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

Average Error: 20.7 → 7.3

Time: 7.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \leq -15060649513884506:\\ \;\;\;\;\frac{1}{z} \cdot \frac{x \cdot \left(9 \cdot y\right) + b}{c} - 4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \mathbf{elif}\;c \leq 5.854139756661663 \cdot 10^{-65}:\\ \;\;\;\;\frac{\left(9 \cdot \left(y \cdot \frac{x}{z}\right) + \frac{b}{z}\right) - 4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;c \leq 1.5280219155113581 \cdot 10^{+193}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + 9 \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{x \cdot \left(9 \cdot y\right) + b}{c} - 4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \end{array}\]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))

↓

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= c -15060649513884506.0)
   (- (* (/ 1.0 z) (/ (+ (* x (* 9.0 y)) b) c)) (* 4.0 (* t (/ a c))))
   (if (<= c 5.854139756661663e-65)
     (/ (- (+ (* 9.0 (* y (/ x z))) (/ b z)) (* 4.0 (* t a))) c)
     (if (<= c 1.5280219155113581e+193)
       (- (+ (/ b (* c z)) (* 9.0 (* (/ x z) (/ y c)))) (* 4.0 (* t (/ a c))))
       (- (* (/ 1.0 z) (/ (+ (* x (* 9.0 y)) b) c)) (* 4.0 (* t (/ a c))))))))

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 tmp;
	if ((c <= -15060649513884506.0)) {
		tmp = ((double) (((double) ((1.0 / z) * (((double) (((double) (x * ((double) (9.0 * y)))) + b)) / c))) - ((double) (4.0 * ((double) (t * (a / c)))))));
	} else {
		double tmp_1;
		if ((c <= 5.854139756661663e-65)) {
			tmp_1 = (((double) (((double) (((double) (9.0 * ((double) (y * (x / z))))) + (b / z))) - ((double) (4.0 * ((double) (t * a)))))) / c);
		} else {
			double tmp_2;
			if ((c <= 1.5280219155113581e+193)) {
				tmp_2 = ((double) (((double) ((b / ((double) (c * z))) + ((double) (9.0 * ((double) ((x / z) * (y / c))))))) - ((double) (4.0 * ((double) (t * (a / c)))))));
			} else {
				tmp_2 = ((double) (((double) ((1.0 / z) * (((double) (((double) (x * ((double) (9.0 * y)))) + b)) / c))) - ((double) (4.0 * ((double) (t * (a / c)))))));
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	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

Bits error versus c

Target

Original	20.7
Target	15.2
Herbie	7.3

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < -1.1001567408041051 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < -0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if c < -15060649513884506 or 1.5280219155113581e193 < c

   1. Initial program Error: 25.0 bits

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

   2. Simplified Error: 19.8 bits

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

   4. Applied div-sub Error: 19.8 bits

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

   5. Simplified Error: 17.1 bits

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

   6. Simplified Error: 12.8 bits

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

   8. Applied *-un-lft-identity Error: 12.8 bits

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

   9. Applied times-frac Error: 9.6 bits

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

   if -15060649513884506 < c < 5.85413975666166294e-65

   1. Initial program Error: 14.9 bits

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

   2. Simplified Error: 3.4 bits

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

   3. Taylor expanded around 0 Error: 3.3 bits

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

   4. Simplified Error: 3.1 bits

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

   if 5.85413975666166294e-65 < c < 1.5280219155113581e193

   1. Initial program Error: 19.1 bits

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

   2. Simplified Error: 13.2 bits

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

   4. Applied div-sub Error: 13.2 bits

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

   5. Simplified Error: 10.2 bits

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

   6. Simplified Error: 8.1 bits

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

   7. Taylor expanded around 0 Error: 8.0 bits

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

   8. Simplified Error: 7.4 bits

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

4. Final simplification Error: 7.3 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -15060649513884506:\\ \;\;\;\;\frac{1}{z} \cdot \frac{x \cdot \left(9 \cdot y\right) + b}{c} - 4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \mathbf{elif}\;c \leq 5.854139756661663 \cdot 10^{-65}:\\ \;\;\;\;\frac{\left(9 \cdot \left(y \cdot \frac{x}{z}\right) + \frac{b}{z}\right) - 4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;c \leq 1.5280219155113581 \cdot 10^{+193}:\\ \;\;\;\;\left(\frac{b}{c \cdot z} + 9 \cdot \left(\frac{x}{z} \cdot \frac{y}{c}\right)\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{x \cdot \left(9 \cdot y\right) + b}{c} - 4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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)))