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

Average Error: 20.6 → 4.3

Time: 15.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -9.300232518812952249570457999506207707385 \cdot 10^{238}:\\ \;\;\;\;\left(\frac{1}{z} \cdot \frac{b}{c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.982946141337434241738306830170385159844 \cdot 10^{-208}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -0.0:\\ \;\;\;\;\left(\sqrt[3]{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t} \cdot \sqrt[3]{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t}\right) \cdot \frac{\sqrt[3]{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 4.299811555402724123181084098955168983672 \cdot 10^{307}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \left(9 \cdot \frac{x}{z}\right) \cdot \frac{y}{c}\right) - 4 \cdot \frac{t}{\frac{c}{a}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r579569 = x;
        double r579570 = 9.0;
        double r579571 = r579569 * r579570;
        double r579572 = y;
        double r579573 = r579571 * r579572;
        double r579574 = z;
        double r579575 = 4.0;
        double r579576 = r579574 * r579575;
        double r579577 = t;
        double r579578 = r579576 * r579577;
        double r579579 = a;
        double r579580 = r579578 * r579579;
        double r579581 = r579573 - r579580;
        double r579582 = b;
        double r579583 = r579581 + r579582;
        double r579584 = c;
        double r579585 = r579574 * r579584;
        double r579586 = r579583 / r579585;
        return r579586;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c) {
        double r579587 = x;
        double r579588 = 9.0;
        double r579589 = r579587 * r579588;
        double r579590 = y;
        double r579591 = r579589 * r579590;
        double r579592 = z;
        double r579593 = 4.0;
        double r579594 = r579592 * r579593;
        double r579595 = t;
        double r579596 = r579594 * r579595;
        double r579597 = a;
        double r579598 = r579596 * r579597;
        double r579599 = r579591 - r579598;
        double r579600 = b;
        double r579601 = r579599 + r579600;
        double r579602 = c;
        double r579603 = r579592 * r579602;
        double r579604 = r579601 / r579603;
        double r579605 = -9.300232518812952e+238;
        bool r579606 = r579604 <= r579605;
        double r579607 = 1.0;
        double r579608 = r579607 / r579592;
        double r579609 = r579600 / r579602;
        double r579610 = r579608 * r579609;
        double r579611 = r579603 / r579590;
        double r579612 = r579587 / r579611;
        double r579613 = r579588 * r579612;
        double r579614 = r579610 + r579613;
        double r579615 = r579602 / r579597;
        double r579616 = r579595 / r579615;
        double r579617 = r579593 * r579616;
        double r579618 = r579614 - r579617;
        double r579619 = -2.9829461413374342e-208;
        bool r579620 = r579604 <= r579619;
        double r579621 = -0.0;
        bool r579622 = r579604 <= r579621;
        double r579623 = r579591 + r579600;
        double r579624 = r579623 / r579592;
        double r579625 = r579597 * r579593;
        double r579626 = r579625 * r579595;
        double r579627 = r579624 - r579626;
        double r579628 = cbrt(r579627);
        double r579629 = r579628 * r579628;
        double r579630 = r579628 / r579602;
        double r579631 = r579629 * r579630;
        double r579632 = 4.299811555402724e+307;
        bool r579633 = r579604 <= r579632;
        double r579634 = r579600 / r579603;
        double r579635 = r579587 / r579592;
        double r579636 = r579588 * r579635;
        double r579637 = r579590 / r579602;
        double r579638 = r579636 * r579637;
        double r579639 = r579634 + r579638;
        double r579640 = r579639 - r579617;
        double r579641 = r579633 ? r579604 : r579640;
        double r579642 = r579622 ? r579631 : r579641;
        double r579643 = r579620 ? r579604 : r579642;
        double r579644 = r579606 ? r579618 : r579643;
        return r579644;
}

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	14.3
Herbie	4.3

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

Derivation

1. Split input into 4 regimes

2. if (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -9.300232518812952e+238

   1. Initial program 38.3

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

   2. Simplified 22.7

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

   3. Taylor expanded around 0 19.2

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

   5. Applied associate-/l* 17.1

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

   7. Applied associate-/l* 10.4

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

   9. Applied *-un-lft-identity 10.4

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

   10. Applied times-frac 15.0

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

   if -9.300232518812952e+238 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -2.9829461413374342e-208 or -0.0 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < 4.299811555402724e+307

   1. Initial program 4.0

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

   if -2.9829461413374342e-208 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) < -0.0

   1. Initial program 30.2

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

   2. Simplified 0.7

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

   4. Applied *-un-lft-identity 0.7

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

   5. Applied add-cube-cbrt 1.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t} \cdot \sqrt[3]{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t}\right) \cdot \sqrt[3]{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t}}}{1 \cdot c}\]

   6. Applied times-frac 1.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t} \cdot \sqrt[3]{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t}}{1} \cdot \frac{\sqrt[3]{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t}}{c}}\]

   7. Simplified 1.4

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

   if 4.299811555402724e+307 < (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))

   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 26.8

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

   3. Taylor expanded around 0 30.3

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

   5. Applied associate-/l* 25.1

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

   7. Applied times-frac 10.7

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

   8. Applied associate-*r* 10.7

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

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -9.300232518812952249570457999506207707385 \cdot 10^{238}:\\ \;\;\;\;\left(\frac{1}{z} \cdot \frac{b}{c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - 4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -2.982946141337434241738306830170385159844 \cdot 10^{-208}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le -0.0:\\ \;\;\;\;\left(\sqrt[3]{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t} \cdot \sqrt[3]{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t}\right) \cdot \frac{\sqrt[3]{\frac{\left(x \cdot 9\right) \cdot y + b}{z} - \left(a \cdot 4\right) \cdot t}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \le 4.299811555402724123181084098955168983672 \cdot 10^{307}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + \left(9 \cdot \frac{x}{z}\right) \cdot \frac{y}{c}\right) - 4 \cdot \frac{t}{\frac{c}{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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)) -1.1001567408041049e-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)) 1.17088779117474882e-53) (/ (+ (- (* (* x 9) y) (* (* z 4) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 2.8768236795461372e130) (- (+ (* (* 9 (/ y c)) (/ x z)) (/ b (* c z))) (* 4 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9) y) (* (* (* z 4) t) a)) b) (* z c)) 1.3838515042456319e158) (/ (+ (- (* (* 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)))