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

Average Error: 20.6 → 9.6

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -2.32197868210335224 \cdot 10^{70}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) - 4 \cdot \left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{t}{\sqrt[3]{c}}\right)\\ \mathbf{elif}\;c \le 6.25600452679530668 \cdot 10^{-263}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \frac{9 \cdot \left(x \cdot y\right)}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;c \le 6.236497692806451 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;c \le 2.59094135237992291 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\left(z \cdot c\right) \cdot \frac{1}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;c \le 2.0127568981310699 \cdot 10^{165}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) - 4 \cdot \left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{t}{\sqrt[3]{c}}\right)\\ \mathbf{elif}\;c \le 2.6783770738007185 \cdot 10^{199}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z \cdot c} \cdot y\right)\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;c \le 1.151269571816795 \cdot 10^{231}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) - 4 \cdot \left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{t}{\sqrt[3]{c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \frac{9 \cdot \left(x \cdot y\right)}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r822471 = x;
        double r822472 = 9.0;
        double r822473 = r822471 * r822472;
        double r822474 = y;
        double r822475 = r822473 * r822474;
        double r822476 = z;
        double r822477 = 4.0;
        double r822478 = r822476 * r822477;
        double r822479 = t;
        double r822480 = r822478 * r822479;
        double r822481 = a;
        double r822482 = r822480 * r822481;
        double r822483 = r822475 - r822482;
        double r822484 = b;
        double r822485 = r822483 + r822484;
        double r822486 = c;
        double r822487 = r822476 * r822486;
        double r822488 = r822485 / r822487;
        return r822488;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r822489 = c;
        double r822490 = -2.3219786821033522e+70;
        bool r822491 = r822489 <= r822490;
        double r822492 = b;
        double r822493 = z;
        double r822494 = r822493 * r822489;
        double r822495 = r822492 / r822494;
        double r822496 = 9.0;
        double r822497 = x;
        double r822498 = r822497 / r822493;
        double r822499 = y;
        double r822500 = r822499 / r822489;
        double r822501 = r822498 * r822500;
        double r822502 = r822496 * r822501;
        double r822503 = r822495 + r822502;
        double r822504 = 4.0;
        double r822505 = a;
        double r822506 = cbrt(r822489);
        double r822507 = r822506 * r822506;
        double r822508 = r822505 / r822507;
        double r822509 = t;
        double r822510 = r822509 / r822506;
        double r822511 = r822508 * r822510;
        double r822512 = r822504 * r822511;
        double r822513 = r822503 - r822512;
        double r822514 = 6.256004526795307e-263;
        bool r822515 = r822489 <= r822514;
        double r822516 = r822497 * r822499;
        double r822517 = r822496 * r822516;
        double r822518 = r822517 / r822494;
        double r822519 = r822495 + r822518;
        double r822520 = r822505 * r822509;
        double r822521 = r822520 / r822489;
        double r822522 = r822504 * r822521;
        double r822523 = r822519 - r822522;
        double r822524 = 6.236497692806451e-151;
        bool r822525 = r822489 <= r822524;
        double r822526 = r822497 * r822496;
        double r822527 = r822526 * r822499;
        double r822528 = r822493 * r822504;
        double r822529 = r822528 * r822509;
        double r822530 = r822529 * r822505;
        double r822531 = r822527 - r822530;
        double r822532 = r822531 + r822492;
        double r822533 = r822532 / r822493;
        double r822534 = r822533 / r822489;
        double r822535 = 2.590941352379923e-20;
        bool r822536 = r822489 <= r822535;
        double r822537 = 1.0;
        double r822538 = r822537 / r822499;
        double r822539 = r822494 * r822538;
        double r822540 = r822497 / r822539;
        double r822541 = r822496 * r822540;
        double r822542 = r822495 + r822541;
        double r822543 = r822542 - r822522;
        double r822544 = 2.01275689813107e+165;
        bool r822545 = r822489 <= r822544;
        double r822546 = 2.6783770738007185e+199;
        bool r822547 = r822489 <= r822546;
        double r822548 = r822497 / r822494;
        double r822549 = r822548 * r822499;
        double r822550 = r822496 * r822549;
        double r822551 = r822495 + r822550;
        double r822552 = r822551 - r822522;
        double r822553 = 1.151269571816795e+231;
        bool r822554 = r822489 <= r822553;
        double r822555 = r822554 ? r822513 : r822523;
        double r822556 = r822547 ? r822552 : r822555;
        double r822557 = r822545 ? r822513 : r822556;
        double r822558 = r822536 ? r822543 : r822557;
        double r822559 = r822525 ? r822534 : r822558;
        double r822560 = r822515 ? r822523 : r822559;
        double r822561 = r822491 ? r822513 : r822560;
        return r822561;
}

