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

Average Error: 20.7 → 8.4

Time: 25.1s

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 r413290 = x;
        double r413291 = 9.0;
        double r413292 = r413290 * r413291;
        double r413293 = y;
        double r413294 = r413292 * r413293;
        double r413295 = z;
        double r413296 = 4.0;
        double r413297 = r413295 * r413296;
        double r413298 = t;
        double r413299 = r413297 * r413298;
        double r413300 = a;
        double r413301 = r413299 * r413300;
        double r413302 = r413294 - r413301;
        double r413303 = b;
        double r413304 = r413302 + r413303;
        double r413305 = c;
        double r413306 = r413295 * r413305;
        double r413307 = r413304 / r413306;
        return r413307;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r413308 = x;
        double r413309 = 9.0;
        double r413310 = r413308 * r413309;
        double r413311 = y;
        double r413312 = r413310 * r413311;
        double r413313 = z;
        double r413314 = 4.0;
        double r413315 = r413313 * r413314;
        double r413316 = t;
        double r413317 = r413315 * r413316;
        double r413318 = a;
        double r413319 = r413317 * r413318;
        double r413320 = r413312 - r413319;
        double r413321 = b;
        double r413322 = r413320 + r413321;
        double r413323 = c;
        double r413324 = r413313 * r413323;
        double r413325 = r413322 / r413324;
        double r413326 = -inf.0;
        bool r413327 = r413325 <= r413326;
        double r413328 = -4.0;
        double r413329 = r413316 * r413318;
        double r413330 = r413329 / r413323;
        double r413331 = r413328 * r413330;
        double r413332 = -2.0497509152462728e-94;
        bool r413333 = r413325 <= r413332;
        double r413334 = 9.418504667764353e+21;
        bool r413335 = r413325 <= r413334;
        double r413336 = 1.0;
        double r413337 = r413308 * r413311;
        double r413338 = fma(r413337, r413309, r413321);
        double r413339 = r413313 / r413338;
        double r413340 = r413336 / r413339;
        double r413341 = r413318 * r413314;
        double r413342 = r413341 * r413316;
        double r413343 = r413340 - r413342;
        double r413344 = r413343 / r413323;
        double r413345 = 1.2629325833394883e+299;
        bool r413346 = r413325 <= r413345;
        double r413347 = r413338 / r413313;
        double r413348 = r413314 * r413316;
        double r413349 = r413318 * r413348;
        double r413350 = r413347 - r413349;
        double r413351 = r413323 / r413350;
        double r413352 = r413336 / r413351;
        double r413353 = r413346 ? r413325 : r413352;
        double r413354 = r413335 ? r413344 : r413353;
        double r413355 = r413333 ? r413325 : r413354;
        double r413356 = r413327 ? r413331 : r413355;
        return r413356;
}

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)))