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

Average Error: 20.0 → 3.3

Time: 22.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9.0\right) - \frac{1}{\frac{\frac{c}{a}}{t}} \cdot 4.0\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -6.191094076037901 \cdot 10^{-156}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 1.0167058707604506 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{c}{\frac{\mathsf{fma}\left(y \cdot 9.0, x, b\right)}{z} - \left(t \cdot a\right) \cdot 4.0}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 3.996505146895602 \cdot 10^{+300}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{z}{\frac{y}{c}}} \cdot 9.0\right) - 4.0 \cdot \frac{t}{\frac{c}{a}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r33722901 = x;
        double r33722902 = 9.0;
        double r33722903 = r33722901 * r33722902;
        double r33722904 = y;
        double r33722905 = r33722903 * r33722904;
        double r33722906 = z;
        double r33722907 = 4.0;
        double r33722908 = r33722906 * r33722907;
        double r33722909 = t;
        double r33722910 = r33722908 * r33722909;
        double r33722911 = a;
        double r33722912 = r33722910 * r33722911;
        double r33722913 = r33722905 - r33722912;
        double r33722914 = b;
        double r33722915 = r33722913 + r33722914;
        double r33722916 = c;
        double r33722917 = r33722906 * r33722916;
        double r33722918 = r33722915 / r33722917;
        return r33722918;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r33722919 = x;
        double r33722920 = 9.0;
        double r33722921 = r33722919 * r33722920;
        double r33722922 = y;
        double r33722923 = r33722921 * r33722922;
        double r33722924 = z;
        double r33722925 = 4.0;
        double r33722926 = r33722924 * r33722925;
        double r33722927 = t;
        double r33722928 = r33722926 * r33722927;
        double r33722929 = a;
        double r33722930 = r33722928 * r33722929;
        double r33722931 = r33722923 - r33722930;
        double r33722932 = b;
        double r33722933 = r33722931 + r33722932;
        double r33722934 = c;
        double r33722935 = r33722934 * r33722924;
        double r33722936 = r33722933 / r33722935;
        double r33722937 = -inf.0;
        bool r33722938 = r33722936 <= r33722937;
        double r33722939 = r33722932 / r33722935;
        double r33722940 = r33722935 / r33722922;
        double r33722941 = r33722919 / r33722940;
        double r33722942 = r33722941 * r33722920;
        double r33722943 = r33722939 + r33722942;
        double r33722944 = 1.0;
        double r33722945 = r33722934 / r33722929;
        double r33722946 = r33722945 / r33722927;
        double r33722947 = r33722944 / r33722946;
        double r33722948 = r33722947 * r33722925;
        double r33722949 = r33722943 - r33722948;
        double r33722950 = -6.191094076037901e-156;
        bool r33722951 = r33722936 <= r33722950;
        double r33722952 = 1.0167058707604506e-195;
        bool r33722953 = r33722936 <= r33722952;
        double r33722954 = r33722922 * r33722920;
        double r33722955 = fma(r33722954, r33722919, r33722932);
        double r33722956 = r33722955 / r33722924;
        double r33722957 = r33722927 * r33722929;
        double r33722958 = r33722957 * r33722925;
        double r33722959 = r33722956 - r33722958;
        double r33722960 = r33722934 / r33722959;
        double r33722961 = r33722944 / r33722960;
        double r33722962 = 3.996505146895602e+300;
        bool r33722963 = r33722936 <= r33722962;
        double r33722964 = r33722922 / r33722934;
        double r33722965 = r33722924 / r33722964;
        double r33722966 = r33722919 / r33722965;
        double r33722967 = r33722966 * r33722920;
        double r33722968 = r33722939 + r33722967;
        double r33722969 = r33722927 / r33722945;
        double r33722970 = r33722925 * r33722969;
        double r33722971 = r33722968 - r33722970;
        double r33722972 = r33722963 ? r33722936 : r33722971;
        double r33722973 = r33722953 ? r33722961 : r33722972;
        double r33722974 = r33722951 ? r33722936 : r33722973;
        double r33722975 = r33722938 ? r33722949 : r33722974;
        return r33722975;
}

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	20.0
Target	14.0
Herbie	3.3

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

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 21.3

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

   3. Taylor expanded around 0 26.6

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

   5. Applied associate-/l* 22.0

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

   7. Applied associate-/l* 9.2

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

   9. Applied clear-num 9.2

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

   if -inf.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -6.191094076037901e-156 or 1.0167058707604506e-195 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 3.996505146895602e+300

   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 -6.191094076037901e-156 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 1.0167058707604506e-195

   1. Initial program 28.3

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

   2. Simplified 0.8

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

   4. Applied clear-num 1.6

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

   if 3.996505146895602e+300 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   1. Initial program 60.3

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

   2. Simplified 25.4

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

   3. Taylor expanded around 0 28.8

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

   5. Applied associate-/l* 24.2

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

   7. Applied associate-/l* 15.4

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

   9. Applied associate-/l* 10.9

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

4. Final simplification 3.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} = -\infty:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{c \cdot z}{y}} \cdot 9.0\right) - \frac{1}{\frac{\frac{c}{a}}{t}} \cdot 4.0\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le -6.191094076037901 \cdot 10^{-156}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 1.0167058707604506 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{c}{\frac{\mathsf{fma}\left(y \cdot 9.0, x, b\right)}{z} - \left(t \cdot a\right) \cdot 4.0}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z} \le 3.996505146895602 \cdot 10^{+300}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9.0\right) \cdot y - \left(\left(z \cdot 4.0\right) \cdot t\right) \cdot a\right) + b}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + \frac{x}{\frac{z}{\frac{y}{c}}} \cdot 9.0\right) - 4.0 \cdot \frac{t}{\frac{c}{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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)))