Development.Shake.Progress:decay from shake-0.15.5

Average Error: 22.7 → 14.9

Time: 16.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} = -\infty \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -2.524192835395849763162285085577468898691 \cdot 10^{-308} \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 0.0\right) \land \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 9.569050410089291249341673112864676119324 \cdot 10^{303}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r934576 = x;
        double r934577 = y;
        double r934578 = r934576 * r934577;
        double r934579 = z;
        double r934580 = t;
        double r934581 = a;
        double r934582 = r934580 - r934581;
        double r934583 = r934579 * r934582;
        double r934584 = r934578 + r934583;
        double r934585 = b;
        double r934586 = r934585 - r934577;
        double r934587 = r934579 * r934586;
        double r934588 = r934577 + r934587;
        double r934589 = r934584 / r934588;
        return r934589;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r934590 = x;
        double r934591 = y;
        double r934592 = r934590 * r934591;
        double r934593 = z;
        double r934594 = t;
        double r934595 = a;
        double r934596 = r934594 - r934595;
        double r934597 = r934593 * r934596;
        double r934598 = r934592 + r934597;
        double r934599 = b;
        double r934600 = r934599 - r934591;
        double r934601 = r934593 * r934600;
        double r934602 = r934591 + r934601;
        double r934603 = r934598 / r934602;
        double r934604 = -inf.0;
        bool r934605 = r934603 <= r934604;
        double r934606 = -2.52419283539585e-308;
        bool r934607 = r934603 <= r934606;
        double r934608 = 0.0;
        bool r934609 = r934603 <= r934608;
        double r934610 = !r934609;
        double r934611 = 9.569050410089291e+303;
        bool r934612 = r934603 <= r934611;
        bool r934613 = r934610 && r934612;
        bool r934614 = r934607 || r934613;
        double r934615 = !r934614;
        bool r934616 = r934605 || r934615;
        double r934617 = r934594 / r934599;
        double r934618 = r934595 / r934599;
        double r934619 = r934617 - r934618;
        double r934620 = r934616 ? r934619 : r934603;
        return r934620;
}

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

Target

Original	22.7
Target	17.5
Herbie	14.9

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

Derivation

1. Split input into 2 regimes

2. if (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < -inf.0 or -2.52419283539585e-308 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < 0.0 or 9.569050410089291e+303 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))

   1. Initial program 61.5

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

   3. Applied clear-num 61.5

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

   4. Simplified 61.5

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(z, b - y, y\right)}{\mathsf{fma}\left(z, t - a, x \cdot y\right)}}}\]

   5. Taylor expanded around inf 40.9

      \[\leadsto \color{blue}{\frac{t}{b} - \frac{a}{b}}\]

   if -inf.0 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < -2.52419283539585e-308 or 0.0 < (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) < 9.569050410089291e+303

   1. Initial program 3.3

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 14.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} = -\infty \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le -2.524192835395849763162285085577468898691 \cdot 10^{-308} \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 0.0\right) \land \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \le 9.569050410089291249341673112864676119324 \cdot 10^{303}\right):\\ \;\;\;\;\frac{t}{b} - \frac{a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :herbie-target
  (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z))))

  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))