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

Average Error: 19.6 → 3.1

Time: 22.1s

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 r24100400 = x;
        double r24100401 = 9.0;
        double r24100402 = r24100400 * r24100401;
        double r24100403 = y;
        double r24100404 = r24100402 * r24100403;
        double r24100405 = z;
        double r24100406 = 4.0;
        double r24100407 = r24100405 * r24100406;
        double r24100408 = t;
        double r24100409 = r24100407 * r24100408;
        double r24100410 = a;
        double r24100411 = r24100409 * r24100410;
        double r24100412 = r24100404 - r24100411;
        double r24100413 = b;
        double r24100414 = r24100412 + r24100413;
        double r24100415 = c;
        double r24100416 = r24100405 * r24100415;
        double r24100417 = r24100414 / r24100416;
        return r24100417;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r24100418 = x;
        double r24100419 = 9.0;
        double r24100420 = r24100418 * r24100419;
        double r24100421 = y;
        double r24100422 = r24100420 * r24100421;
        double r24100423 = z;
        double r24100424 = 4.0;
        double r24100425 = r24100423 * r24100424;
        double r24100426 = t;
        double r24100427 = r24100425 * r24100426;
        double r24100428 = a;
        double r24100429 = r24100427 * r24100428;
        double r24100430 = r24100422 - r24100429;
        double r24100431 = b;
        double r24100432 = r24100430 + r24100431;
        double r24100433 = c;
        double r24100434 = r24100433 * r24100423;
        double r24100435 = r24100432 / r24100434;
        double r24100436 = -inf.0;
        bool r24100437 = r24100435 <= r24100436;
        double r24100438 = r24100431 / r24100434;
        double r24100439 = r24100434 / r24100421;
        double r24100440 = r24100418 / r24100439;
        double r24100441 = r24100440 * r24100419;
        double r24100442 = r24100438 + r24100441;
        double r24100443 = r24100428 / r24100433;
        double r24100444 = r24100426 * r24100443;
        double r24100445 = r24100424 * r24100444;
        double r24100446 = r24100442 - r24100445;
        double r24100447 = -1.5906234503561492e-156;
        bool r24100448 = r24100435 <= r24100447;
        double r24100449 = 5.0596648813598755e+87;
        bool r24100450 = r24100435 <= r24100449;
        double r24100451 = r24100421 * r24100418;
        double r24100452 = r24100451 / r24100423;
        double r24100453 = r24100431 / r24100423;
        double r24100454 = fma(r24100419, r24100452, r24100453);
        double r24100455 = r24100426 * r24100428;
        double r24100456 = r24100424 * r24100455;
        double r24100457 = r24100454 - r24100456;
        double r24100458 = r24100457 / r24100433;
        double r24100459 = 7.263326730960966e+307;
        bool r24100460 = r24100435 <= r24100459;
        double r24100461 = cbrt(r24100421);
        double r24100462 = r24100433 / r24100461;
        double r24100463 = r24100461 * r24100461;
        double r24100464 = r24100423 / r24100463;
        double r24100465 = r24100462 * r24100464;
        double r24100466 = r24100418 / r24100465;
        double r24100467 = r24100419 * r24100466;
        double r24100468 = r24100438 + r24100467;
        double r24100469 = r24100433 / r24100428;
        double r24100470 = r24100426 / r24100469;
        double r24100471 = r24100470 * r24100424;
        double r24100472 = r24100468 - r24100471;
        double r24100473 = r24100460 ? r24100435 : r24100472;
        double r24100474 = r24100450 ? r24100458 : r24100473;
        double r24100475 = r24100448 ? r24100435 : r24100474;
        double r24100476 = r24100437 ? r24100446 : r24100475;
        return r24100476;
}

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)))