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

Average Error: 20.4 → 6.0

Time: 8.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7610087338959005 \cdot 10^{+26}:\\ \;\;\;\;\frac{\left(9 \cdot \left(y \cdot \frac{x}{z}\right) + \frac{b}{z}\right) - 4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;z \leq -5.176286705330166 \cdot 10^{-96}:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(9 \cdot \left(y \cdot \frac{x}{z \cdot c}\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\right)\\ \mathbf{elif}\;z \leq 18414931.287040245:\\ \;\;\;\;\frac{b + x \cdot \left(9 \cdot y\right)}{z \cdot c} - \left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right) \cdot \left(t \cdot \left(\frac{a}{c} \cdot \sqrt[3]{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(9 \cdot \left(y \cdot \frac{x}{z}\right) + \frac{b}{z}\right) - 4 \cdot \left(t \cdot a\right)}{c}\\ \end{array}\]

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 (<= z -2.7610087338959005e+26)
   (/ (- (+ (* 9.0 (* y (/ x z))) (/ b z)) (* 4.0 (* t a))) c)
   (if (<= z -5.176286705330166e-96)
     (+ (/ b (* z c)) (- (* 9.0 (* y (/ x (* z c)))) (* 4.0 (* t (/ a c)))))
     (if (<= z 18414931.287040245)
       (-
        (/ (+ b (* x (* 9.0 y))) (* z c))
        (* (* (cbrt 4.0) (cbrt 4.0)) (* t (* (/ a c) (cbrt 4.0)))))
       (/ (- (+ (* 9.0 (* y (/ x z))) (/ b z)) (* 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 ((z <= -2.7610087338959005e+26)) {
		tmp = (((double) (((double) (((double) (9.0 * ((double) (y * (x / z))))) + (b / z))) - ((double) (4.0 * ((double) (t * a)))))) / c);
	} else {
		double tmp_1;
		if ((z <= -5.176286705330166e-96)) {
			tmp_1 = ((double) ((b / ((double) (z * c))) + ((double) (((double) (9.0 * ((double) (y * (x / ((double) (z * c))))))) - ((double) (4.0 * ((double) (t * (a / c)))))))));
		} else {
			double tmp_2;
			if ((z <= 18414931.287040245)) {
				tmp_2 = ((double) ((((double) (b + ((double) (x * ((double) (9.0 * y)))))) / ((double) (z * c))) - ((double) (((double) (((double) cbrt(4.0)) * ((double) cbrt(4.0)))) * ((double) (t * ((double) ((a / c) * ((double) cbrt(4.0))))))))));
			} else {
				tmp_2 = (((double) (((double) (((double) (9.0 * ((double) (y * (x / z))))) + (b / z))) - ((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	20.4
Target	14.3
Herbie	6.0

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

Derivation

1. Split input into 3 regimes

2. if z < -2.76100873389590048e26 or 18414931.2870402448 < z

   1. Initial program Error: 30.5 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: 9.3 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: 9.2 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: 5.9 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 -2.76100873389590048e26 < z < -5.1762867053301664e-96

   1. Initial program Error: 7.5 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: 11.3 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: 8.3 bits

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

   4. Simplified Error: 6.1 bits

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

   if -5.1762867053301664e-96 < z < 18414931.2870402448

   1. Initial program Error: 5.8 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: 20.6 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: 20.6 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: 8.6 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: 5.8 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 add-cube-cbrt Error: 6.0 bits

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

   9. Applied associate-*l* Error: 6.1 bits

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

   10. Simplified Error: 6.1 bits

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

4. Final simplification Error: 6.0 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7610087338959005 \cdot 10^{+26}:\\ \;\;\;\;\frac{\left(9 \cdot \left(y \cdot \frac{x}{z}\right) + \frac{b}{z}\right) - 4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;z \leq -5.176286705330166 \cdot 10^{-96}:\\ \;\;\;\;\frac{b}{z \cdot c} + \left(9 \cdot \left(y \cdot \frac{x}{z \cdot c}\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\right)\\ \mathbf{elif}\;z \leq 18414931.287040245:\\ \;\;\;\;\frac{b + x \cdot \left(9 \cdot y\right)}{z \cdot c} - \left(\sqrt[3]{4} \cdot \sqrt[3]{4}\right) \cdot \left(t \cdot \left(\frac{a}{c} \cdot \sqrt[3]{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(9 \cdot \left(y \cdot \frac{x}{z}\right) + \frac{b}{z}\right) - 4 \cdot \left(t \cdot a\right)}{c}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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)))