Development.Shake.Progress:decay from shake-0.15.5

Average Error: 23.0 → 8.9

Time: 8.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1.514625125026602 \cdot 10^{-302} \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0\right) \land \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 4.602029547023957 \cdot 10^{+304}\right):\\ \;\;\;\;\left(\frac{a}{y - b} + \frac{y \cdot a}{z \cdot {\left(y - b\right)}^{2}}\right) - \left(\frac{t}{y - b} + \frac{y}{z} \cdot \left(\frac{x}{y - b} + \frac{t}{{\left(y - b\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \end{array}\]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) (- INFINITY))
         (not
          (or (<=
               (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))
               -1.514625125026602e-302)
              (and (not
                    (<= (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))) 0.0))
                   (<=
                    (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))
                    4.602029547023957e+304)))))
   (-
    (+ (/ a (- y b)) (/ (* y a) (* z (pow (- y b) 2.0))))
    (+ (/ t (- y b)) (* (/ y z) (+ (/ x (- y b)) (/ t (pow (- y b) 2.0))))))
   (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= -((double) INFINITY)) || !(((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= -1.514625125026602e-302) || (!((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= 0.0) && ((((x * y) + (z * (t - a))) / (y + (z * (b - y)))) <= 4.602029547023957e+304)))) {
		tmp = ((a / (y - b)) + ((y * a) / (z * pow((y - b), 2.0)))) - ((t / (y - b)) + ((y / z) * ((x / (y - b)) + (t / pow((y - b), 2.0)))));
	} else {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	}
	return tmp;
}

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	23.0
Target	17.5
Herbie	8.9

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or -1.51462512502660205e-302 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or 4.602029547023957e304 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 59.4

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

   2. Taylor expanded around -inf 35.2

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

   3. Simplified 22.7

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.51462512502660205e-302 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.602029547023957e304

   1. Initial program 0.3

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1.514625125026602 \cdot 10^{-302} \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0\right) \land \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 4.602029547023957 \cdot 10^{+304}\right):\\ \;\;\;\;\left(\frac{a}{y - b} + \frac{y \cdot a}{z \cdot {\left(y - b\right)}^{2}}\right) - \left(\frac{t}{y - b} + \frac{y}{z} \cdot \left(\frac{x}{y - b} + \frac{t}{{\left(y - b\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020339 
(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)))))