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

Average Error: 7.5 → 0.9

Time: 13.0s

Precision: 64

Math

\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.039215365299041780464230440384741424909 \cdot 10^{257}\right):\\ \;\;\;\;\frac{x}{a} \cdot \frac{y}{2} - \frac{4.5 \cdot t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r651554 = x;
        double r651555 = y;
        double r651556 = r651554 * r651555;
        double r651557 = z;
        double r651558 = 9.0;
        double r651559 = r651557 * r651558;
        double r651560 = t;
        double r651561 = r651559 * r651560;
        double r651562 = r651556 - r651561;
        double r651563 = a;
        double r651564 = 2.0;
        double r651565 = r651563 * r651564;
        double r651566 = r651562 / r651565;
        return r651566;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r651567 = x;
        double r651568 = y;
        double r651569 = r651567 * r651568;
        double r651570 = z;
        double r651571 = 9.0;
        double r651572 = r651570 * r651571;
        double r651573 = t;
        double r651574 = r651572 * r651573;
        double r651575 = r651569 - r651574;
        double r651576 = -inf.0;
        bool r651577 = r651575 <= r651576;
        double r651578 = 1.0392153652990418e+257;
        bool r651579 = r651575 <= r651578;
        double r651580 = !r651579;
        bool r651581 = r651577 || r651580;
        double r651582 = a;
        double r651583 = r651567 / r651582;
        double r651584 = 2.0;
        double r651585 = r651568 / r651584;
        double r651586 = r651583 * r651585;
        double r651587 = 4.5;
        double r651588 = r651587 * r651573;
        double r651589 = cbrt(r651582);
        double r651590 = r651589 * r651589;
        double r651591 = r651588 / r651590;
        double r651592 = r651570 / r651589;
        double r651593 = r651591 * r651592;
        double r651594 = r651586 - r651593;
        double r651595 = 1.0;
        double r651596 = r651582 * r651584;
        double r651597 = r651595 / r651596;
        double r651598 = r651575 * r651597;
        double r651599 = r651581 ? r651594 : r651598;
        return r651599;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.5
Target	5.6
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709043451944897028999329376 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.144030707833976090627817222818061808815 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (* x y) (* (* z 9.0) t)) < -inf.0 or 1.0392153652990418e+257 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 50.2

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

   3. Applied div-sub 50.2

      \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}}\]

   4. Taylor expanded around 0 49.8

      \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{4.5 \cdot \frac{t \cdot z}{a}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 49.9

      \[\leadsto \frac{x \cdot y}{a \cdot 2} - 4.5 \cdot \frac{t \cdot z}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

   7. Applied times-frac 25.4

      \[\leadsto \frac{x \cdot y}{a \cdot 2} - 4.5 \cdot \color{blue}{\left(\frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\right)}\]

   8. Applied associate-*r* 25.4

      \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\left(4.5 \cdot \frac{t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right) \cdot \frac{z}{\sqrt[3]{a}}}\]

   9. Simplified 25.6

      \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\frac{4.5 \cdot t}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{z}{\sqrt[3]{a}}\]
   10. Using strategy rm

   11. Applied times-frac 1.1

       \[\leadsto \color{blue}{\frac{x}{a} \cdot \frac{y}{2}} - \frac{4.5 \cdot t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\]

   if -inf.0 < (- (* x y) (* (* z 9.0) t)) < 1.0392153652990418e+257

   1. Initial program 0.8

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

   3. Applied div-inv 0.9

      \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 1.039215365299041780464230440384741424909 \cdot 10^{257}\right):\\ \;\;\;\;\frac{x}{a} \cdot \frac{y}{2} - \frac{4.5 \cdot t}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{z}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.14403070783397609e99) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))