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	20.6
Target	14.2
Herbie	9.6

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -1.10015674080410512 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt -0.0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.17088779117474882 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 2.8768236795461372 \cdot 10^{130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \lt 1.3838515042456319 \cdot 10^{158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

1. Split input into 5 regimes

2. if c < -2.3219786821033522e+70 or 2.590941352379923e-20 < c < 2.01275689813107e+165 or 2.6783770738007185e+199 < c < 1.151269571816795e+231

   1. Initial program 22.6

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

   2. Taylor expanded around 0 14.2

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

   4. Applied add-cube-cbrt 14.5

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \frac{x \cdot y}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{\color{blue}{\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \sqrt[3]{c}}}\]

   5. Applied times-frac 11.0

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

   7. Applied times-frac 9.5

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\left(\frac{x}{z} \cdot \frac{y}{c}\right)}\right) - 4 \cdot \left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{t}{\sqrt[3]{c}}\right)\]

   if -2.3219786821033522e+70 < c < 6.256004526795307e-263 or 1.151269571816795e+231 < c

   1. Initial program 19.6

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

   2. Taylor expanded around 0 9.8

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

   4. Applied associate-*r/ 9.8

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

   if 6.256004526795307e-263 < c < 6.236497692806451e-151

   1. Initial program 15.0

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

   3. Applied associate-/r* 11.0

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

   if 6.236497692806451e-151 < c < 2.590941352379923e-20

   1. Initial program 14.3

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

   2. Taylor expanded around 0 5.6

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

   4. Applied associate-/l* 6.6

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

   6. Applied div-inv 6.6

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

   if 2.01275689813107e+165 < c < 2.6783770738007185e+199

   1. Initial program 23.9

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

   2. Taylor expanded around 0 15.4

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

   4. Applied associate-/l* 13.8

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

   6. Applied associate-/r/ 13.5

      \[\leadsto \left(\frac{b}{z \cdot c} + 9 \cdot \color{blue}{\left(\frac{x}{z \cdot c} \cdot y\right)}\right) - 4 \cdot \frac{a \cdot t}{c}\]
3. Recombined 5 regimes into one program.

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -2.32197868210335224 \cdot 10^{70}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) - 4 \cdot \left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{t}{\sqrt[3]{c}}\right)\\ \mathbf{elif}\;c \le 6.25600452679530668 \cdot 10^{-263}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \frac{9 \cdot \left(x \cdot y\right)}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;c \le 6.236497692806451 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;c \le 2.59094135237992291 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\left(z \cdot c\right) \cdot \frac{1}{y}}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;c \le 2.0127568981310699 \cdot 10^{165}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) - 4 \cdot \left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{t}{\sqrt[3]{c}}\right)\\ \mathbf{elif}\;c \le 2.6783770738007185 \cdot 10^{199}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z \cdot c} \cdot y\right)\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;c \le 1.151269571816795 \cdot 10^{231}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) - 4 \cdot \left(\frac{a}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{t}{\sqrt[3]{c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \frac{9 \cdot \left(x \cdot y\right)}{z \cdot c}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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)) -1.1001567408041051e-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)))