;
        double r434713 = r434711 * r434712;
        double r434714 = y;
        double r434715 = r434713 * r434714;
        double r434716 = z;
        double r434717 = 4.0;
        double r434718 = r434716 * r434717;
        double r434719 = t;
        double r434720 = r434718 * r434719;
        double r434721 = a;
        double r434722 = r434720 * r434721;
        double r434723 = r434715 - r434722;
        double r434724 = b;
        double r434725 = r434723 + r434724;
        double r434726 = c;
        double r434727 = r434716 * r434726;
        double r434728 = r434725 / r434727;
        double r434729 = -inf.0;
        bool r434730 = r434728 <= r434729;
        double r434731 = -4.0;
        double r434732 = r434719 * r434721;
        double r434733 = r434732 / r434726;
        double r434734 = r434731 * r434733;
        double r434735 = -2.0497509152462728e-94;
        bool r434736 = r434728 <= r434735;
        double r434737 = 9.418504667764353e+21;
        bool r434738 = r434728 <= r434737;
        double r434739 = 1.0;
        double r434740 = r434711 * r434714;
        double r434741 = fma(r434740, r434712, r434724);
        double r434742 = r434716 / r434741;
        double r434743 = r434739 / r434742;
        double r434744 = r434721 * r434717;
        double r434745 = r434744 * r434719;
        double r434746 = r434743 - r434745;
        double r434747 = r434746 / r434726;
        double r434748 = 1.2629325833394883e+299;
        bool r434749 = r434728 <= r434748;
        double r434750 = r434741 / r434716;
        double r434751 = r434717 * r434719;
        double r434752 = r434721 * r434751;
        double r434753 = r434750 - r434752;
        double r434754 = r434726 / r434753;
        double r434755 = r434739 / r434754;
        double r434756 = r434749 ? r434728 : r434755;
        double r434757 = r434738 ? r434747 : r434756;
        double r434758 = r434736 ? r434728 : r434757;
        double r434759 = r434730 ? r434734 : r434758;
        return r434759;
}

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	20.7
Target	14.7
Herbie	8.4

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

Derivation

1. Split input into 4 regimes

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

   1. Initial program 64.0

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

   2. Simplified 24.8

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

   4. Applied *-un-lft-identity 24.8

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

   5. Applied *-un-lft-identity 24.8

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

   6. Applied times-frac 24.8

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

   7. Simplified 24.8

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

   8. Simplified 24.8

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

   10. Applied clear-num 24.9

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

   11. Taylor expanded around inf 33.1

       \[\leadsto \color{blue}{-4 \cdot \frac{t \cdot a}{c}}\]

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -2.0497509152462728e-94 or 9.418504667764353e+21 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 1.2629325833394883e+299

   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 -2.0497509152462728e-94 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 9.418504667764353e+21

   1. Initial program 15.2

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

   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 *-un-lft-identity 1.2

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

   5. Applied *-un-lft-identity 1.2

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

   6. Applied times-frac 1.2

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

   7. Simplified 1.2

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

   8. Simplified 1.2

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

   10. Applied clear-num 1.3

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

   if 1.2629325833394883e+299 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 61.7

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

   2. Simplified 28.3

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

   4. Applied *-un-lft-identity 28.3

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

   5. Applied *-un-lft-identity 28.3

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

   6. Applied times-frac 28.3

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

   7. Simplified 28.3

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

   8. Simplified 28.1

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

   10. Applied clear-num 28.1

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

   12. Applied clear-num 28.2

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

   13. Simplified 28.2

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

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} = -\infty:\\ \;\;\;\;-4 \cdot \frac{t \cdot a}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.049750915246272776707763168040883621167 \cdot 10^{-94}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 9418504667764352876544:\\ \;\;\;\;\frac{\frac{1}{\frac{z}{\mathsf{fma}\left(x \cdot y, 9, b\right)}} - \left(a \cdot 4\right) \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 1.262932583339488317194724427101820494475 \cdot 10^{299}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c}{\frac{\mathsf{fma}\left(x \cdot y, 9, b\right)}{z} - a \cdot \left(4 \cdot t\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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)) -1.100156740804105e-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)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* 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)))