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

Average Error: 20.3 → 7.5

Time: 7.7min

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -3.72420200238695375 \cdot 10^{186}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - \frac{t}{\frac{c}{a}} \cdot 4\\ \mathbf{elif}\;c \le -4.38410046387720475 \cdot 10^{105}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;c \le -1.8804472972176997 \cdot 10^{-32} \lor \neg \left(c \le 7.9322776426014124 \cdot 10^{36}\right):\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - \frac{t}{\frac{c}{a}} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c}{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - 4 \cdot \left(t \cdot a\right)}}\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((double) (((double) (((double) (((double) (x * 9.0)) * y)) - ((double) (((double) (((double) (z * 4.0)) * t)) * a)))) + b)) / ((double) (z * c)));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c) {
	double VAR;
	if ((c <= -3.7242020023869538e+186)) {
		VAR = ((double) (((double) ((b / ((double) (z * c))) + ((double) (9.0 * (x / (((double) (z * c)) / y)))))) - ((double) ((t / (c / a)) * 4.0))));
	} else {
		double VAR_1;
		if ((c <= -4.384100463877205e+105)) {
			VAR_1 = ((double) (((double) ((b / ((double) (z * c))) + ((double) (9.0 * ((double) ((x / z) * (y / c))))))) - ((double) (4.0 * (((double) (a * t)) / c)))));
		} else {
			double VAR_2;
			if (((c <= -1.8804472972176997e-32) || !(c <= 7.932277642601412e+36))) {
				VAR_2 = ((double) (((double) ((b / ((double) (z * c))) + ((double) (9.0 * (x / (((double) (z * c)) / y)))))) - ((double) ((t / (c / a)) * 4.0))));
			} else {
				VAR_2 = (1.0 / (c / ((double) ((((double) (b + ((double) (((double) (x * 9.0)) * y)))) / z) - ((double) (4.0 * ((double) (t * a))))))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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	20.3
Target	14.7
Herbie	7.5

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

Derivation

1. Split input into 3 regimes

2. if c < -3.72420200238695375e186 or -4.38410046387720475e105 < c < -1.8804472972176997e-32 or 7.9322776426014124e36 < c

   1. Initial program 23.3

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

   2. Simplified 18.1

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

   3. Taylor expanded around 0 15.3

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

   5. Applied associate-/l* 13.3

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

   7. Applied add-cube-cbrt 13.6

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

   8. Applied times-frac 9.0

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

   10. Applied pow1 9.0

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

   11. Applied pow1 9.0

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

   12. Applied pow-prod-down 9.0

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

   13. Simplified 9.2

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

   if -3.72420200238695375e186 < c < -4.38410046387720475e105

   1. Initial program 21.0

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

   2. Simplified 18.0

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

   3. Taylor expanded around 0 12.6

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

   5. Applied times-frac 12.0

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

   if -1.8804472972176997e-32 < c < 7.9322776426014124e36

   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. Simplified 2.8

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

   4. Applied clear-num 2.8

      \[\leadsto \color{blue}{\frac{1}{\frac{c}{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - 4 \cdot \left(t \cdot a\right)}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -3.72420200238695375 \cdot 10^{186}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - \frac{t}{\frac{c}{a}} \cdot 4\\ \mathbf{elif}\;c \le -4.38410046387720475 \cdot 10^{105}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;c \le -1.8804472972176997 \cdot 10^{-32} \lor \neg \left(c \le 7.9322776426014124 \cdot 10^{36}\right):\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{x}{\frac{z \cdot c}{y}}\right) - \frac{t}{\frac{c}{a}} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c}{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - 4 \cdot \left(t \cdot a\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -1.1001567408041051e-171) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -0.0) (/ (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9.0 (/ y c)) (/ x z)) (/ b (* c z))) (* 4.0 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (- (+ (* 9.0 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4.0 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))